# 1. Thermodynamic Equilibrium

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2       The Thermodynamic Substance

The thermodynamic systems comprise of fluids, i.e. liquids and gases and a
thermodynamic process can be defined in terms of the fluid properties within a system. The
operational parts are excluded from thermodynamic analysis through a proper choice of the
boundaries. A study of the behaviour of the working fluid (fluid properties) is, therefore
necessary for a thermodynamic analysis of energy and matter transformations. We consider
working substances at equilibrium state in form of fluid phases.

There are two points of view from which the behaviour of matter can be studied: the
microscopic and the macroscopic.

From the microscopic point of view, matter is composed of an extremely large
number of molecules, which are themselves built from atoms and are of complicated
structure. For the simplest case of gases, it may be assumed that each molecule at a given
instant has a certain position, velocity, and energy, and for each molecule these change very
frequently, as a result of collisions. The behaviour of the gas is described by summing up the
behaviour of each molecule. Such a study is made in statistical thermodynamics.

In classical thermodynamics, which is only concerned with the effects of the action of
many molecules, a macroscopic approach is adopted. Clearly speaking, a certain quantity of
matter (water in a boiler or gas in a combustion engine) is considered without the events
occurring at the molecular level being taken into account. For example, the macroscopic
quantity, pressure, is the average rate of change of momentum due to all the molecular
collisions which occur on a unit area. The effects of pressure can be felt and measured by
using, for example a pressure gauge. For thermodynamic analysis the behaviour of matter will
be described in terms of macroscopic observable properties. Engineering thermodynamics
uses the classical model. However, sometimes we will use molecular picture (microscopic
model) for a better understanding of some phenomenon.

2.1    Quantity of Matter
The thermodynamic studies normally include the quantity of matter, e.g. the quantity
of water in a boiler or the amount of gas in the cylinder of a combustion engine. The quantity
of matter is characterized by mass m and number of moles n. The amount of matter of a
system is given by the mass enclosed within its boundary.

In common language the mass of a substance is also called its weight. But they are
different conceptionally. Weight is actually the force G with which a body of mass m will be
attracted from the surface of earth at a particular place having acceleration due to gravity g:

G=mg                                                        (2.1)
The acceleration due to gravity is not a universal constant. It depends on the place of
observation. At earth´s surface g = 9.81 m s-2. As the acceleration due to gravity depends on
altitude, the weight of a particular mass m will also vary with altitude.

The mass is determined through weighing by comparing with a standard mass. The
kilogram is the SI ( SI from the french System International ) unit of mass; it is equal to the
mass of the international prototype of the kilogram mass of a lump of platinum-iridium.
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An alternative way to represent the amount of matter is as the number n of moles. The
mole is the amount of substance of a system which contains as many elementary entities as
there are atoms in 0.012 kilogram of Carbon-12. When the mole is used, the elementary
species must be specified. These may be atoms, molecules, ions, electrons, other particles, or
specified groups of such particles.

Examples of the use of mole:

1 mol of H2 contains about 6.022  1023 H2 molecules, or 12.044  1023 H atoms.
1 mol of Hg2Cl2 has a mass of 472.08 g.
1 mol of Hg22+ has a mass of 401.18 g.
1 mol of e- has a mass of 548.60 g.

2.2    The Thermal Variables
For a quantitative description of phases and of thermodynamic systems certain state
variables (properties) are needed. These properties should be measurable or computable from
other measured properties. The most common properties for fluid phases are temperature,
pressure and volume. These are known as thermal variables.

2.2.1 Volume

The volume V is the space occupied by a substance and is measured in cubic meters
(m³). The meter is the basic unit of length. Thus we write V = 1m³ if the system occupies 1 m³
of space. One meter is defined as the length of path travelled by light in vacuum during a time
interval of 1/299792458 of a second. The second, represented by the symbol ' s ', is the unit of
time and is defined as the duration of 9192631770 periods of the radiation corresponding to
the transition between the hyperfine levels of the ground state of Caesium -133 atom.
Sometimes instead of absolute volume, the term specific volume is used. Specific volume  is
the space occupied by unit mass of a substance and is measured in m³/kg.

V
                                                                   [unit is m³/kg]
m

Similarly the volume of one mole of a substance is called molar specific volume or simply the
molar volume and is also denoted by small 

V VM
                                                                  [unit is m³/mol]
n     m
where M [unit is g/mol] denotes the molar mass (molecular weight) of the substance, i.e. the
mass of 1 mol of substance. M = m / n.

2.2.2 Pressure

Pressure is the force exerted by a system (fluid) on a unit area. The SI unit of pressure
is defined as
pressure = force / area
= N / m²
3

This composite unit is called ' Pascal ' and denoted by the symbol Pa

1Pa = 1N m-2

The pressure of the atmosphere is of the order of 105 Pa. This shows that Pascal is
rather a small unit of pressure. It is, therefore convenient to describe the pressure in terms of
kPa or MPa.

Other common unit for pressure is ' bar ', which nearly equals the pressure of one
atmosphere. The unit ' bar ' is defined as

1 bar = 105 Pa = 105 Nm-2 = 100 kPa.

The atmospheric pressure varies from region to region and is not constant throughout.
A standard atmospheric pressure is defined as

1 standard atmosphere = 1 atm = 101.325 kPa = 1.01325 bar

Figure 2.1 shows two blocks of matter which have the same mass. They exert the same force
on the surface on which they are standing, but the narrow block exerts a higher pressure
because it exerts the force on a smaller area than the fatter block.

Figure 2.1 : Comparison of pressure exerted by the same force.

The instrument to measure the pressure is known as manometer. If atmospheric
pressure (which varies with altitude and weather) is measured the instrument is called
barometer. Most instruments indicate pressure relative to the atmospheric pressure pU,
whereas the pressure of a system is its pressure above zero, or relative to a perfect vacuum.
The pressure relative to the atmospheric pressure is called gauge pressure (pG). The pressure
relative to a perfect vacuum is called absolute pressure p which is given by:

Absolute pressure = Gauge pressure + Atmospheric pressure

p = pG + pU                                    (2.2)

Figure 2.2 shows a few pressure measuring devices (manometers). Figure (a) shows
an open u-tube manometer indicating gauge pressure, and Figure (b) shows an open u-tube
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indicating vacuum, i.e. pressure below the atmospheric pressure. Figure (c) shows a closed u-
tube indicating absolute pressure.

pu
pu
Evacuated
p                                   p                             p

h                                                                h
h

Hg                             Hg                                   Hg

a)                                b)                             c)

Figure 2.2 : Some U-Tube Manometers.

If h is the difference in the heights of the fluid (mercury) columns in the two limbs of
the u-tube [cf. Fig. (a) and Fig. (b)],  the density of the fluid (here, mercury) and g the
acceleration due to gravity, then the gauge pressure pG is given by

F m g A h g  
pG                      h g                      (2.3)
A   A      A

where F is the force of the mercury column on the gas over an area A. For mercury with a
density = 13.6*103 kg/m³ a 760 mm (0.760 meter) column of mercury is equivalent to

1.013*105 N/m² = 1.013 bar = 1 atm.

Sometimes the pressure is also expressed in kgf/m2 ( kg force per square meter ) or, as this
unit is large, in kgf/cm2 ( also known as ´ata´, atmosphere technical absolute ).

1 standard atmosphere = 1 atm = 1.0332 kgf/cm2 = 1.0332 ata

2.2.3 Temperature

The temperature is a very familiar concept in everyday life as it is a measure of the
'hotness' or 'coldness' of a body or fluid. But it is quite difficult to give the concept a precise
definition. Temperature is associated with the ability to distinguish hot from cold. When two
bodies at different temperatures are brought into contact, energy (heat) flows from the higher
temperature object (hot body) to the lower temperature object (cold body). After some time
they attain a common temperature and are then said to be in thermal equilibrium. Hence, the
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temperature may be defined as the property whose value is identical in the two systems which
are in thermal equilibrium.

The temperature of a fluid is one of the properties of that fluid, along with pressure
and specific volume. It is necessary to have a precise value of temperature in order to define
the state of a fluid. The basis for the temperature measurement is provided by the zeroth law
of thermodynamics: 'When a body A is in thermal equilibrium with a body B, and also
separately with a body C, then B and C will be in thermal equilibrium with each other'.

In order to obtain a quantitative measure of temperature, a reference body, say
thermometer, is used. A column of liquid inside the thermometer expands or contracts with
changes of temperature. The thermometric property is the expansibility of the liquid. When a
thermometer is brought into contact with a body at a temperature higher (greater) than itself,
then heat is transferred from that body to the glass of the thermometer and from there to the
liquid inside the thermometer raising its temperature. The heat transfer stops when a thermal
equilibrium is reached. Now the thermometer has the same temperature as that of the body
outside. The increase in temperature of the liquid results in its expansion. As the expansibility
of liquids is much higher than that of the glass, the expansion of glass may be neglected in
comparison to that of liquids. The extent of expansion of a liquid in a glass column is hence a
direct measure of the increase in the temperature of the liquid.

It is necessary to have a common temperature scale for measurement. The most
common temperature scale is the Celsius scale. This uses two arbitrary fixed points, the
freezing point and the boiling point of water under standard atmospheric conditions, to define
0 0C and 100 0C respectively. For a mercury thermometer, the difference in the length of
mercury column is divided in 100 equal parts. Each part then represents 1 0C.

This temperature scale is not satisfactory in many aspects. As each substance has a
different coefficient of thermal expansion this temperature scale will depend on the nature of
the liquid in the thermometer. Even if the thermometers made of mercury and alcohol, with
linear scales, are made to agree at two temperatures, they will not agree at some intermediate
temperature. Secondly, the division in 100 equal parts is only applicable to a linear expansion
with temperature. Strictly, the coefficient of thermal expansion is not a linear function of
temperature. Such a thermometer for temperature measurements is therefore not accurate.
Also the Celsius temperature scale shows negative values for temperatures which are below
the freezing point of water, e.g. the boiling point of liquid oxygen is -183°C. This seems
physically unsound as the other properties of state, namely pressure and volume have only
positive values. It is therefore very much desirable to look for another scale for temperature.

An empirical temperature scale, which is also an absolute scale, is that of an ideal gas
thermometer which is described here in detail.

A schematic diagram of a constant volume gas thermometer is shown in Figure 2.3. A
small amount of gas is enclosed in a bulb which is connected via a capillary tube with one
limb of the mercury manometer. The other limb of the mercury manometer is open to the
atmosphere and can be moved vertically to adjust the mercury levels so that the mercury just
touches the reference mark on the capillary. The sensor is hereby the gas filled in the constant
volume bulb. The pressure in the bulb is measured through a height of the mercury column.
The thermometric property of this constant volume gas thermometer is the pressure of the
gas. It changes in a characteristic way with the temperature of the bulb and is given by the
difference in level z of the mercury column:
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p = pU + z  g                           (2.4)

where pU is the atmospheric pressure and  the density of mercury.

pu

V = const.                                                           z = p
^

Mark

Hg tubing
Figure 2.3 : Diagram showing the working principle of an ideal gas thermometer.

If the pressure in the bulb is equal to the atmospheric pressure then z = 0. Since the
volume of the bulb is constant, a definite amount of gas (e.g. n mol) in the bulb under
thermal equilibrium conditions at a certain empirical temperature  will result in a definite
value for the product (p), where  V/n. For a definite temperature the value of (p)
depends on the nature of the gas. The value of  can be varied by filling the bulb (of constant
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volume V) with different amounts n of a gas. Figure 2.4 shows a plot of the values of (p) at
a known temperature z = const.) against 1/ for different gases.

 = const.
pv
water gas

air
(pv)ig

Helium

1/v
Figure 2.4 : (p) versus 1/ plot for various gases.

It is seen from the graph that although the readings of a constant volume gas
thermometer depend upon the nature of the gas, all gases indicate the same temperature, i.e.
the product (p), as 1/ is lowered and made to approach zero. This empirical experimental
finding is of great significance for defining a temperature scale which is based on physical
grounds. It holds:

lim( p )  ( p )ig  const .                       for =const.
1
0


This relation describes an ideal gas. The instrument is called an ideal gas
thermometer. Each ideal gas thermometer, irrespective of the gas filling indicates the same
value for (pig and hence the same empirical temperature if it is in thermal equilibrium
with a system. The value of (pig can be used to define an empirical temperature ig of the
ideal gas thermometer

ig = C  (pig                     (2.5)

It may be pointed out that the linear relation in Equation (2.5) is chosen for the sake of
simplicity. Any other relationship between pand , e.g. quadratic could also be taken.

This definition has many advantages. The first is, that the temperature can not attain
negative values as p and  have positive values only. Secondly, we need only one fixed point
to determine the constant C and hence to define the temperature scale. Thirdly, this
temperature scale does not depend on the nature of the gas (filling medium).

The scale is established by arbitrarily selecting a fixed point. The temperature of a
mixture of pure water, water vapor, and ice in thermal equilibrium is selected for this
purpose. Such a mixture exists at only one temperature (called the triple point), and the
temperature of this mixture provides an easily reproducible standard. The gas thermometer is
brought in thermal equilibrium with pure water at its triple point, the corresponding value of
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(pig is determined. The value of (pig is determined by measuring the values of (p) at
different  (i.e. with different fillings of a gas) and then extrapolating these to 1/ = 0. It
holds then
1                  Nm
( p ) ig.,H 2O  ig,H 2O  2271.2
tr          tr                                       (2.6)
C                  mol

It is agreed that the triple point of water shall be 273.1600 K. Here K is used for
'Kelvin', which is the unit of temperature in this scale (of ideal gas thermometer). The value
of C follows
molK
C = 0.12027                                 (2.7)
Nm

and the unit of temperature, Kelvin:
 ig, H 2 O
1K 
tr
(2.8)
.
27316

Hence 1 Kelvin is one 273.16 th part of the empirical temperature of ideal gas thermometer at
the triple point of water.

By an international agreement in 1954 the value of 273.16 was selected to make the
unit of the new scale the same size as that of the Celsius (formally Centigrade) scale. Since
the ordinary ice point (freezing point of water) is 0.01 K lower than the triple point of water it
follows that 0°C=273.15 K.

To measure the temperature of a system the gas thermometer is brought in thermal
equilibrium with the system and the value of (pig is determined. The ideal gas thermometer
temperature of the system is then
molK             Nm
ig  C  ( p ) ig  0.12027       ( p ) ig
Nm              mol
This empirical ideal gas thermometer temperature satisfies the conditions of a
thermodynamic state property. Later on in the course of our studies of entropy an absolute
thermodynamic temperature scale will be defined. Fortunately it proves to be identical with
the ideal gas scale. It will be shown that the point which we have defined as the absolute zero
of temperature is not an arbitrary point. At present we take it for granted that the
thermodynamic scale of temperature is identical with the ideal gas thermometer temperature
scale. We write:

ig = T                                        (2.9)

The thermodynamic temperature T is therefore a measurable quantity. The Equation
(2.5) may then be written as

pig = Rig = RT                              (2.10)

where R = 1/C = 1/0.12027=8.315 Nm/(molK) and is called the molar gas constant (universal
gas constant).

In everyday life the temperature is expressed in Celcius scale. The temperature in
Celcius (°C) is simply related to the thermodynamic temperature T via
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t ( in °C ) = T ( in K ) - T0                                 (2.11)

where T0 = 273.15 K represents the ordinary ice point. The Celcius temperature scale
employs a degree of the same magnitude as that of the thermodynamic temperature, but its
zero point is shifted.

A second absolute temperature scale, used with the old English system, is the Rankine
scale, defined by

T(Rankins) = 9/5 T (Kelvins)

This makes the water triple-point temperature correspond to 491.69 R. The corresponding
relative temperature scale is Fahrenheit scale. The temperature differences are identical in
degrees Fahrenheit (°F) and Rankins (R), but a level of 0°F corresponds to 459.67 R.

It follows, that the relationship between the Celsius and Fahrenheit scale is
°F = °C * (9/5)+32             (2.12)
The temperature of the ice point is about 32°F and that of the steam point about 212°F.

The measurement of temperature with the help of ideal gas thermometer is a tedious
process [extrapolation of many measured (pto (pig]. The temperatures are, therefore
generally measured with other types of thermometers such as thermocouple and platinum
resistance thermometers. Naturally, these have to be calibrated against absolute temperature
(measured with ideal gas thermometers). For this purpose a practical temperature scale
(International Temperature Scale-ITS) was adopted first of all in 1927 and then revised from
time to time (latest revision 1990). The ITS is defined by:

1- Assigning values to certain accurately reproducible temperatures such as boiling and melting points.
2- Specifying the type of thermometer to be used in each range of the scale.
3- Specifying the interpolation formula to be used for each thermometer between the assigned values.

This scale is a practical scale and it is based on a number of fixed and easily reproducible
points which are assigned definite numerical values of temperature, and on specified formulae
which relate temperature to the readings on certain temperature measuring instruments. The
scale was so defined that it conforms closely to the ideal gas temperature scale. The
temperature interval from the oxygen point to the gold point is divided into three main parts,
as given below in International Temperature Scale of 1968 (ITS-68).

Table 2.1: Temperatures of Fixed Points
Temperature °C
Normal boiling point of oxygen                                   -182.97
Triple point of water ( Standard )                                 + 0.01
Normal boiling point of water                                      100.00
Normal boiling point of sulphur                                    444.60
(Normal melting point of zinc-suggested as an
alternative to the sulphur point)                                   419.50
Normal melting point of antimony                                     630.50
Normal melting point of silver                                       960.80
Normal melting point of gold                                        1063.00
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(a) From 0 to 660°C:
A platinum resistance thermometer with a platinum wire whose diameter must lie between
0.05 and 0.02 mm is used, and the temperature is given by equation

R = R0 (1+At +Bt²)

where the constants R0, A, and B are computed by measurements at ice point, steam point and
sulphur point. The temperature t is in °C.

(b) From -190 to 0°C:
The same platinum resistance thermometer is used, and the temperature is given by

R = R0 [1+At +Bt²+ C(t-100)t³]

where R0, A and B are the same as before, and C is determined from a measurement at the
oxygen point.

(c) From 660 to 1063°C:
A thermocouple, one wire of which is made of platinum and the other of an alloy of 90%
platinum and 10% rhodium, is used with one junction at 0°C. The temperature is given by the
formula

 = a + bt + ct²

where a,b and c are computed from measurements at the antimony point, silver point, and
gold point. The diameter of each wire of the thermocouple must lie between 0.35 and 0.65mm

An optical method is adopted for measuring temperatures higher than the gold point.
The intensity of radiation of any convenient wavelength is compared with the intensity of
radiation of the same wavelength emitted by a black body at the gold point. The temperature
is then determined with the help of Planck´s law of radiation.

2.3    Pure Substances

The practical application of thermodynamics lies in the working of heat power plants
or refrigerating machinery etc., which operate with the help of some working fluid or system
(thermodynamic system). The processes carried out in these applications are analysed by
considering the effects produced on the systems. It is therefore, necessary to have information
on the properties of these working fluids or systems. In many cases the working fluid is a
pure substance which may change its phase (A phase is any homogeneous part of a system
that is physically distinct). The important characteristic of a pure substance is that its
chemical composition is same throughout its mass. It may exist in one or more phases. Thus,
a system consisting of a mixture of various phases of water, namely water and ice or water
and steam is a pure substance. A system consisting of oxygen as a vapor, a liquid, or a solid
or a combination of these is also a pure substance. Sometimes air, even though made up of
several gases, namely nitrogen, oxygen, carbon dioxide etc. and thus a gas mixture, is
considered and mathematically treated as a pure substance as long as there is no change of
phase. This is due to the fact that under normal conditions its composition remains constant
and that none of the constituent gases undergoes a chemical change during a thermodynamic
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process. For many applications it will be treated like a pure substance. Strictly speaking, this
is not true. We should rather say that air exhibits the characteristics of a pure substance.

Pure substances are used as the working medium in many energy transformation
devices. Water is used as the working fluid in steam power plants. In heat pumps and
refrigerators the use of pure organic fluids is very common. Water vapor is used as energy
carrier for the heating of devices in certain matter conversion processes. It is, therefore very
much necessary to learn about the properties of pure substances.

2.3.1 The Thermal Properties and their Technical Use

Experience has shown that there is an interrelation between the various properties of a
substance. Thus, there is a relationship between the pressure, specific volume, and
temperature, which may be expressed by

F(p, ,T)=0

This mathematical relationship between the thermal state variables pressure (p), specific
volume () and absolute temperature (T) is conventionally known as the thermal equation of
state. It may also be expressed as

p = p(T, )
= (T,p)
T = T(p, )

The relation between the three variables may be a simple one or very complicated
depending on the substance and the range of observation. In general, the properties of every
pure substance may be represented by an equation of state.

2.3.2 Thermal Properties of Gases

We have seen in section 2.2.3 that the products (p) of all real gases at given
temperature approach the same value, as the pressure approaches zero (1/ approaches zero).
1
At the state of zero pressure , i.e. at vanishing density (  0 ) all real gases behave in a

similar manner. We can characterize this identical limiting behaviour of gases as ideal
behaviour and call the state of zero pressure as an ideal gas state.

The behaviour of real gases at the ideal state (state of p  0) suggests the concept of
an ideal gas (perfect gas) - a hypothetical gas that will behave in an ideal manner at all
pressures. Note that the ideal (perfect) gas is not real; it is an imaginary gas that displays at all
pressures the same behaviour which the real gases display only at the state of p  0. For a
perfect (ideal) gas holds

(p)ig = RT                                             (2.13)
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where R = 8.315 Nm/(molK), is the universal gas constant and , the molar volume. So in the
1
range of low pressures the molar density ( ) increases linearly with pressure at a constant

temperature. These constant temperature curves are called isotherm ( isotherm = same
temperature ).

The isotherms appear as hyperbola in a p diagram. Figure 2.5 shows the p-
diagram for a perfect gas.

1.5

bar
p

1.0

873.15 K

0.5
573.15 K
373.15 K
T = 273.1 5 K
0.1
0     0.05          0.10   m3 /mol      0.20
v
Figure 2.5 : p- Diagram of an ideal gas.

If  is used for representing the specific volume in Equation (2.13) then it becomes

R
(p)ig = (     )T                       (2.14)
M

where M is the molecular weight of the substance. The p- diagram is not universal but
individual (different) for every gas. This difference is not due to the fact that the function is
R
different for every gas. It is because the factor ( ) is different for every gas and hence the
M
R
individuality. This factor (    ) is called the specific gas constant.
M

A study of the p- diagram explains why the power plants use gases as the working
medium. The product p has the dimensions of energy, namely Nm/mol or Nm/kg. By
heating at a constant volume the temperature and the pressure (c.f. Fig. 2.5) will increase.
Thus the energy supplied in the form of heat will be stored in gas (in form of a higher
pressure). This can be converted partly into work through the relaxation of pressure
(decreasing the pressure) and may be used, e.g. to push back a piston. This forms the basis for
a spark ignition internal combustion engine (Otto engine). The increase of temperature and
the consequent increase of pressure is realized practically through combustion.

Figure 2.6 shows the schematic p- diagram of an idealized Otto cycle. Process 1-2
represents the compression of the gas without any heat transfer. Heat is then added at constant
volume through combustion process. This is represented in 2-3 as the increase of pressure at
constant volume. Process 3-4 is an expansion without heat transfer. Process 4-1 is the
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rejection of heat from the gas while it is disposed into the atmosphere (it is shown as cooling).
The p- behaviour of the gas therefore presents the relevant (appropriate) model for the
thermodynamic analysis of the energy transformations in an Otto process. The hatched
(marked) area in the p- diagram then represents the obtainable work from an idealized Otto
process 1-2-3-4-1. It is the difference between the work done by the system during the
expansion from 3-4 and the work done on the system during the compression from 1-2 ( this
process will be discussed in detail in chapter 5 ).

p          3
combustion

expansion

2

4
cooling

compression
1

v

Figure 2.6 : Idealized Otto process in p- diagram.

2.3.3 Thermal Properties of Fluids

Simple relations (2.13) and (2.14) describe the behaviour of gases at low densities and
pressures, e.g. water vapor at 500 oC and 1 atm. Now we look at the behaviour of this vapor
at other temperatures. We consider a cylinder and frictionless piston assembly, as shown in
Figure 2.7a ( situation 1 ) containing a definite amount of water vapor at 500 oC at a constant
pressure [ state 1 represents this point in a t- diagram (b) ]. The fluid enclosed between the
cylinder and the piston forms a closed system. The pressure inside the system will remain
constant throughout the process. If the water vapor is cooled down by bringing the assembly
in contact with a surrounding of 20 oC ( e.g. room temperature ), its specific volume will
decrease. This decrease in the beginning will be given by relation (2.13) or (2.14). With
further cooling it will deviate from this relation and at a temperature of nearly 100 oC ( state
2 ) a new phenomenon (event) will be observed, namely the beginning of condensation (
appearence of first water drop ). This event in which the volume decreases very sharply
(cosiderably) at almost constant temperature can not be described with Equation (2.13) or
(2.14). State 2 represents the situation where the first drop of liquid is formed. The water
drops represent the case of boiling water. The vapor at the beginning of condensation is
known as saturated vapor. As the system further cools down we observe that both the liquid
water and the water vapor are present at the same time (coexistence). However, the amount of
vapor as well as the total volume decreases and that of liquid increases. The states from 2 to 4
represent the range for the coexistence of liquid water and water vapor. The temperature
remains constant during this period. At state 4 the last bubble of vapor disappears and at state
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4 we find all the water vapor of state 1 in the form of boiling liquid. The process of cooling
goes on further, the liquid stops to boil and finally reaches the temperature of the
surroundings, namely 20 oC ( state 5 ). We observe further that the decrease in volume
between 4 and 5 is smaller as compared to that between 1 and 2.

1
2
3
4
5
Gas            Gas
Gas

liquid                                                                        (a)
liquid       liquid
last vapor                       first liquid
bubble                           drop

t             p=const.                                              1

4                       3            2
(b)
5
saturated                         saturated
liquid state                      vapor state

V

Figure 2.7 : Condensation of a pure substance ( schematic t-V Diagram ).

This process can be reversed if we place the cylinder on a hot plate . The temperature of the
liquid rises from state 5 to 4 from 20 oC to 100 oC with a small increase of volume. At 4 the
water begins to boil ( building of vapor bubbles - evaporation ). If more heat is supplied the
process goes on further to state 2, where the last drop of liquid disappears. If the supply of
heat continues the temperature as well as the volume of water vapor increase and we arrive at
state 1.

At state 4 a liquid reaches the saturation temperature and is called saturated liquid.
Similarly, at state 2 where all the saturated liquid is converted to vapor, the vapor at
saturation temperature is called the saturated vapor. Between the saturated liquid and
saturated vapor states ( between 2 and 4 ), the fluid will be at saturated temperature but will
consist of a proportion of liquid and a proportion of vapor in equilibrium. This condition is
known as wet vapor.

The specific volume in wet vapor range, i.e. in the coexistence range of liquid and gas,
depends not only on the saturation temperature/pressure but also on the relative masses of
vapor and liquid. The relative mass of vapor and liquid is expressed in terms of vapor content
x ( mass fraction of the vapor also called vapor quality or dryness fraction ):
15

m/ /
x                                                            (2.15)
m/  m/ /

where m/ is the mass of the liquid and m// is the mass of the vapor at saturation. The total mass
m = m/ + m//.

Obviously the mass fraction of the liquid is (1-x). The wet vapor region at x = 0 ( m// = 0 )
then represents the state of boiling liquid ( saturated liquid state ) which lies on the saturated
liquid line. The point at x = 1 ( m/ = 0 ) represents the saturated vapor state and lies on the
saturated vapor line. The total volume of the system in wet vapor region is an extensive state
property and is made up of the liquid and vapor parts.

V = V/ + V// = m/ / + m// //

The volume per unit mass ( specific volume  ) is

V m / / m // //
           (1  x ) /  x //
m m     m

= / + x (// - / )                                       (2.16)

Equation (2.16) states that the average specific volume of the two phase mixture, , is
a linear function of x, and equals the specific volume of the liquid plus the volume increase
upon vaporization (// - / ) times the mass fraction of the system vaporized. The vapor
content , x, can be expressed explicitly by

  /
x  //                                                        (2.17)
  /

The temperature between the states 2 and 4, i.e. the temperature at which both liquid
and vapor coexist depends on pressure. It has higher values at higher pressures and smaller
values at smaller pressures. This temperature is called saturation temperature or boiling point
temperature and is defined as that temperature at which vaporization takes place at a given
pressure. The corresponding pressure is called the saturation (vapor) pressure. For a pure
substance there is a definite relation between the saturated temperature and saturated pressure.
This relation between temperature and pressure during evaporation and condensation is given
by the so called vapor pressure curve ( vapor pressure line ). Figure 2.8 shows the vapor
pressure curve for water and some other pure substances in p-t diagram above 0 oC.

The dependence of boiling temperature or condensation temperature on pressure, i.e.
the shape of vapor pressure curve is very important from technical point of view and forms
the basis of the steam power plants, heat pumps and refrigerators. We will have a brief look
into these. A detailed study is left for next semester.
16

H2O
200
p

bar
150

NH3
100
CO2
R12
50

0
0          100          200     °C   300
t
Figure 2.8 : Vapor pressure curves for some pure substances.

A simple steam power plant working on the vapor power cycle is schematically shown
in Figure 2.9. In its simplest version the steam power plant consists of four elements- a boiler,
a turbine, a condenser, and a pump.

.
Heat Q is tranferred from an external source ( furnace, where fuel is continuously burnt ) to
water in the boiler to produce steam having a pressure of 200 bar and at nearly 550 oC
(superheated steam; the boiling point of water at 200 bar is 365.81 oC). During this process
the water takes up energy which is stored in the form of high pressure and high temperature.
Hence the steam leaving the boiler has the potential (capacity) for doing work and rejecting
(giving) heat.


Q

1
boiler
2

PSP                                                                           PT
turbine
pump

condenser                              3
0


Q0

Figure 2.9 : Schematic diagram of a simple steam power plant.
17

The high pressure, high temperature steam leaving the boiler expands in the turbine ( to a
pressure 0.0426 bar ) to produce shaft work PT which in turn produces electric current via a
generator. The low end pressure at the turbine corresponds to a condensation temperature of
.
30 oC according to the vapor pressure curve of water. So the vapor can reject heat Q 0 when it
is brought in contact with a cold surrounding ( condenser, where cooling water at 15 oC
circulates ). Hereby it is condensed completely. Then the liquid water is pumped back to the
boiler (pressure level 200 bar) with the help of a condensate pump by supplying energy PSP in
the form of work. Here one should note that the working of a steam power plant is possible
only because of the characteristic ( namely steep ) vapor pressure curve of water.

The steam power plant works only because the different pressures at the entry and exit
of the turbine correspond to technically attainable temperatures (550 °C and 30 °C) for the
heat supply and the heat rejection. A flat vapor pressure curve in comparison to that of water
would be unsuitable for the technical realization of the power plant cycle. Because then after
the expansion in the turbine, the vapor will attain a saturation Temperature of 30 °C at a
much higher pressure. As the expansion only up to a relatively higher value for pressure
would be possible, the obtainable work will be smaller (lower work production). Also in the
case of a flat vapor pressure curve the saturation temperature corresponding to desired high
pressure at the entrance of the turbine would also be very high and difficult to attain. The
quantitative p-t behaviour of water is therefore technically very much important for the
functioning of steam power plants. One sees here the importance of the study of the properties
of substances (e.g., the vapor pressure curve).

The working of a very common domestic appliance, namely the refrigerator working on the
vapor-compression refrigeration cycle, is also based on the vapor pressure curve. A simple
refrigerator consists of a compressor, a condenser, an expansion valve (or capillary), and an
evaporator, as shown in Figure 2.10.


Q

3
condenser
2

expansion                    compressor             PV
valve

evaporator                      1
4


Q0
Figure 2.10 : Schematic diagram for the working of a refrigerator.
18

A suitable refrigerant (e.g. Dichlorodifluoromethane, R12, Refrigerant 12), which has
a saturation vapor pressure of 3 bar at 0 °C, enters the compressor as a slightly superheated
.
vapor (the liquid refrigerant absorbs heat Q 0 in evaporator from the food storage
compartment) at a low pressure. It will be brought to a higher pressure by the application of
compression work Pv. Now it is also at a higher temperature (corresponding to its vapor
.
pressure curve) and has the potential to give heat Q during condensation. The fluid
(refrigerant) then leaves the compressor and enters the condenser as a vapor at some elevated
pressure (e.g., 10 bar), where it is condensed as a result of heat transfer to the surroundings
(For a condensation temperature of 40 °C, surroundings in the kitchen, R12 has a pressure of
about 10 bars). The refrigerant then leaves the condenser as a higher pressure liquid. As the
liquid flows through the expansion valve (generally in the form of a long capillary tube) its
pressure is reduced from the condenser pressure to the evaporator pressure. This vapor then
enters the compressor. The refrigerator rejects in this way heat from the cool storage room (of
refrigerator) to the surroundings and keeps the temperature of storage room low. A machine
which works on similar principles may be used for heating purposes. It is then known as heat
pump. In a heat pump the heat is taken from the surroundings so as to evaporate a suitable
working substance. The vapor is compressed and gives (rejects) heat to the heating system at
a suitable temperature. A refrigerator and a heat pump are thus distinguished only by the
desired effect, i.e. cooling or heating.

2.3.4 Thermal Properties in the entire state region

Till now our interest has been confined to fluid region (liquid - vapor region) for the
simple reason that the working substance in power cycles etc. is a fluid. We saw the
importance of vapor pressure curve of water for its use as working fluid in steam power plant.
We now look at the behaviour of pure substances over the entire range of its existence. We
study the properties of the most common substance water. Consider a unit mass of ice (water
in solid state) at -20 °C and 1 atm (1.01325 bar) contained in a cylinder and piston. Let the ice
be heated slowly so that its temperature is always uniform. If the pressure is held constant
(isobaric process) and heat is supplied continuously, the following changes in the mass of
water are observed:

1 atm                     1 atm
1 atm                 1 atm

ice                     steam                     steam
ice                   liquid water
liquid water
-20 °C                 0 °C                   100 °C                    250 °C
heated                heated                  heated                    heated
(a)                   (b)                     (c)                       (d)

Figure 2.11: Heating of H2O at constant pressure

(a) The temperature of ice rises and approaches 0 °C as a limit. There will be small
increase in volume.

(b) On further heating the temperature remains constant at 0 °C and a change of phase
takes place from solid to liquid state. The two phases exist in equilibrium. During this
19

melting of ice there is a small decrease in volume,which is a peculiarity of water
(almost all the other substances show a small increase in volume during melting).The
energy required to completely convert the ice to liquid water is known as enthalpy of
fusion or latent heat of fusion.

(c) When more heat is supplied the temperature of water rises until the temperature of
vaporization is attained (100 °C). This is called the saturated liquid state because any
further heat addition causes vaporization to start.

(d) With further addition of heat the liquid water will boil and another change of
phase occurs at constant temperature [Fig. (c)]. The liquid becomes vapor (steam) as
the heat is continued to be supplied. There is a considerable increase in volume (wet
vapor region). By the time the last drop of liquid is vaporized (and attains saturated
vapor state) the liquid has absorbed energy equal to the enthalpy of vaporization
during the change of phase (liquid to vapor) at constant temperature. This energy is
also known as latent heat of vaporization.

(e) If more heat is added to the saturated vapor (steam), the rise in its temperature will
be resumed and it will be accompanied by increase in volume [Fig. (d)]. The process
is known as superheating and the resulting vapor is called superheated vapor.

If the heating of ice at -20 °C to steam at 250 °C were done at a constant pressure of 2 bar
similar regimes of heating would have been obtained with similar saturation states.

If the initial pressure of the ice at -20 °C is 0.2602 kPa, heat transfer to the ice first results in
an increase in temperature to -10 °C. At this point, however, the ice would pass directly from
the solid phase to the vapor phase in the process known as sublimation. Further heat transfer
would result in superheating of the vapor.

If we start with ice with an initial pressure of 0.6113 kPa and a temperature of -20 °C and
heat it slowly. Then as a result of heat transfer the temperature will increase until it reaches
0.01 °C. At this point if further heat is supplied some of the ice will become vapor and some
will become liquid, because at this point it is possible to have the three phases in equilibrium
(triple point).

2.3.4.1 Pressure-temperature Diagram

The state changes of pure water, upon slow heating at different constant pressures, are
plotted on p-T coordinates in Figure 2.12. This diagram shows how the solid, liquid and
vapor phases may exist together in equilibrium.
20

Pressure
G                                H

Fus
ion
Critical point
Liquid
line      phase

e
li n
n
io
at
E                                                 F

ri z
po
Solid                                                Vapor
phase                           Va                   phase
C                         D
Triple point
A                            B
o   n
m   at i
bl i
Su          lin
e

Temperature

Figure 2.12 : Pressure-temperature diagram for a substance like water.

The curve passing through the melting points at different pressures is called the fusion
curve ( fusion line ), and the curve passing the vaporization or condensation temperatures at
different pressure is called the vaporization curve (line). The sublimation curve (line) gives
the vapor pressure of solid at different temperatures. The fusion curve, the vaporization
curve, and the sublimation curve meet at the triple point (triple phase state).

Along the fusion curve, solid and liquid phases are in equilibrium, along the
sublimation curve solid and vapor phases are in equilibrium and along the vaporization curve
the liquid and the vapor phases are in equilibrium. The intersection of these three curves at
the triple point in a p-T diagram represents the conditions under which three phases can
coexist in equilibrium. These conditions are represented by a point only on a pressure-
temperature diagram; on other property diagrams they are represented by a line or an area.
If three phases of a pure substance exist together in a system, only one value of pressure and
only one value of temperature are possible. The state of each phase present in the system is
fixed. However, more information is required (namely, the proportions of the phases present)
to determine the state of the system (mixture of liquid + vapor) at the triple point temperature
and pressure. For example, the specific volume of the system with 1 per cent vapor content is
much less than that of the system with 10 per cent vapor content.

The triple point of water is at 0.6113 kPa and 273.16 K. The vaporization curve ends
at the critical point because there is no distinct change from the liquid phase to the vapor
phase above the critical point. The liquid and vapor are indistinguishable from each other.
There is no sudden change from the liquid state to the vapor state. The critical pressure and
the critical temperature are the highest pressure and temperature at which distinguisable
liquid and vapor phases can coexist together in equilibrium.

A look at the Figure 2.12 shows that a solid at the state A will pass directly to vapor
at state B if it is heated at constant pressure (which is less than the triple point pressure). At a
constant pressure above the triple point pressure ( from state E ) the substance first passes
from the solid to the liquid state at a definite temperature and then passes to vapor state F at a
21

higher temperature. The constant pressure line C-D passes through the triple point. At a
pressure above the point the change from the state G to H is not distinguishable.

Water is one of a very few substances that expand on freezing. Most substances
contract on freezing and the freezing temperature increases as the pressure increases, so that
the solid-liquid saturation curve or fusion (melting) line appears as in Figure 2.13, i.e. the
slope of the fusion line is positive. On the other hand, this slope is negative for water (see
Figure 2.12). This is also the reason that ice melts at constant temperature if the pressure
increases.

Figure 2.13 : Pressure-temperature diagram for a substance that contracts on freezing.

2.3.4.2 Pressure-volume Diagram

The pressure-specific volume diagram for a substance that contracts on freezing is
shown in Figure 2.14. The line h-l is a plot of the specific volume of saturated liquid versus
pressure and is called the saturated liquid line. As the pressure increases (and consequently
22

the saturation temperature increases), the specific volume of saturated liquid increases
slightly. The region immediately to the left of the saturated liquid line represents compressed
or subcooled liquid states. The line l-g is a plot of the specific volume of saturated vapor
versus pressure and is called the saturated vapor line. As the pressure increases, the specific
volume of saturated vapor decreases. The region immediately to the right of the saturated
vapor line represents superheated vapor states. Solid, liquid , and vapor states are represented
by points in three different regions on a pressure-volume diagram just as they are on a
pressure-temperature diagram. However, mixtures of two phases are represented by areas on a
p- diagram while they are represented by lines on a p-T diagram. Mixtures of three phases
are represented by a line on a p- diagram (by a point on a p-T diagram).

Figure 2.14 : Pressure-volume diagram for a substance that contracts on freezing.

The area beneath the dome formed by the saturated liquid line h-l and the part of the
saturated vapor line l-g down to the line h-i represents mixtures of liquid and vapor. All the
states represented by this area on the p- diagram are represented by the line on a p-T diagram
between the triple point and the critical point.
23

In Figure 2.14 saturated solid states at pressures higher than the triple phase pressure
are represented by points along the line r-s, called the saturated solid line (curve). The liquid
at the freezing temperature (saturated liquid) is represented by points along the line h-m,
called the freezing liquid line to avoid the confusion with the saturated liquid line (liquid at
the boiling point). Mixtures of solid and liquid are represented by points in the region
bounded by lines r-s, h-m and r-h. All points within this region lie on the fusion line on the p-
T diagram. Line u-w represents states of minimum specific volume for the solid at various
pressures, that is, the solid can not be compressed further. Below the triple phase pressure,
liquid can not exist. This area beneath the line u-r-h-i-g represents solid vapor mixtures. The
states in this region of p- diagram lie on the sublimation line of a p-T diagram. The line u-r
represents saturated solid states and the line i-g represents saturated vapor states. The line r-h-
i on the p- diagram represents the states in which solid, liquid, and vapor can exist together.
The triple point state is thus represented by a line on a p- diagram as against a point on a p-T
diagram. Point h represents the saturated liquid at the triple phase pressure, point i represents
saturated vapor and the point r represents saturated solid at the same pressure. The points
between r and h represent solid-liquid, solid-vapor or solid-liquid-vapor mixtures. The points
between h and i represent solid-vapor, liquid-vapor or solid-liquid-vapor mixtures. The phase
transition from liquid to solid at constant pressure is accompanied by a decrease in specific
volume. It is to be pointed out again that at constant pressure the phase transition from liquid
to solid water (ice) is accompanied by an increase of specific volume.

The isotherms (lines of constant temperature) are also shown in Figure 2.14 as broken
lines. In all two phase regions the constant temperature lines coincide with constant pressure
lines.

It is to be kept in mind that the specific volume scale is greatly distorted in p-
diagram. For all substances the change in specific volume between points h and i is many
times greater than between r and h and also the critical specific volume is only slightly greater
than  at r or h. To emphasize this, only parts of the p-v diagrams of carbon dioxide and
water are shown in Figure 2.15 with true linear scales. Usually the volume axis is plotted
logarithmically.

Figure 2.15 : p- diagrams to true linear scale for carbon dioxide and water.
24

2.3.4.3 p--T Surfaces

The relationship between pressure, specific volume and temperature can be
represented in a three-dimensional diagram. It is conventional to assign the p--T surfaces,
the vertical ordinate as pressure axis while the volume and temperature axes lie in horizontal
plane. Figure 2.16 shows the p--T diagram for water (a substance that expands on freezing)
and Figure 2.17 for other substances (substances in which the specific volume decreases
during freezing). The p-T and p- projections are also shown in these diagrams. The p-T and
p- diagrams have been discussed in detail. On the p--T surface the triple point appears as
the triple line since the pressure and temperature of the triple point are fixed, but the specific
volume may vary, depending on the proportion of each phase. The saturated liquid curve with
respect to the vaporization and the saturated vapor curve incline towards each other and join
what is known as the saturation or vapor dome. The two lines meet at the critical point. The
temperature, pressure and specific volume at this point are called the critical temperature
(Tc), critical pressure (pc) and critical volume (c) respectively.

It will be noted that there are six distinct regions in each of the p--T surfaces.

Single Phase Region:

(a) solid:      Volume is very small; temperature is below triple point; pressure has
any value.
(b) liquid:     Volume is not much different from that of solids; temperature is higher
than triple point; pressure has any value.
(c) vapor:      Volume is large; temperature is below critical; pressure is high or low.
If the temperature is above critical the state is referred to as gas.

Two Phase Regions:

(a) solid - liquid              These are regions (between two single phases) where
(b) liquid - vapor              the two phases can exist simultaneously in quilibrium.
(c) solid - vapor               

A p--T diagram provides a better picture of the relationship among pressure,
specific volume and temeperature because here each point represents only one equilibrium
state. Various regions of a p--T diagram overlap on a p- projection. Any given point on a
p- diagram can represent a solid, a solid-liquid, a liquid or a liquid-vapor mixture
depending on the temperature
25

Figure 2.16 : Pressure-volume-temperature surface for a substance that expands on freezing
(e.g. water).
26

Figure 2.17 : Pressure-volume-temperature surface for a substance that contracts on freezing.
27

2.3.5 Fluid Models

A quantitative description of the properties of fluids is necessary for quantitative
thermodynamic analysis. The properties of fluids may be described by using fluid models.
The properties of pure substances are arranged in the form of tables as functions of pressure
and temperature. Separate tables are provided to give the properties of substances in the
saturation states and in the liquid and vapor phases. The properties of water are arranged in
the so called steam tables. Tables 2.1 and 2.2 give the properties of each phase in the vapor-
liquid saturation region, namely saturated liquid water and saturated steam. When a liquid and
its vapor are in equilibrium at certain temperature and pressure, only the pressure or the
temperature is sufficient to identify the saturation state. If the temperature is given, the
saturation pressure gets fixed, or if the pressure is given, the temperature gets fixed. For
saturated liquid or the saturated vapor only one property is required to be known to fix up the
state. Steam Tables 2.1 and 2.2 give besides thermal properties temperature, pressure and
specific volume other properties internal energy, enthalpy and entropy too, which will be
introduced in a later chapter. From Table 2.1 one can read the saturation pressure and the
specific volume of saturated liquid and saturated vapor for a particular temperature. Similarly,
Table 2.2 may be used to read other saturation properties at a given pressure. If data are
required for intermediate temperatures or pressures, linear interpolation is done.

Table 2.3 gives the values of the properties (volume, internal energy, enthalpy and
entropy) of superheated vapor for each tabulated pair of values of pressure and temperature,
both of which are now independent. Now the interpolation or extrapolation is to be done for
pairs of values of pressure and temperature. Table 2.4 lists the properties of subcooled water
(compressed liquid water ). Table 2.5 lists the ice-vapor saturation (sublimation) values.
Similar tables for other technically important substances are also available.

However, now with the development of computer based calculation it is more
convenient to use analytical equations, e.g. equation of state in place of these table. The
model of ideal gas (perfect/ideal gas equation) is used for the calculation of thermal
properties of gases at low densities (pressures up to a few bars). This model is based on the
universal behaviour of gases irrespective of their nature at low densities.

(p)ig = R T
or
(pV)ig = n R T                                        (2.18)
R
(pV)ig = m ( ) T                                      (2.19)
M

This model is quite accurate and is used for describing the behaviour of gases at low
pressures. A real gas obeys ideal gas law at temperatures that are high relative to the critical
temperature and at pressures that are low relative to the critical pressure. The gas oxygen,
nitrogen, air or hydrogen may be treated as ideal gases at ordinary temperature and pressure.

The ideal gas equation of state p = RT is valid only under the following assumptions:
- the volume occupied by the molecules themselves is negligibly small compared to
the volume of the gas.
- there is no attraction between the molecules of the gas.

So at low pressures and high temperatures, where these conditions are fulfilled the real gases
follow ideal gas equation. But as pressure increases, the intermolecular forces of attraction
28

and repulsion increase, and also the volume of the molecules becomes appreciable compared
to the total gas volume. Then the real gases deviate considerably from the ideal gas equation.

To describe the behaviour of gases at higher pressures some other equation of states
may be used. A well known equation is Van der Waals Equation, which is given below:

a
(p         )(  b)  RT                             (2.20)
2

where ‘a’ and ‘b’ are the constants specific to the substance. The coefficient a is introduced to
account for the existence of mutual attraction between the molecules. The coefficient b
accounts for the finite size of molecules.

Real gases conform more closely with Van der Waals equation of state than the ideal
gas equation of state, particularly at higher pressures. A widely used equation of state with
good accuracy is the Redlich-Kwong Equation:

RT               a
p                                                   (2.21)
 b        1
T  (  b)
2

The constant a and b are evaluated from the critical data.

0.4275R 2 Tc2.5
a                                                    (2.22)
pc

0.0866 RTc
b                                                    (2.23)
pc

Another widely used equation for describing the properties of gases is the virial
equation of state:

p
 1  B' p  C ' p ²  D' p ³  ....               (2.24)
RT

or an alternative expression
p      B C D
 1   2   ...                                  (2.25)
RT          ³

B´,C´, B, C etc. are called virial coefficients. B´ and B are called second virial coefficient, C´
and C are called third virial coefficients, and so on. For a given gas, these coefficients are
p
functions of temperature only. The ratio           is called the compressibility factor z. For an
RT
ideal gas z = 1. The magnitude of z for a certain gas at a particular pressure and temperature
gives an indication of the extent of deviation of gas from the ideal gas behaviour. The virial
equations (2.24) and (2.25) may also be written as:

z  1  B' p  C' p²  D' p³....                     (2.26)
29

or
B       C           D
z  1                         ...          (2.27)
          2
³

B C D
The terms       , , ... etc. arise on account of molecular interactions. If no such interactions
 2 ³
exist ( e.g. at very low pressures ) then B = 0, C = 0 etc. Hence z = 1, i.e. p = RT.

A similar and accurate model for liquid region does not exist. For calculation the fluid
model of ideal liquid is used:

il = const.

According to this model (ideal liquid model) the volume (molar or specific) of a
liquid is supposed to be constant. The small dependence of volume on temperature and
pressure is neglected. Figure 2.18 shows the t- diagram of water. We see that the specific
volume is almost independent of temperature and pressure up to 200 °C. The model of ideal
liquid will not describe the real behaviour of liquids exactly but may be used for a reasonable
and good approximation.

500
r
ba

°C
0
50

a    r
0b
p =
t

25
B                          p =                  A
400
p = 200 bar

p = 100 bar
300

p = 25 bar

200
critical point
boiling liquid
saturated vapor

p = 1 bar
100

0
3
0                      5         cm /g                   10
v

Figure 2.18 : t- diagram of water.
30

The vapor pressure curve of pure substances in a given temperature region may be
described by an equation of form:

ps(T) = p0 exp [A(1-To/T)]                            (2.28)

where ps is the saturated vapor pressure at temperature T; p0, T0 and A are three substance
specific constants. According to this equation a log(p) versus 1/T plot should be linear. In
Figure 2.19 log(p) is plotted against inverse Temperature [see the logrithmic scale for p] for
some substances and we see a linear relationship. The critical temperature and triple point
temperature for the substances are also indicated. There are of course some substances which
do not show linear behaviour.

1000
374°C
31°C
100
112°C
bar                 -147°C       -240°C
p

10                 CO2
-56°C
H2
1                       N2
-210°C
0.1
-259°C
H2O
0.01
0°C
R12
0.001

0.0001
0                  0.2            0.4        0.6   1/K      0.8
1/T

Figure 2.19 : Vapor pressure curve of some substances in log p versus 1/T diagram.
31

2.4    Mixtures
Raw materials are generally found in nature as mixtures. Pure substances as well
mixtures with desired properties are produced through appropriate material transformations of
substances available in nature. Mixtures of desired properties may also be produced by
mixing the pure substances in appropriate ratios. For example, the useful product ammonia is
produced by mixing hydrogen and nitrogen under suitable conditions. In many other
engineering applications we encounter thermodynamic systems, which are mixtures of pure
substances. The mixtures are sometimes referred to as solutions, particularly a solid dissolved
in liquid. A knowledge of the qualitative and quantitative behaviour of mixtures is thus
necessary for the thermodynamic analysis of the material transformation.

A mixture is defined as any collection of molecules, ions, electrons. Each of these is
called a species (or a constituent of the mixture) if it is distinguishable from the others by
virtue of its chemical structure. In reacting mixtures the amount of each constituent can not
necessarily be varied independently. In nonreacting mixtures each and every constituent can
be varied independently and every constituent is also a component. Components are those
constituents the amount of which can be independently varied. By varying the amount of
components the composition may be varied. The composition of a mixture is expressed in
various forms which are described below.

Mass fraction:
The ratio of the mass of a constituent a to that of the total mixture is known as the
mass fraction ( wa ) of the constituent. For instance, the mass fraction of a component a in a
mixture is:

ma
wa                    where mT = ma + mb + mc + ....         (2.29)
mT
If wa is multiplied by 100 we get weight per cent. The sum of all w´s is equal to 1, i.e.

wi = 1.

Mole fraction:
The mole fraction xa of a component a is the amount of a expressed as a fraction of
the total amount of molecules ( i.e. moles ). It is the ratio of the number of moles of a
constituent to the total number of moles in the mixture.

na
xa            nT = total number of moles = na + nb + nc +.. (2.30)
nT
If xa is multiplied by 100 we get mol per cent. The sum of all x´s is equal to 1, i.e.

xi = 1.

Volume fraction:
The ratio of the volume occupied by a constituent if theoretically assumed to be
separated (at the pressure and temperature of the mixture) to the total volume of the mixture
is known as volume fraction a.
32

Va
a                where VT = Va + Vb + ...                   (2.31)
VT

If a is multiplied by 100 we get volume per cent. The sum of all ´s is equal to 1, i.e.

i = 1.

The mass fraction and the mole fraction are related to each other as under:

mi    M i  ni    M i  xi
wi                                             (2.32)
 mi  M i  ni  M i  xi
mi       wi
n     Mi       Mi
xi  i                                          (2.33)
 ni  mi  wi
Mi        Mi

2.4.1 The Thermal Properties and their Technical Applications

The mixtures undergo changes when they are heated and cooled or when their
compositions are changed. During heating or cooling phase changes may also take place. We
will first see, how the intensive phase properties (also called the degrees of freedom) are
related to the number of phases coexisting in equilibrium.

2.4.2 The Phase Rule

We have seen in our studies of the properties of pure substances that a system
consisting of a single component (for which we write C=1), when two phases are present
(e.g., liquid and vapor phases; we write P=2) only pressure or temperature could be changed
independently. For example, for reading a property (say the molar volume) from the steam
table in a two phase system we needed only one state property (temperature or pressure). We
say that such a system has only one degree of freedom (variance F=1). In saturated vapor-
liquid range we could change only one property (t or p) and the other was automatically fixed.
Similarly, at the triple point of a pure substance where three phases exist simultaneously, we
could not change any property without disturbing the equilibrium between the three phases.
The three phase equilibrium is given by a single fixed value of temperature and of pressure
(No degree of freedom for this system; variance of the system is zero). For a single phase one
component system both the temperature and the pressure could be changed independently.

We write here the general relation between the degrees of freedom of a system
(variance, F), the number of components (C) and the number of phases at equilibrium (P) for
a system of any composition:
F=C-P+2                                           (2.34)

This is the phase rule applicable to non reacting systems. It was first formulated by Gibbs. For
systems of two components, i.e. for a binary mixture, C=2 and hence F = 4 - P.
33

In a single phase region this system has three degrees of freedom i.e. its temperature,
pressure and composition (e.g. mole fraction) can be varied independently. We therefore note
that the properties of a fluid mixture depend not only on the pressure and temperature but also
on its composition.

 =  (t, p, x2)                                                                        (2.35)

where x2 is the mole fraction of say component no. 2. The dependence of  on the mole
fraction of component 1 (x1) is already taken into account by considering x2 because the
relation x1 = (1 - x2 ) holds.

A similar relationship for specific volume is

 = (t, p, w2)                                                                         (2.36)

The thermal properties of fluid mixtures in the two phase region (liquid-vapor region)
are very much important because many operations performed in chemical, petroleum, and
related industries involve the contact between phases. The analysis or design of processes
such as distillation, liquid-liquid extraction requires an understanding of principles of phase
equilibrium and an availability of equilibrium data.

For a binary system in a two phase region the number of degrees of freedom i.e. the
number of intensive phase properties is 2 (2 - 2 + 2 = 2). So if the composition and pressure
are fixed, the temperature is automatically fixed.

Now we look at the vapor-liquid equilibrium of a binary system by considering a
mixture of toluene (component 1) and cyclohexane (component 2).

120
Constant p = 101.3 kPa                                                                 Constant t = 90 °C
°C                                                                                  140
A                                                                                                                                                       C
110                                                                             kPa
Subcoole d                                   Saturate d
Superhea ted
120          liquid                                     liquid
Saturate d
t                 Tw
o-
ph
vapor           v apor          p
100                          as
e
B      E
re
gi                                       100                                                             D
on
Saturate d
liquid
on
E      D                                                             gi
90                                                                                  80                               e
re
B                                                            as
ph
o-
Tw                                              Superhea ted
C                                                                            v apor
Subcoole d                                                               60 A            Saturate d
80           liquid                                                                                 vapor

40
70
0.0                                                0.5 0.65           1.0              0.0                                            0.5 0.65              1.0
x2                                                                                   x2

Figure 2.20 : Phase diagrams for toluene (1) + cyclohexane (2) system.
34

Both toluene and cyclohexane exist as liquids at room temperature (vapor pressure of
toluene at 25 °C = 28.47 mm Hg = 3.796 kPa ; vapor pressure of cyclohexane at 25 °C=
97.77 mm Hg = 13.035 kPa). We consider a mixture of toluene and cyclohexane at 115 °C at
a constant pressure of 1 atm (101.3 kPa). At this temperature both toluene and cyclohexane
exist as superheated vapors (b.p. of toluene = 110.56 °C ; b.p. of cyclohexane = 80.75 °C)
and form a gaseous mixture (only one phase, namely vapor phase). Applying the phase rule
we get the number of degrees of freedom F for this situation : F = C - P - 2 = 2-1+2 = 3. We
have fixed the temperature and pressure and hence there remains one more variable
(composition) which is necessary to completely describe the system. The composition can be
selected in the entire composition range ( xcyclohexane = 0.0 to 1.0 ). We select the composition
of the gas phase as xcyclohexane = 0.65 and xtoluene = 0.35. If we cool down this mixture then at
a certain temperature the vapor will start to condense as liquid and will do so over a range of
temperature. The temperature range lies between the b.p. of cyclohexane and toluene
(80.75°C to 110.56°C). Within this range the mixture will exist as a two phase system (one
vapor phase and one liquid phase) [Fig. 2.20 shows a t-x diagram]. For the two phase region
the number of degrees of freedom F = 2-2+2 = 2. So if the pressure and temperature are given
(e.g. p = 101.3 kPa, t = 90°C) then there is no more variable left. The composition of the
vapor phase and of the liquid phase is fixed automatically ( points B and D ). The
compositions of liquid phase and of the vapor phase are not identical. The vapor phase is
richer in the more volatile component, namely cyclohexane (having lower boiling point and
higher vapor pressure) and the liquid phase is richer in toluene (higher b.p. and lower vapor
pressure). This fact is the basis for separation of liquid mixtures by distillation. We will learn
more about it when we study ideal systems in the forthcoming chapter.

The above mentioned mixture ( x2 = 0.65 ) of toluene and cyclohexane at 90°C and at
40 kPa pressure (low pressure) will exist as a homogeneous gaseous mixture. On increasing
the pressure we will come across a two phase region (vapor-liquid region) where the
composition of the phases is different. The liquid phase will be richer in toluene and the
vapor phase will be richer in cyclohexane. So by lowering the temperature or increasing the
pressure the components of a mixture of superheated vapors may be condensed.

2.4.3 The Chemical Properties and their Technical Applications

An important property of mixtures is their ability to react chemically and produce new
substances. Chemical reactions are often used for matter conversions. Substances of desirable
properties, which do not occur in nature as such, may be produced through chemical
reactions. The most common example is ammonia, which is used for the production of
fertilizers. Ammonia may be synthesized from the reaction of hydrogen and nitrogen under
suitable conditions.

N2 + 3H2  2NH3

One molecule of nitrogen reacts with three molecules of hydrogen to produce two molecules
of ammonia. The symbol indicates the equilibrium reaction. In reaction equilibrium the
maximum technically possible yield should be reached. That is on complete reaction two
moles of ammonia should be found at chemical equilibrium if one starts with 1 mole of
nitrogen and three moles of hydrogen. However, the composition at chemical equilibrium
depends on the thermodynamic states. Figure 2.21 shows the effect of pressure and
temperature on the chemical equilibrium of ammonia synthesis.
35

It is seen that the total yield of two moles of ammonia may only be achieved at low
temperatures. However, at low temperatures the rate of reaction will be very low. Reasonable
rate of reactions are found only at higher temperatures and using suitable catalysts. The yield
increases with the increase in pressure. And so after an analysis the parameters could be
optimized. The technical synthesis of ammonia is performed in the presence of catalysts at
around 450°C and at 250 bars.

2.0

mol                                                p
=
40
0
nNH3

1.5                                                             ba
r

20
0
10
0
25

1.0
10
2 .5
1 .0

0.5

0
0    100        200                 300                        400     °C   500
t

Figure 2.21 : Chemical equilibrium of ammonia synthesis ( Initial composition: 1 mol N2 + 3
mol H2 ).

2.4.4 Models for Fluid Mixtures

For a quantitative description of fluid mixtures we need suitable models. Basically,
any state property of mixture depends on temperature, pressure and its composition. For
example, the molar volume of a fluid mixture will depend on T, p and the mole fractions of
all the components and may be written as:

 (T, p, {xj}) =  xii                                           (2.37)

where {xj} represents the mole fraction of all components. xi represents the mole fraction of a
component i and i, the partial molar volume of the component. The partial molar volume of
a component i, i is the molar volume of that component in the mixture. In general, it is
different from the molar volume of the pure component i. It is a complex function of
temperature, pressure and the composition.

i = i ( T, p, {xj})                                             (2.38)

Relations similar to Equations (2.37) and (2.38) may be written for the corresponding
specific volumes.
36

The difference between the partial molar volume and the molar volume may be
understood from the following experiment:
At a given temperature and pressure (say room temperature and atmospheric pressure)
we add a small amount of water (say 1 mol) to a very large volume of pure water. The
increase in volume is found to be:
18 cm3 (= the molar volume of water, ow)

However if we add the same amount of water (i.e. 1 mol) to a large volume of pure ethanol,
the increase in volume is not 18 cm³ (i.e. ow ) but only 14 cm³ (the molar volume of water in
the mixture, w). Obviously, in this case the effective molar volume of water in ethanol is
smaller than the molar volume of pure water (w < ow ). The reason for the smaller increase
is that the volume a given number of water molecules (e.g. 1 mol) occupy, depends on the
molecules that surround them. In this case each water molecule is surrounded by ethanol
molecules and such a packing of molecules results in an increase of only 14 cm³. The quantity
14 cm³ mol-1 is the partial molar volume of water in pure ethanol. In general this quantity is
denoted by w (w = 18 at xethanol = 0 and w = 14 at xethanol = 1) and is a mixture property. The
partial molar volumes of the components of a mixture vary with composition ( That is i is a
function of the composition). The partial molar volumes of water and ethanol in (water +
ethanol) mixture for the entire composition range at 25°C and 1atm are shown in Figure 2.22.

58

Ethanol
Water
-1
v (water) /cm mol

18.0
v (ethanol) /cm mol

56
3

3

16.0
54
-1

14.0

0.1       0.2        0.4        0.6        0.8        1.0
x (ethanol)
Figure 2.22 : The partial molar volumes of water and ethanol in water + ethanol mixture.

In future the symbol „oi” will be used in subscript to denote the properties of the pure
components.

oi = molar or specific volume of pure component i
i = partial molar or partial specific volume of component i
37

Simple models for the mixture set the partial molar properties or the partial specific
properties equal to the properties of pure substances, the properties which are generally
known.

For example
i ( T, p, {xj}) = oi ( T, p)                               (2.39)

2.4.4.1 Ideal Gas Mixture

For gases the model of ideal gas mixture is used. As in the case of pure gases, it is
assumed that the molecules themselves have negligible volumes and the molecule of type i is
not influenced by the presence of molecule of type j. Each component k behaves as if it were
alone in the available space (volume) and has the pressure (its partial pressure):

RT
pk  nk
ig
(2.40)
V
RT           RT           RT
i.e. p1  n1
ig
, p2  n2
ig
, p3  n3
ig
....
V            V            V

Adding these equations, we obtain

RT        RT    RT
p1  p2  p3  ...  (n1  n2  n3  .....)
ig   ig   ig
  nk    n    p               (2.41)
V         V     V

Equation (2.41) is known as Dalton´s law of partial pressures.

Substituting the value of V from Eq. (2.41) in Eq. (2.40) for pkig we get
n
pk  k p  xk p
ig
(2.42)
 nk
For an ideal gas mixture, the partial pressure of a component is therefore equivalent to
the mole fraction.

Further
ni RT n j RT nRT
V ig (T , p, {n j })               
pi    pj    p
(2.43)
ni RT
V ig              Voi (T , p, ni )   ni oi (T , p )
ig                     ig

i      p    i                   i

Division through the total number of moles n yields

 ig (T , p,{x j })   xioi (T , p)
ig
(2.44)

One can write, therefore, for the partial molar volume of a component i in an ideal gas
mixture:
38

RT
 iig (T , p, {x j })   oi (T , p ) 
ig
(2.45)
p
and for the partial specific volume

R T
 iig (T , p, {w j })   oi (T , p )  (
ig
)                   (2.46)
M p
The ideal gas mixture model describes the thermodynamic behavior of gas mixtures in the
same range in which the ideal (perfect) gas model describes the behavior of pure gases,
namely at low pressures up to a few bars and at a distance from the saturation vapor curve.

2.4.4.2 Moist Air

In addition to mixtures consisting of ideal gases, we may frequently come across such
mixtures where one of the constituents is a vapor. The vapor in the mixtures is at such a low
partial pressure that we can treat it as an ideal gas. Hence the general rules applicable to ideal
gas mixtures apply to these mixtures as well.

A significant difference in the behavior of gas-vapor mixtures as compared to ideal
gas mixtures, however, lies in the fact that the composition of gas-vapor mixtures may change
on heating or cooling while that of ideal gas mixture does not change normally. When the
temperature of the gas-vapor mixture is lowered below a certain limit, it may condense or
solidify and thus the composition of the mixture may change. The vapor condenses to liquid
form if its partial pressure is above triple point pressure and solidifies directly if the pressure
is below the triple point pressure.

Completely dry air does not exist in nature. Water vapor in varying amounts is
diffused through it. We usually come across moist air which is actually a mixture of dry air
and water vapor (a gas-vapor mixture). A study of properties of moist air has become
important due to its wide application in air-conditioning, cooling towers, evaporative
condensers and weather forecasting etc.

We have seen in chapter 2.3.4 that every pure substance has a characteristic saturation
curve (see also Figure 2.23). At any given temperature, the equilibrium pressure of a pure
substance, in the vapor phase, can not exceed its saturation pressure. As the pressure reaches
the saturation pressure, the vapor condenses into liquid. The same rules hold for the
condensable component in a gaseous mixture, i.e. the partial pressure of a component
(namely water vapor) in air can not exceed its saturation pressure for the temperature of air.
Any attempt to increase the partial pressure of water vapor above saturation value will result
in its condensation.

pV < psV (T)                                           (2.47)

where pV is the partial pressure of water vapor and psV is the saturation pressure of pure water
vapor at temperature T.
39

T             vapor
saturation line

1               2

dew 3
point
liquid

p

Figure 2.23 : T-p diagram for moist air.

This may be seen by considering Figure 2.23. Assume that the moisture in the system is at
point 1, inside the superheated vapor range. State 1 of water vapor represents a partial
pressurep1 = pV = pW at a temperature T1. The pressure at point 2 which is situated on the
saturation line at the same temperature (T2=T1) gives the corresponding saturation pressure
psV.

The condensation of water vapor can also occur if the gaseous mixture is cooled at
constant pressure up to the saturation temperature (point 3).

The properties of gas-vapor mixture (moist air) may be quantitatively modelled by
using the model of ideal gas-vapor mixture. In this model the unsaturated gas-vapor mixture
is considered as an ideal gas mixture of components: gas (dry air) and vapor (water vapor).
Dry air which is a mixture of non-condensable gases is treated as a single component. If water
vapor is mixed with an unsaturated gas-water vapor mixture it reaches saturated state at a
definite vapor content above which some water condenses. The ideal gas-vapor mixture then
consists of an ideal gas mixture of (dry air + water vapor) and a condensed phase of pure
vapor (liquid water). The vapor content in a gas-vapor mixture up to saturation point is
generally represented as the partial pressure of the vapor according to equation (2.40):

RT      RT
pV  nV       mV
V       M VV

This is the pressure exerted by the vapor on the container wall if it were alone in the
volume V. Thus in an ideal gas-vapor mixture the saturated state is defined so that the partial
pressure of vapor is equal to the saturation vapor pressure of the pure vapor at the temperature
of the mixture:

pV = psV(T) = psW(T)                                  (2.48)

[psW for saturation pressure of pure water vapor]

The values of psV = psW can be read from the steam tables. The standard tables and diagrams
for water vapor are also applicable to the moisture, provided the partial pressure is used
instead of pressure. The saturation partial pressure gives the maximum possible vapor content
in the gas phase. The unsaturated ideal gas-vapor mixture which does not contain condensed
phase (liquid) is characterized by Eq. (2.47). The saturation sate may be arrived either by
40

supplying more water vapor to the gaseous mixture or by decreasing the temperature. The
temperature at which water vapor starts condensing is called the dew point temperature or
simply dew point, tdp, of the mixture (see Fig. 2.23; the temperature at point 3 represent the
dew point for state 1). It is equal to the saturation temperature of the vapor corresponding to
its partial pressure in the mixture. At dew point holds:

pV = psV (tdp)                                        (2.49)

On further cooling more condensate is formed and the gas phase will contain less
vapor, i.e. it will be dried. The dew point temperature of a gas-vapor mixture (moist air) has a
definite value depending on the partial pressure of water vapor in the moist air. Hence the
relation between the dew point and the corresponding partial pressure of water vapor in moist
air is provided according to the model of „ideal gas / water vapor mixture” through the vapor
pressure curve of pure water.

We now look at some other definitions relevant to moist air. Specific humidity or
humidity ratio x, is defined as the mass of water vapor (or moisture) per unit mass of dry air
in a mixture of air and water vapor. If mG is the mass of dry air in mixture and mV the mass of
water vapor then
m
x V                                                 (2.50)
mG

Specific humidity is a suitable property to describe the state of unsaturated and
saturated gas-vapor mixture. The advantage of describing the relative humidity in terms of the
mass of dry air instead of mass of the mixture is that the mass of dry air does not change
when water condenses or evaporates into the air (mass of dry air remains constant whereas
the mass of moist air changes with the amount of moisture).

Another important property of moist air is the relative humidity,  The relative
humidity is defined as the ratio of the partial pressure of water vapor, pV, in a gas-vapor
mixture to the saturation pressure, psV, of pure water, at the same temperature of the mixture:

pV
                                       
p sV ( T )

The specific humidity may be related to the relative humidity and to the partial
pressure. For unsaturated gas-vapor mixture using the ideal gas law we can write:

V
mV  pV
R
(    )T
MV

V
and     mG  p G
R
(    )T
MG

as p = pV + pG it follows:
41

mV   p M      M V pV
x       V V                                     (2.52)
mG  pG M G M G ( p  pV )

18        pV                 pV                 p sV ( T )
or              x                     0.622             0.622                       (2.53)
28 .96 ( p  pV )         ( p  pV )         ( p   p sV ( T ))

For a given specific humidity, the partial pressure of water vapor is

xp
pV                                                (2.54)
(0.622  x)

The saturated specific humidity xs is found when =1, i.e.

MV      p sV (T )                p sV (T )
x s (T , p )                         0.622                           (2.55)
M G ( p  p sV (T ))         ( p  p sV (T ))

The volume of an ideal gas-vapor mixture is given by the addition of the volume of
gas and the volume of vapor

V = VG + VV

In practice the mass of gas (air) remains constant in moist air and it is only the mass of
vapor (moisture, water vapor) which changes. It is, therefore practice to define the specific
properties of gas-vapor mixtures in terms of the mass of the gas mG. For example, the specific
volume of an unsaturated gas-vapor mixture or a just saturated gas-vapor mixtures is defined
as the volume per kg of gas:
V     m   mV V
1 x        G G             G  xV
mG          mG
(2.56)
RT      RT       R   R      T
1 x         x      [        x]
MG p    MV p    M G MV      p

The subscript (1+x) in 1+x denotes that the specific volume is for a mixture of 1 kg gas and x
kg vapor.

For the specific volume of saturated gas-vapor mixture the volume of the condensed
phase (liquid) should also be considered. But as the specific volume of the condensate is very
small as compared to specific volume of gas, it may be neglected and the above equation
(2.56) may also be used for saturated ideal gas-vapor mixtures.
42

2.4.4.3 Ideal Solutions

The term solution is generally used for solids, liquids or gases dissolved in liquids. As
the thermodynamic system is a fluid or a fluid mixture we will be more interested in the
properties of liquid solutions and gas solutions. First we look into liquid solutions. The
behavior of real liquid mixtures (solutions) is complex. However, to simplify thermodynamic
and mathematical treatment of real solutions the fluid model „Ideal Solution / Ideal Liquid
Solution” is used. The model „Ideal Liquid Solution” serves as a first step into the analysis of
real liquid systems.

As in the case of gas mixtures we set the partial molar volume of a component i in
ideal solution equal to its molar volume as pure substance i.e.

iil (T , p, {x j })  oi (T , p)
l
(2.57)

where loi (T,p) is the molar volume of pure liquid i at the temperature and pressure of the
solution (liquid mixture). We write, therefore, for the molar volume of an ideal solution
(liquid mixture)

il ( T, p, {xj}) = xi iil =  xi oil ( T, p)   (2.58)

Similar relation holds for the partial specific volume of an ideal liquid mixture.

il ( T, p, {wj}) = wi oil (T,p)                  (2.59)

With xi and wi′ we denote the mole fraction and the weight fraction of any component i in the
liquid phase. {xj} and {wj} represent the mole fractions and weight fractions of all
components.

2.4.4.4 Ideal Systems

We have described the properties of gas mixture with the help of a model of „Ideal
gas mixture”. The properties of liquid mixtures could be described with the help of „Ideal
Solution” model. Now we consider the vapor-liquid phase equilibrium of a mixture of
volatile liquids. To describe this equilibrium we define an „Ideal System”. We denote it with
superscript „is”.

Consider a solution composed of several volatile substances in a container which is
initially evacuated. Since the components are volatile, some of the solution evaporates to fill
the space above the liquid with vapor. After the attainment of thermodynamic equilibrium at a
temperature T, the total pressure p within the container is the sum of the partial pressures of
all the components, i.e.

p = p1 + p2 + ...                                     (2.60)

These partial pressures are measurable, as are the equilibrium mole fractions x1′, x2′,.... in the
liquid. Let one of the components, i, be present in relatively large amount as compared to any
of the others. Then it is found experimentally that

pi = xi′ ps0i                                         (2.61)
43

where ps0i is the vapor pressure of the pure component i. This is Raoult´s law. It states that the
partial pressure of a substance in a mixture is proportional to its mole fraction and its vapor
pressure when pure. It is a limiting law and it is followed in any solution as xi approaches
unity. Raoult´s law is obeyed very well over a wide range of concentrations, when the
components are similar in structure and chemical nature. An example is the mixture of n-
hexane and n-heptane. Similarly, a mixture of benzene and toluene (methylbenzene) obeys
the Raoult´s law throughout the composition range and is termed as an ideal system. The
ideal system is defined as one in which each component obey Raoult´s law over the entire
range of composition. According to equation (2.61) the partial pressure of species i in the
vapor phase, written on the left hand side, is equal to the product of the liquid phase mole
fraction xi′ of species i and its vapor pressure ps0i at equilibrium temperature T.

We now apply Raoult´s law to binary mixtures in which both components are volatile.
The partial vapor pressures of the two components of an ideal binary mixture are proportional
to the mole fractions of the components in the liquid. The total pressure of the vapor is the
sum of the two partial vapor pressures. So we have in a binary liquid solution

x1′ + x2′ = 1

p1 = x1′ ps01 = (1-x2′ ) ps01                (2.62)

and also                                p2 = x2′ ps02                                (2.63)

If the total pressure over the solution is p, then

p = p1 + p2 = (1-x2′ ) ps01 + x2′ ps02

or                                      p = ps01 + x2′ (ps02 – ps01)                 (2.64)

Equation (2.64) relates the total pressure over the mixture to the mole fraction of component
2 in the liquid at constant temperature. It further shows that p is a linear function of x2′. This
relationship is shown in Figure 2.24. The curve is called bubble-point curve. It is clear from
Fig. 2.24 that the bubble-point curve in an ideal system is a straight line in p-x diagram ( also
called vapor pressure diagram). The bubble-point curve connects the vapor pressures of the
pure components. The terms x2′ ps02 and (1-x2′ ) ps01 are also plotted in Figure 2.24 and show
the Raoult´s law.

The expression for the dew-point curve may be derived as follows:

x2″ = p2 / p                                 (2.65)

[ We use "double slash" in the superscript to represent the composition (here, mole fraction)
of the vapor phase. When the liquid and the vapor phase exist together we will consistently
use "single slash" in the superscript for liquid phase and "double slash" in the superscript for
the vapor phase composition. For overall composition no superscript will be used]
44

150
Constant t = 90 °C
pS02
kPa
bubble-p oint
curve
120
p

100
°C Constant p = 101.3 kPa
t S01
B                     dew-poin t
curve
D'   dew-poin t
pS01
D                                                    100          B'                  curve
is

t
'
p2 = x2 pS02
50                                                                                  bubble-p oint
is                                                90        curve
'
p1 = (1-x 2 )pS01

80                                                t S02

0                                                                       70
0              0.5                      1.0                               0               0.5                   1.0
Mole fraction benzene                                                         Mole fraction benzene
',
x2 x2''                                                                       x2', x2''

Figure 2.24 : p-x and T-x Diagrams for an ideal binary system.

Using the values of p2 and p from equations (2.63) and (2.64) we obtain

x'2 p s 02              x'2 p s 02
x 
''
                                        (2.66)
p s 01  x'2 ( p s 02  p s 01 )
2
p

solving for x2′ yields

x2′ ps02=x2″ ps01 + x2′ x2″ (ps02-ps01)

'
x2' ps 01
or                                 x2 
'
(2.67)
ps 02     x2' ( ps 02  ps 01 )
'

Putting the value of x2′ from (2.67) in (2.64) we get

x2' ps 01 ( ps 02  ps 01 )
'
p  ps 01 
ps 02  x2' ( ps 02  ps 01 )
'

p s 01  p s 02
p                                                                            (2.68)
p s 02    x2' ( p s 02  p s 01 )
'

Equation (2.68) expresses p as a function of x2″, the mole fraction of component 2 in vapor.
This function is also plotted in Figure (2.24). It is called dew-point curve. One sees that the
dew-point curve is not a straight line but a curve.
45

Equation (2.68) can also be rearranged in another form.

1   x '' x ''
 1  2                                                 (2.69)
p ps 01 ps 02

Equation (2.68) and (2.69) give VL-Equilibria of an ideal system at constant
temperature (that means under isothermal conditions).

The solution of equation (2.64) for equilibrium temperatures at specified x2′ is
inconvenient. Hence the isobaric VL-Equilibria cannot be represented by such explicit
equations. By rearranging equation (2.64) follows the expression for the bubble point curve of
a binary system

p s 01 (T )    ' p    (T )
1  (1  x2 )
'
 x2 s 02                    (2.70)
p              p

For the dew-point curve on rearrangement of equation (2.69) we get:

p                 p
1  (1  x2' )
'
 x2'
'
(2.71)
p s 01 (T )       p s 02 (T )

Equations (2.70) and (2.71) can only be solved iteratively as the vapor pressure is a function
of temperature.

The bubble-point curve ( T- x2′ ) is most readily constructed by solving (2.64) for the
bubble-point compositions x2′ at representative values of T between the saturation
temperatures of the pure components

p  p s 01 (T )
x2 
'
(2.72)
p s 02 (T )  p s 01 (T )

The corresponding equilibrium compositions on the dew-point curve are obtained by
substituting the value of x2′ from equation (2.72) in equation (2.66).

p s 02 (T ) p s 02 (T )        [ p  p s 01 (T )]
x2'  x2
'     '
                                            (2.73)
p           p        [ p s 02 (T )  p s 01 (T )]

The VLE (vapor-liquid equilibrium, i.e. x2″ vs. x2′) diagram of an ideal system at a given
temperature may be constructed most conveniently using equation (2.66). The isobaric VLE
diagram may be constructed from the corresponding T-x diagram. Figure 2.25 shows
schematic VLE diagram for an ideal system. The VLE diagrams at constant temperature and
at constant pressure do not differ much from one another. One can see from Eq. (2.66) that
for an ideal system if ps02 =ps01, i.e. if the vapor pressures of the pure components are equal,
then no material separation can be achieved as the bubble-point curve and the dew-point
curve merge into one another.
46

Figure 2.25 : VLE Diagram for an ideal system.

The enrichment of more volatile component in the vapor phase is represented by the
so called "relative volatility" or "volatility ratio" defined as

x´´ / x´2
12     2
´´     ´
(2.74)
x1 / x1

For a binary ideal system the relative volatility is then

'
( x2 p s 02 (T )) '
/ x2
p              p (T )
12  '
is
 s 02                     (2.75)
( x1 p s 01 (T )) '    ps 01 (T )
/ x1
p

Equation (2.75) shows clearly that the relative volatility in a binary ideal system is the ratio of
vapor pressure of the two components, i.e. it depends only on temperature. It does not depend
on the composition of the mixture. Its temperature dependence is weak and for small
temperature ranges the relative volatility may be considered as constant.

There are only a few systems which may be considered as ideal systems. However, the
model of ideal systems helps very much to understand the behavior of real solutions. Because
now only the deviations from the ideal system behavior is to be considered. Both liquid-phase
and vapor-phase behavior may contribute to non-ideality of a system. In most of the cases it is
the non-ideal behavior of liquid phase which causes the deviations from an ideal system. The
non-ideality of the liquid phase is interpreted in terms of the deviations from the Raoult´s law.
A significant result of Raoult´s law is that the bubble-point curve of an ideal system is a
straight line. Systems for which the bubble-point and partial pressure curves lie above the
47

Raoult´s law lines, are said to exhibit positive deviations from the Raoult´s law. Similarly, if
these curves lie below the Raoult´s law lines, the system is said to exhibit negative deviations
from the Raoult´s law.

Figure 2.26 : Positive and negative deviations from Raoult´s law.

Figure 2.26 shows different types of deviations. Even though the course of bubble-point curve
and dew-point curve for non-ideal systems is different from the ideal system curves, the basic
finding for ideal systems that in the two phase region the vapor phase will become richer in
the more volatile component is also applicable to real systems and forms the basis for the
separation of liquid mixtures by distillation.

2.4.5 Interpretation of Phase Diagrams
Now we look at the vapor-liquid phase equilibrium of a binary mixture of substance A
(component number 1) and substance B (component number 2) maintained at constant
pressure p (e.g., 101.3 kPa) inside a container (e.g., in a piston-cylinder assembly). We follow
an isobaric process in which the temperature is gradually increased from a low value (state 1),
to a high value (state 5), as shown in Figure 2.27. State 1 represents a mixture of subcooled
liquids A and B. Substance B is more volatile. This mixture has a mole fraction x2′ of B
[subscript 2 for component no 2 i.e. substance B and superscript ‚single slash‘ for liquid
phase]. There is only one phase and hence for a binary system three degrees of freedom. Thus
the temperature may be changed although pressure and composition remain fixed. At state 2
the temperature reaches the liquid line and the first bubble of vapor appears which has a
composition of x2″ (at f) [here superscript ‚double slash‘ for vapor phase]. The vapor is much
richer in the lower boiling point substance B than the liquid [the value of x2″ (at f) is greater
than x2′ (at 2)]. This fact is basis for the separation of liquid mixtures by distillation. As the
temperature is increased further, the vapor phase grows at the expense of the liquid phase.
State 3 lies in vapor-liquid two phase region. Here the liquid phase composition is x2′ (at c)
and the vapor phase composition is x2″ (at d). In this region there are only two degrees of
freedom and hence the temperature and pressure fix the state of each phase. At state 4 the last
48

traces of liquid, of composition x2′ (at e) disappear and the vapor has a composition of x2″ (at
4). At state 5 the system exists entirely as a vapor.

p = const. = 101.3 kPa

Superhea ted vapor
T
5

T0,1 a                              Two-phase
region

e                  4     Saturate d vapor

3
c                      d
T2                                             f
2

Saturate d liquid
1                               b T0,2

Subcoole d liquid

0                                                   1.0
x2
Figure 2.27 : Vapor-liquid phase equilibrium for substance A(1) + substance B(2) at p =
const.

We look into the T-x diagram of this system (Figure 2.27). The lower curve a-e-c-b is
called the bubble-point curve and describes the liquid composition. The upper curve a-d-f-b,
called the dew-point curve, describes the vapor composition. The curves a-e-c-b and a-d-f-b
intersect the T-axes at T0,1 and T0,2. These are the boiling points of substance A and substance
B respectively at a constant pressure of 101.3 kPa. The region on and above a-d-f-b represents
the superheated vapor. Here only vapors of component 1 (i.e.,A) and component 2 (i.e.,B) are
present and the overall mole fraction x2 of B is the same as x2″ (at state 5). On and below the
bubble-point curve (a-e-c-b) only liquid phase is present and the overall mole fraction x2 of B
is x2′ (at state 1). Within the area enclosed by the two curves both liquid and vapor are present
and any point within this area represents quantitatively the relative amounts of each (liquid
and vapor). We note that the temperature varies from T (state 2) to T (state 4) during the
49

phase transition from liquid to vapor. This is in contrast to the phase transition in a pure
substance that takes place at a constant temperature (if pressure is fixed).

Mixtures within the area enclosed by a-e-c-b and a-d-f-b will consist of both liquid
and vapor parts. The equilibrium compositions of the phases are determined from the phase
diagram by drawing a horizontal line at constant temperature (at equilibrium state the
temperature and pressure in the two phases should be the same). The intersection of this line
with the dew-point curve yields the the value of x2″ which describes the composition of the
vapor, while the intersection with bubble-point curve yields the value of x2′ which describes
the composition of the liquid. Straight lines such as e-4, c-d and 2-f, which connect states in
phase equilibrium, are known as tie lines.

Tie lines are useful for deriving the overall composition in the vapor-liquid two-phase
region. Let n be the total number of moles of a mixture having a mole fraction x2 of
component 2 which separates into n′ moles of liquid with a mole fraction x2′ and n″ moles of
vapor of mole fraction x2″. A material balance (mole number balance) on component 2 gives

x2′ n′ + x2″n″ = x2 n                                        (2.76)

Similarly, an overall mol number balance gives:

n′ + n″ = n                                                  (2.77)

Solving the two equations we obtain
n ' x 2'  x 2
'
                                                        (2.78)
n '' x 2  x 2
'

Derivation of lever principle

Dividing (2.76) by n″

n'              n
´
x2     ''
 x2'  x2 ''
'
(a)
n              n
Similary dividing (2.77) by n´´
n'         n
''
 1  ''                          (b)
n         n
n
Putting the value of          from (b) in (a)
n ''
n '  x2''
n'
 x2'  x2  x2 ''
'

n ''                      n
'
n
''
( x2  x2 )  x2'  x2
'     '

n

We see from Figure (2.27) that (x2″-x2) and (x2-x2′) are the lengths of the segments 3-d and c-3
of the tie line c-d. Thus we have
n' 3  d

n '' c  3
50

This is known as the lever principle. It states that the ratio of the numbers of moles of
the equilibrium phases is inversely proportional to the ratio of the lengths of the
corresponding segments of the tie line.

The lever principle also holds for mass units if the analogous quantities are substituted for
mole numbers and mole fraction in the above relation.

m ' w2'  w2
'
                                        (2.79)
m '' w2  w2
'

2.4.6 The Distillation of Mixtures

The sequence of events shown in section 2.4.5 is observed if no material is removed
from the system as the temperature is increased. We now see what happens when some of the
vapor formed is removed from the system.

T
T0,1

T2                             3
2

T4                                               5
4

T6                                                        7
6
8         T0,2
1

R1             D1               D2

0                                                        1.0
x2
Figure 2.28 : The process of distillation.
51

It may be seen from Figure 2.28 that when a liquid mixture of composition x2′ (at
state 1) is heated, it boils when the temperature reaches T2 (at 2) and the vapor has a
composition x2″ (at 3), which is different from the liquid composition at state 2. The vapor is
richer in the more volatile component B. If more heat is supplied at this fixed temperature we
move along the tie line. If the vapor at state 3 is drawn off and completely condensed, then
the first drop gives a liquid of composition x2′ (at 4) [the same as x2″ (at 3)], which is richer in
the more volatile component B than the original liquid. We call it distillate D1. The rest is
known as residue R1. This is the process of simple distillation.

In the process of fractional distillation, the boiling and condensation cycle is repeated
successively. When the condensate of composition x2′ (at 4) is reheated, it boils at T4 (at state
5) and yields a vapor of composition x2″ (at 5) which is even richer in the more volatile
component. The vapor (distillate D2) is drawn off, and the first drop condenses to a liquid of
composition x2′ (at 6). The cycle can be repeated until almost pure B is obtained. The time
and labor involved in this batch type of separation is very high. Hence in practice the so
called fractionating column is used for continuous separation. The vapor passes up a vertical
fractionating column packed with glass rings or beads to give a large surface area. The
successive cycles of vaporization and condensation take place on their surfaces.

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