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1 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. 2 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 4 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 5 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: 6 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 7 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 (pig and hence the same empirical temperature if it is in thermal equilibrium with a system. The value of (pig can be used to define an empirical temperature ig of the ideal gas thermometer ig = C (pig (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 pand , 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 8 (pig is determined. The value of (pig 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 (pig 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 pig = Rig = 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 9 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 (pto (pig]. 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 10 (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 11 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) 12 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 13 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 14 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 ( xcyclohexane = 0.0 to 1.0 ). We select the composition of the gas phase as xcyclohexane = 0.65 and xtoluene = 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}) = xii (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 }) xioi (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 pressurep1 = 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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thermodynamic equilibrium, thermal equilibrium, local thermodynamic equilibrium, internal energy, equilibrium state, two systems, laws of thermodynamics, equilibrium constant, equation of state, first law of thermodynamics, zeroth law of thermodynamics, classical thermodynamics, chemical equilibrium, non-equilibrium thermodynamics, thermodynamic system

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