# Thermodynamics - PDF by shariq1309

VIEWS: 12,009 PAGES: 27

• pg 1
```									MasteringPhysics                                                                                                                           11/19/08 6:13 PM

Assignment Display Mode:    View Printable Answers

Physics 5D Fall 2008
Assignment 5

Due at 2:00pm on Tuesday, November 18, 2008

Isobaric, Isochoric, Isothermal, and Adiabatic Processes
Description: Isobaric, isochoric, isothermal, and adiabatic processes on an ideal gas are examined using pV diagrams.
Questions are posed about heat, work, and how they explain internal energy change using the first law of thermodynamics.

Learning Goal: To recognize various types of processes on            diagrams and to understand the relationship between           -
diagram geometry and the quantities       ,    , and      .

The first law of thermodynamics is an expression of conservation of energy. This law states that changes in the internal energy of
a system       can be explained in terms of energy transfer into or out of the system in the form of heat and/or work       . In
this problem, we will write the first law of thermodynamics as

.

Here "in" means that energy is being transferred into the system, thereby raising its internal energy, and "out" means that energy
is leaving the system, thereby reducing its internal energy. You will determine the sizes of these energy transfers and classify
their effect on the system as energy in or energy out.

Consider a system consisting of an ideal gas confined within a container, one wall of which is a movable piston. Energy can be
added to the gas in the form of heat by applying a flame to the outside of the container. Conversely, energy can also be
removed from the gas in the form of heat by immersing the container in ice water. Energy can be added to the system in the
form of work by pushing the piston in, thereby compressing the gas. Conversely, if the gas pushes the piston out, thereby
pushing some atmosphere aside, the internal energy of the gas is reduced by the amount of work done.

The internal energy of an ideal gas is directly proportional to its absolute temperature       . An ideal gas also obeys the ideal gas
law

,

so the absolute temperature       is directly proportional to the product of the absolute pressure    and the volume     . Here
denotes the amount of gas in moles, which is a constant because the gas is confined, and          is the universal gas constant.

A       diagram is a convenient way to track the pressure and
volume of a system. Energy transfers by heat and/or work are
associated with processes, which are lines or curves on the
diagram taking the system from one state (i.e., one point on the
diagram) to another. Work corresponds geometrically to the area
under the curve on a    diagram. If the volume increases (i.e.,
the system expands) the work will be classified as an energy
output from the system.

Part A
What is the sign of        as the system of ideal gas goes from point A to point B on the graph? Recall that        is proportional to
.

Hint A.1     How to approach the problem
Use the ideal gas law to figure out how the absolute temperature of the gas in state A compares to its absolute temperature
in state B. Since the internal energy of an ideal gas is proportional to its absolute temperature, this will tell you how
changes from state A to state B.

ANSWER:               The internal energy of the system increases, so         is positive.
The internal energy of the system decreases, so         is negative.
The states A and B have the same internal energy, so          is zero.
cannot be determined without knowing the process used (i.e., the path taken) to get from state A to

http://session.masteringphysics.com/myct                                                                                                        Page 1 of 27
MasteringPhysics                                                                                                                   11/19/08 6:13 PM

cannot be determined without knowing the process used (i.e., the path taken) to get from state A to
state B

The value of      depends only on the state of the system. Thus          depends only on the endpoint states, not on the
process followed that determines the path between the endpoint states.

One possible way for the system to get from state A to state B is to follow a hyperbolic curve through point C, along which
the product of   is a constant. Temperature is proportional to
the product      , so this is a constant-temperature path, also
known as an isothermal process.

Part B
How are       and    related during this isothermal expansion?

Part B.1      Find the sign of

Recall that the magnitude of the work done in going from one state to another is the area underneath the curve defined by the
path. If the work is nonzero, then the sign of  is determined by the direction of the path. Which of the following describes
for the path in Part B?

From Part A, you know that               . Using the first law of thermodynamics, you can see that this is equivalent to
saying that                 . Now that you know that        is positive, what does this tell you about   ?

Both       and    provide energy input.
Both       and    provide energy output.
provides energy output, while      provides energy input. They are equal in magnitude.
provides energy input, while      provides energy output. They are equal in magnitude.

You can tell that the system is losing internal energy due to work because its volume is increasing. The internal energy
change during any isothermal process involving an ideal gas is zero, so here the system must gain as much energy in the
form of heat as it loses by doing work during this process.

Another way to get from state A to state B is to go vertically from A to point D, holding volume constant, and then go
horizontally to point B, holding pressure constant. A constant-
volume path is called an isochoric process. A constant-pressure
path is called an isobaric process.

Part C
How are       and    related during the isochoric part of the overall path from state A to state D?

Hint C.1      How to approach the problem

http://session.masteringphysics.com/myct                                                                                                Page 2 of 27
MasteringPhysics                                                                                                                        11/19/08 6:13 PM

Use the ideal gas law to determine how the absolute temperature of the gas in state A compares to its absolute temperature
in state D. This will help you determine whether the net energy transfer is in or out, since the internal energy of an ideal gas
is proportional to its absolute temperature.

provides energy input, while       equals zero.
provides energy output, while       equals zero.
provides energy input, while       provides energy output.
provides energy output, while       provides energy input.

You can tell that the system is losing internal energy since its temperature goes down (since        goes down). No work is
done during any isochoric process, since no area accumulates under a vertical curve. Hence energy transfer in the form of
heat must account for the entire internal energy change.

Part D
How are     and      related during the isobaric part of the overall path from state D to state B?

ANSWER:            Both       and    provide energy input.
Both       and    provide energy output.
provides energy output, while     provides energy input. They are equal in magnitude.
provides energy output, while     provides energy input;      is larger.
provides energy output, while     provides energy input;     is larger.

In going from state A to state D the system loses internal energy. Since the overall change of internal energy from state A
to state B is zero, during the isobaric part of the overall process the system internal energy must increase. Since the
system is expanding, internal energy is lost from the system due to work. Hence         must exceed      (in magnitude) to
explain the net increase in internal energy.

Another way to get from state A to state B is to follow an adiabatic path from state A to state E, in which no heat energy
transfer is allowed, and then to follow an isochoric path from state E vertically to state B. Notice that during the adiabatic part
of this path, from state A to state E,        by definition and
internal energy is lost due to work since the system is expanding.

Part E
Which of the following statements are true about the isochoric part of the overall path, from state E to state B?

Hint E.1     How to approach the problem
Recall that the total internal energy change from state A to state B is zero. This means that the isochoric process must undo

Check all that apply.

provides energy input.
decreases.
increases.

Since no work is allowed in isochoric processes,       must serve as an energy input to explain the increase in both absolute
temperature and internal energy.

One more way to get from state A to state B is to follow a direct path through state F. This process is not isobaric,
isochoric, isothermal, or adiabatic, yet you can draw some
conclusions about its energetics using the first law of
thermodynamics.

http://session.masteringphysics.com/myct                                                                                                     Page 3 of 27
MasteringPhysics                                                                                                                            11/19/08 6:13 PM

Part F
Which of the following statements are true about the first half of this process, just going from state A to state F?

Check all that apply.

provides energy input.
provides energy input.
is larger (in magnitude) than    .

State F has a larger        value than state A, so the internal energy increases in this part of the process. Since the system is
expanding, internal energy is lost from the system due to work. Hence           must exceed          (in magnitude) to explain the
net increase in internal energy.

Understanding what happens during the second half of the process, going from state F to state B, is more subtle. The
temperature and the internal energy both go down. Since the system continues to expand,      provides energy output.
However, it is challenging to determine whether         provides energy input or energy output from state F to state B. Can
you figure it out?

Understanding pV Diagrams
Description: Several qualitative and conceptual questions related to pV-diagrams.

Learning Goal: To understand the meaning and the basic applications of pV diagrams for an ideal gas.

As you know, the parameters of an ideal gas are described by the equation

,

where    is the pressure of the gas,     is the volume of the gas,   is the number of moles,          is the universal gas constant, and
is the absolute temperature of the gas. It follows that, for a portion of an ideal gas,

.

One can see that, if the amount of gas remains constant, it is impossible to change just one parameter of the gas: At least one
more parameter would also change. For instance, if the pressure of the gas is changed, we can be sure that either the volume or
the temperature of the gas (or, maybe, both!) would also change.

To explore these changes, it is often convenient to draw a graph showing one parameter as a function of the other. Although
there are many choices of axes, the most common one is a plot of pressure as a function of volume: a pV diagram.

In this problem, you will be asked a series of questions related to different processes shown on a pV diagram . They will help
you become familiar with such diagrams and to understand what
information may be obtained from them.

One important use for pV diagrams is in calculating work. The product             has the units of

http://session.masteringphysics.com/myct                                                                                                         Page 4 of 27
MasteringPhysics                                                                                                                     11/19/08 6:13 PM

; in fact, the absolute value of the work done by the gas (or on the gas) during any
process equals the area under the graph corresponding to that process on the pV diagram. If the gas increases in volume, it does
positive work; if the volume decreases, the gas does negative work (or, in other words, work is being done on the gas). If the
volume does not change, the work done is zero.

The following questions may seem repetitive; however, they will provide practice. Also, the results of these calculations may
be helpful in the final section of the problem.

Part A
Calculate the work       done by the gas during process        .

=

Part B
Calculate the work       done by the gas during process        .

=

Compare your result with that from part A. The work              done during a process        is equal to          , the work
done during the reverse process            .

Part C
Calculate the work       done by the gas during process        .

=

Part D
Calculate the work       done by the gas during process                .

=

Part E
Calculate the work       done by the gas during process        .

=

No work is done during a process, if the gas does not experience a change in volume.

The absolute value of the work done by the gas during a cycle (a process in which the gas returns to its original state)
equals the area of the loop corresponding to the cycle. One must be careful, though, in judging whether the work done by the
gas is positive or negative. One way to determine the total work is to calculate directly the work done by the gas during each
step for the cycle and then add the results with their respective signs.

Part F
Calculate the work       done by the gas during process                          .

http://session.masteringphysics.com/myct                                                                                                  Page 5 of 27
MasteringPhysics                                                                                                                          11/19/08 6:13 PM

=

This result can be obtained either by calculating the area of the region 1265 or by adding the amounts of work done by
the gas during each process of the cycle. The latter method helps verify that the net work done by the gas is, indeed,
positive.

As discovered earlier, The work             done during a process                             is equal to            , the work
done during the reverse process                               .

Part G
Calculate the work        done by the gas during process                            .

=

A Law for Scuba Divers
Description: Find the increase in the concentration of air in a scuba diver's lungs. Find the number of moles of air exhaled.
Also, consider the isothermal expansion of air as a freediver diver surfaces. Identify the proper pV graph. Associated medical
conditions discussed.

SCUBA is an acronym for self-contained underwater breathing apparatus. Scuba diving has become an increasingly popular
sport, but it requires training and certification owing to its many dangers. In this problem you will explore the biophysics that
underlies the two main conditions that may result from diving in an incorrect or unsafe manner.

While underwater, a scuba diver must breathe compressed air to compensate for the increased underwater pressure. There are a
couple of reasons for this:

1. If the air were not at the same pressure as the water, the pipe carrying the air might close off or collapse under the
external water pressure.
2. Compressed air is also more concentrated than the air we normally breathe, so the diver can afford to breathe more
shallowly and less often.

A mechanical device called a regulator dispenses air at the proper (higher than atmospheric) pressure so that the diver can inhale.

Part A
Suppose Gabor, a scuba diver, is at a depth of            . Assume that:

1. The air pressure in his air tract is the same as the net water pressure at this depth. This prevents water from coming in
through his nose.
2. The temperature of the air is constant (body temperature).
3. The air acts as an ideal gas.
4. Salt water has an average density of around 1.03            , which translates to an increase in pressure of 1.00    for
every 10.0   of depth below the surface. Therefore, for example, at 10.0         , the net pressure is 2.00    .

What is the ratio of the molar concentration of gases in Gabor's lungs at the depth of 15 meters to that at the surface?

The molar concentration refers to      , i.e., the number of moles per unit volume. So you are asked to calculate                   .

Part A.1     Find an equation for calculating concentrations

What is the equation for the molar concentration          of the air in Gabor's lungs?

Hint A.1.a The ideal gas law

The ideal gas law states that                . From this, determine       .

=

Now recall that the temperature is assumed to be the same at the surface and at this depth.

http://session.masteringphysics.com/myct                                                                                                       Page 6 of 27
MasteringPhysics                                                                                                                         11/19/08 6:13 PM

Part A.2      Find the pressure underwater
What is the     , the pressure on the diver at 15       ?

Hint A.2.a How to approach the problem
Using the information given in the problem introduction, find the increase in pressure at an underwater depth of 15
from that at the surface. Add the pressure increase to the air pressure at the water's surface to find the total pressure.

Part A.2.b Find an expression for the increase in pressure underwater
Given that for every 10      of depth below the surface, the pressure increases by 1         , find the increase in pressure    at
a given depth     (measured in meters).

=

=

The increased concentration of air in the lungs, through diffusion, leads to an increased concentration of air in the
bloodstream. While the increase in the oxygen concentration achieved at typical depths is not toxic, the increase in
nitrogen concentration can lead to a condition called nitrogen narcosis.

"The bends" refer to another condition associated with incorrect scuba diving. If scuba divers rise to the surface from a
depth while attempting to hold their breath, or too fast, air bubbles may form in their bloodstream. These bubbles may
then get stuck at the joints, causing great pain and possible death. This condition is called decompression sickness, often
referred to as the bends.

Why it happens: The exact mechanism of formation of these bubbles is not completely understood. However, it is
believed that the nature of the mechanism is as follows. When the concentration of air in the lungs decreases very
rapidly, a high concentration of air in some parts of the body is produced as the air starts to diffuse back into the lungs.

Prevention: By surfacing slowly, a diver allows the concentration of gases dissolved in the blood to reduce slowly
(through exchange with the lungs), which prevents the formation of bubbles.

Treatment: One of the possible treatments is to repressurize the diver in a pressure chamber, and then slowly decompress
him or her over hours, or even days.

Part B
If the temperature of air in Gabor's lungs is 37     (98.6       ), and the volume is    , how many moles of air    must be released

by the time he reaches the surface? Let the molar gas constant be given by          = 8.31         .

Hint B.1      How to approach the problem
Compute the number of moles of air in 6      at the underwater pressure and again at the surface pressure. The difference is
the number of moles of air that must be exhaled.

Part B.2      Solve the ideal gas law for moles
If you have a sample of gas occupying volume            at pressure   and temperature    , how many moles of gas do you have?

=

Part C

http://session.masteringphysics.com/myct                                                                                                      Page 7 of 27
MasteringPhysics                                                                                                                           11/19/08 6:13 PM

Now let's briefly discuss diving without gear. If Gabor takes a deep dive without scuba gear, and without inhaling or exhaling,
which of the graphs in the figure best describes the pressure-
volume relationship of the air in his lungs?

Again, assume that the temperature of the air is constant (body
temperature) and that the air acts as an ideal gas.

Note that the units and scales are not shown on the graphs
because they depend on the number of moles of the gas.

Hint C.1       How to approach the problem
The ideal gas law               describes the behavior of a gas given the macroscopic parameters of pressure , volume               ,
and temperature     , the amount of gas in moles      , and the gas constant   , where
. Use the ideal gas law to analyze the behavior of the air in the
diver's lungs as he approaches the surface from a given depth.

Part C.2    Express the pressure in terms of the volume and temperature
To determine what the graph should look like, solve the ideal gas law for pressure.
Express the answer in terms of      ,   ,     , and    .

=

Hint C.3 Importance of constant temperature
Assuming that the temperature of the gas is constant simplifies the problem because we can then express               as a constant
. The ideal gas law becomes                . This inverse relationship determines the basic shape of the pV graph.

Choose the letter associated with the graph that best depicts the pressure-volume relationship.

B
C
D

If you didn't seal your nose and mouth shut, to keep the water from getting into your nose, you would need the air
pressure in your lungs to be the same as the water pressure outside. As the graph shows, this means that your lungs would
need to contract. Of course, they cannot do so indefinitely. This is why you may need an air tank or a snorkel even for a
short, but deep, dive. For longer dives, you simply need the air for physiological reasons, regardless of depth.

Part D
What type of expansion does the air in the "freediver's" (no gear) lungs undergo as the diver ascends?

Hint D.1    How to approach the problem
Determine which of the thermodynamic properties of the air in the diver's lungs does not change value for this problem.
Think about a prefix that means "staying the same."

Part D.2 Determine the proper prefix
Which of the following prefixes means constant or staying the same?

uni
sub
trans

Hint D.3    Possible terms for description of processes
The key terms for the macroscopic properties of a gas are -baric for pressure, -choric for volume, and -thermal for
temperature.

Give the answer as one word.

http://session.masteringphysics.com/myct                                                                                                        Page 8 of 27
MasteringPhysics                                                                                                                          11/19/08 6:13 PM

Cylinder of Ideal Gas with a Piston
Description: Students calculate the work done, change in internal energy, and amount of heat supplied for ideal gas in a
cylinder which is kept at constant pressure (via a piston) as the temperature is raised by heating the system.
A cylinder contains 0.250 mol of carbon dioxide               gas at a temperature of 27.0    . The cylinder is provided with a
frictionless piston, which maintains a constant pressure of 1.00         on the gas. The gas is heated until its temperature increases
to 127.0     . Assume that the        may be treated as an ideal gas.

Part A
How much work          is done by the gas in this process?

Hint A.1      Definition of the work done by an ideal gas
The work done by an ideal gas when its volume is changed is given by the expression

.

where      is the initial volume,    is the final volume, and    is the pressure (which is constant in this particular case).

Part A.2      Calculate the initial volume
Using the ideal gas law and the constants given, determine the initial volume         of the gas. Note that
.

Hint A.2.a The ideal gas law
Recall that the ideal gas law is given by

,

where      is pressure,    is volume,    is the number of moles of gas,                                is the gas constant, and
is the gas temperature in kelvins.

Part A.3      Calculate the final volume
Using the ideal gas law and the constants given, determine the final volume         of the gas. Note that
.

Hint A.3.a The ideal gas law
Recall that the ideal gas law is given by

,

where      is pressure,    is volume,    is the number of moles of gas,                                is the gas constant, and
is the gas temperature in kelvins.

Part B
What is the change in internal energy         of the gas?

Hint B.1      Dependence of internal energy on temperature
One of the unique features of an ideal gas is that its internal energy is independent of volume or pressure. It depends only on
the temperature of the gas. This can be seen if we look at the internal energy       of one atom, which, from equipartition,
has internal energy given by                       , where                              is the Boltzmann constant and       is the

http://session.masteringphysics.com/myct                                                                                                       Page 9 of 27
MasteringPhysics                                                                                                                            11/19/08 6:13 PM

temperature in kelvins.

The 3/2 comes from the three translational degrees of freedom (x, y, and z). If the gas consists of diatomic molecules there
can also be two independent degrees of rotational freedom (                      ) and, if the gas is polyatomic, there can also
be two independent degrees of vibrational freedom from bending (                               ). At higher temperatures both the
diatomic and polyatomic gases can have additional modes due to longitudinal vibrations. (At low temperatures there is not
enough energy to excite these modes.)

Hint B.2     The molar heat capacities of

For        (which is a polyatomic gas) at low pressure the molar heat capacity at constant volume is
.

The molar heat capacity at constant pressure is                                   .

Hint B.3     Internal energy and heat capacity
The relation between the internal energy and the heat capacity of an ideal gas is given by the equation                          ,
where      is the differential energy,    is the number of moles,         is the heat capacity at constant volume, and        is the
differential temperature.

The the heat capacity at constant volume is used instead of that at constant pressure because of the first law of
thermodynamics, which states that                     .

The heat capacity basically means that the heat goes as                          (constant volume) or                    (constant
pressure). If the system is at constant volume then             and thus                              . Now, since the internal energy
for any process depends only on temperature, then its change in one type of process (constant volume) must be the same as
that in another type of process (constant pressure).

Part C
How much heat       was supplied to the gas?

Hint C.1     The first law of thermodynamics
The first law of thermodynamics states that                       . In the case we are dealing with, it can also be written as
, where        is the total change in internal energy,        is the total amount of heat put in, and    is the total
amount of work that the gas does on its surroundings. Since we know               from Part A and        from Part B, it should be
relatively straightforward to find    .

=

Part D
How much work         would have been done if the pressure had been 0.50               ?

Hint D.1     How work depends on pressure
Recall that this gas obeys the ideal gas law,

,

and that the equation for work for a system under constant pressure is given by

.

As a result, one can eliminate the dependence on pressure of the amount of work a gas does, as long as the pressure is kept
constant.

http://session.masteringphysics.com/myct                                                                                                        Page 10 of 27
MasteringPhysics                                                                                                                             11/19/08 6:13 PM

Expansion and Compression of a Gas
Description: Short conceptual problem on the internal energy and work associated with adiabatic expansions and
compressions. This problem is based on Young/Geller Conceptual Analysis 15.5.

Part A
An ideal gas expands through an adiabatic process. Which of the following statements is/are true?

Hint A.1       How to approach the problem
To determine the correct statement(s) you need to apply the first law of thermodynamics. Note that when a gas expands it
does work on its surroundings.

Hint A.2       First law of thermodynamics
When heat         is added to a system, some of this added energy goes to increase the internal energy of the system by an
amount         . The remaining energy leaves the system as the system does work          on its surroundings. Thus, we have

.

Since       and     may be positive, negative, or zero, we can also expect        to be positive, negative or zero, depending on
the process.

An adiabatic process is a thermodynamic process in which no heat exchange occurs.

Check all that apply.

ANSWER:               The work done by the gas is negative, and heat must be added to the system.
The work done by the gas is positive, and no heat exchange occurs.
The internal energy of the system has increased.
The internal energy of the system has decreased.

Part B
After the adiabatic expansion described in the previous part, the system undergoes a compression that brings it back to its
original state. Which of the following statements is/are true?

Hint B.1       Internal energy in cyclic processes
A process, or a sequence of processes, that brings the system back to its original state is called a cyclic process. In a cyclic
process the total internal energy change is zero.

Check all that apply.

ANSWER:               The total change in internal energy of the system after the entire process of expansion and compression
must be zero.
The total change in internal energy of the system after the entire process of expansion and compression
must be negative.
The total change in temperature of the system after the entire process of expansion and compression must
be positive.
The total work done by the system must equal the amount of heat exchanged during the entire process of
expansion and compression.

Fast vs. Slow Tire Pumping
Description: Compare pumping a tire using a fast (adiabatic) pump cycle versus a slow (isothermal) cycle. Which transfers
more air into the tire per pump cycle? Algebraic with numeric parts to get reinforce the conclusion.

Imagine the following design for a simple tire pump. The pump is filled with a volume           of air at atmospheric pressure        and
ambient temperature        . When you push the pump handle, the air is compressed to a new (smaller) volume           , raising its
pressure. A valve is then opened, allowing air to flow from the pump into the tire until the remaining air in the pump reaches the
pressure of the air in the tire, .

In this problem, you will consider whether you can get more air into the tire per pump cycle by pushing the pump handle quickly
or slowly.

We will make the following simplifying assumptions:

The pressure of the air in the tire, , does not change significantly as air flows into the tire from the pump.
The temperature of the air in the pump does not change significantly while air is flowing into the tire (i.e., this is an
isothermal process).
The air is mainly composed of diatomic molecules with               .

Part A
First imagine that you push the pump handle quickly, so that the compression of air in the pump occurs adiabatically. Find the

http://session.masteringphysics.com/myct                                                                                                         Page 11 of 27
MasteringPhysics                                                                                                                           11/19/08 6:13 PM

absolute temperature          of the air inside the pump after a rapid compression from volume       to volume     , assuming an
ambient temperature of         .

Hint A.1      Properties of an adiabatic process
For an adiabatic process the product             is constant. Using the ideal gas law, you can derive the equivalent condition that
the product            is constant throughout the adiabatic process.

=

Part B
Once the tire and pump pressures have equilibrated at          , what fraction    of the gas particles initially in the pump will have
ended up in the tire?

Hint B.1      How to approach the problem
Find an expression for the number of particles in the pump initially (          ) and another for the number in the pump after the
pump and tire have come into pressure equilibrium (             ). Use these to determine the fraction of the particles in the pump
that are transferred into the tire:

.

Part B.2      Find an expression for the number of particles in a gas
Using the ideal gas law, find an expression for the number of particles           in a gas at pressure , volume     , and
temperature     .

Use      for Boltzmann's constant.

=

Part B.3            Find

What is the ratio of the number of particles in the pump after it comes into pressure equilibrium with the tire to the number
of gas particles initially in the pump (before pushing the handle)?

Part B.3.a Find the initial number of particles in the pump
How many gas particles           were in the pump initially?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's constant                      .

=

Part B.3.b Find the final number of particles in the pump
How many gas particles           are in the pump after the pump and tire have equilibrated?

Recall that the final pressure is      and the temperature continues to be                  .

Express your answer in terms of quantities given in the problem introduction and Boltzmann's constant                      .

=

http://session.masteringphysics.com/myct                                                                                                       Page 12 of 27
MasteringPhysics                                                                                                                                       11/19/08 6:13 PM

=

Express the ratio in terms of         ,       ,       ,       , and     . The temperatures         and    should not appear in your answer.

=

Express the fraction in terms of           ,       ,       ,     , and     . The temperatures        and     should not appear in your answer.

=

Part C
Now imagine that you push the pump handle slowly, so that the compression of air in the pump occurs isothermally. Find the
absolute temperature   of the air inside the pump assuming an ambient temperature of .

Hint C.1      Properties of an isothermal process
Isothermal means "at constant temperature."

Part D
Once the tire and pump pressures have equilibrated at                      , what fraction    of the gas particles initially in the pump will have
ended up in the tire?

Hint D.1      How to approach the problem
Find an expression for the number of particles in the pump initially (                      ) and another for the number in the pump after the
pump and tire have come into pressure equilibrium (                        ). Use these to determine the fraction of the particles in the pump
that are transferred into the tire:

.

Part D.2      Find an expression for the number of particles in a gas
Using the ideal gas law, find an expression for the number of particles                       in a gas at pressure , volume     , and
temperature     .

Use      for Boltzmann's constant.

=

Part D.3       Find

What is the ratio of the number of particles in the pump after it comes into pressure equilibrium with the tire to the number
of gas particles initially in the pump (before pushing the handle)?

Part D.3.a Find the initial number of particles in the pump
How many gas particles         were in the pump initially?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's constant                                  .

=

Part D.3.b Find the final number of particles in the pump

http://session.masteringphysics.com/myct                                                                                                                   Page 13 of 27
MasteringPhysics                                                                                                                                  11/19/08 6:13 PM

How many gas particles         are in the pump after the pump and tire have equilibrated?

Express your answer in terms of quantities given in the problem introduction and Boltzmann's constant                           .

=

Express the ratio in terms of       ,       ,       ,       , and     . The temperatures     and     should not appear in your answer.

=

Express the fraction in terms of         ,       ,       ,     , and     . The temperatures     and     should not appear in your answer.

=

Part E
Assume that               and                . Which method, fast pumping (adiabatic process) or slow pumping (isothermal
process) will put a larger amount of air into the tire per pump cycle?

What is the numerical value of      for the fast (adiabatic) process?

Part E.2     Isothermal process
What is the numerical value of      for the slow (isothermal) process?

slow pumping

So when you pump quickly, not only do you get more pump cycles per unit time, but you also put more air in the tire per
cycle (at least according to this simplified model). Of course you have to pump not only faster but also harder, since the
average pressure you will pump against will be higher. In fact this higher pressure is the main reason that more particles
are transferred to the tire each time.

Piston in Water Bath Conceptual Work-Energy Problem
Description: A piston containing ideal gas sits in a large water bath. Find the sign of the work done by the gas, heat transfer to
the gas, and change in internal energy of the gas for four different situations (conceptual).
Imagine a piston containing a sample of ideal gas in thermal equilibrium with a large water bath. Assume that the piston head is
perfectly free to move unless locked in place, and the walls of the piston readily allow the transfer of energy via heat unless
wrapped in insulation. The piston head is unlocked and the gas is in an equilibrium state.

For each of the actions described below, state whether the work                    done by the gas, the heat energy   transferred to the gas,
and the change in the internal energy                of the gas are positive (     ), negative ( ), or zero (0).

After each action the piston is reset to its initial equilibrium state.

Part A
Action: Lock the piston head in place. Hold the piston above a very hot flame.

Part A.1     Find the sign of the work done by the gas
With the piston head locked in place, will the work done by the gas be positive, negative, or zero?

Hint A.1.a Work done by a gas

http://session.masteringphysics.com/myct                                                                                                              Page 14 of 27
MasteringPhysics                                                                                                                           11/19/08 6:13 PM

When a gas expands, it does work on its surroundings. When a gas is compressed, its surroundings do work on it.
Considering the gas as the system of interest, when the gas does work, this work is considered positive, and when the work
is done on the gas, this work is considered negative.

Give the sign of     . Answer with +, -, or 0.

Part A.2     Find the sign of the heat transferred to the gas
With the piston held above a flame, will the heat transferred to the gas be positive, negative, or zero?

Hint A.2.a Heat energy transferred to a gas
Heat energy can either flow into or out of a gas sample. When the energy flows into the gas, it is considered positive heat;
when the energy flows out of the gas it is considered negative heat.

Give the sign of    . Answer with +, -, or 0.

Part A.3     Find the sign of the change in internal energy of the gas
Based on the signs of work        and heat    , what must be the sign of change in internal energy      ?

Hint A.3.a First law of thermodynamics
The first law of thermodynamics states that the change          in internal energy of a gas is equal to the heat      added to the
gas minus the work        done by the gas:

.

This relationship must hold for each of the actions described.

Give the sign of      . Answer with +, -, or 0.

The first law of thermodynamics states that the change in internal energy is positive. This means that the temperature of
the gas must increase. This makes sense since the piston is held over a hot flame.

Enter the signs of     ,   , and       . Use    ,   , or 0 separated by commas. For example, if         is positive,      is negative,
and       is zero, you would type +,-,0.

Part B
Action: Very slowly push the piston head down.

Part B.1     Find the sign of the work done by the gas
If the piston head is pushed down, will the work done by the gas be positive, negative, or zero?

Hint B.1.a Work done by a gas
When a gas expands, it does work on its surroundings. When a gas is compressed, its surroundings do work on it.
Considering the gas as the system of interest, when the gas does work this work is considered positive, and when the work
is done on the gas this work is considered negative.

Give the sign of     . Answer with +, -, or 0.

Part B.2     Find the sign of the change in internal energy of the gas
Since the piston is in constant thermal equilibrium with a large water bath, will the change in internal energy of the gas be
positive, negative, or zero?

Hint B.2.a Change in internal energy of a gas
The internal energy of a gas is directly proportional to its temperature. Therefore, if the gas temperature increases, its
change in internal energy is positive. If the temperature decreases, the change in internal energy of the gas is negative.

Give the sign of      . Answer with +, -, or 0.

http://session.masteringphysics.com/myct                                                                                                       Page 15 of 27
MasteringPhysics                                                                                                                              11/19/08 6:13 PM

Part B.3     Find the sign of the heat transferred to the gas
Based on the signs of       and       , what must be the sign of     ?

Hint B.3.a First law of thermodynamics
The first law of thermodynamics states that the change             in internal energy of a gas is equal to the heat      added to the
gas minus the work          done by the gas:

.

This relationship must hold for each of the actions described.

Give the sign of    . Answer with +, -, or 0.

The first law of thermodynamics states that the heat flow is negative (i.e., out of the gas). This makes sense because as
the gas is compressed, the energy flowing into the gas from the work done on it must go somewhere. Since the
temperature is held constant by the thermal bath, this energy will leave the gas as heat flow.

Enter the signs of       ,   , and      . Use    ,   , or 0 separated by commas. For example, if           is positive,      is negative,
and       is zero, you would type +,-,0.

Part C
Action: Lock the piston head in place. Plunge the piston into very cold water.

Part C.1     Find the sign of the work done by the gas
With the piston head locked in place, will the work done by the gas be positive, negative, or zero?
Give the sign of     . Answer with +, -, or 0.

Part C.2     Find the sign of the heat transferred to the gas
After the piston is plunged into cold water, will the heat transferred to the gas be positive, negative, or zero?
Give the sign of    . Answer with +, -, or 0.

Part C.3     Find the sign of the change in internal energy of the gas
Based on the signs of       and     , what must be the sign of       ?

Give the sign of        . Answer with +, -, or 0.

The first law of thermodynamics states that the change in internal energy is negative. This means that the temperature
of the gas will decrease. This makes sense since the piston is plunged into cold water.

Enter the signs of       ,   , and      . Use    ,   , or 0 separated by commas. For example, if           is positive,      is negative,
and       is zero, you would type +,-,0.

Part D
Action: Wrap the piston in insulation. Pull the piston head up.

Part D.1     Find the sign of the work done by the gas
If the piston head is pulled up, will the work done by the gas be positive, negative, or zero?
Give the sign of     . Answer with +, -, or 0.

http://session.masteringphysics.com/myct                                                                                                          Page 16 of 27
MasteringPhysics                                                                                                                       11/19/08 6:13 PM

Part D.2     Find the sign of the heat transferred to the gas
The piston is wrapped in insulation; will the heat transferred to the gas be positive, negative, or zero?
Give the sign of    . Answer with +, -, or 0.

Part D.3     Find the sign of the change in internal energy of the gas
Based on the signs of       and     , what must be the sign of     ?

Give the sign of        . Answer with +, -, or 0.

Enter the signs of       ,   , and      . Use    ,    , or 0 separated by commas. For example, if      is positive,   is negative,
and       is zero, you would type +,-,0.

pV Diagram for a Piston
Description: Determine the path on a pV diagram for an ideal gas undergoing various thermodynamic processes.
A container holds a sample of ideal gas in thermal equilibrium, as shown in the figure.
One end of the container is sealed with a piston whose head is
perfectly free to move, unless it is locked in place. The walls of the
container readily allow the transfer of energy via heat, unless the
piston is wrapped in insulation.

Refer to the pV diagram presented to answer the questions below. In each case, the piston head is initially unlocked and the gas
is in equilibrium at the pressure and volume indicated by point 0
on the diagram.

Part A
Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Lock the piston head in place, and hold the container above a very hot flame."

Part A.1     Understand the graph
The graph axes represent the pressure and volume of the gas sample. If either of these quantities changes, the position of the
point representing the gas sample must also change. Moreover, since temperature is proportional to the product of pressure
and volume, the temperature of the gas can also be determined from the graph.

A common way of denoting temperature on a pV graph is by including curves representing constant temperature, curves on
which the product of pressure and volume are constant. These are called isotherms (from the Greek iso, equal, and therme,
heat), and three isotherms are indicated on the graph. Of the three isotherms, which one designates the largest temperature,
, , or ?

http://session.masteringphysics.com/myct                                                                                                   Page 17 of 27
MasteringPhysics                                                                                                                     11/19/08 6:13 PM

Part A.2     Find the change in volume
With the piston head locked in place, will the volume of the gas increase, decrease, or stay the same when the piston is placed
above the flame?

The volume decreases.
The volume stays the same.

Part B
Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Immerse the container into a large water bath at the same temperature, and very slowly push the piston head further into the
container."

Part B.1     Find the change in volume
If the piston head is pushed further into the container, will the volume of the gas increase, decrease, or stay the same?

The volume will decrease.
The volume will stay the same.

Part B.2     Find the change in temperature
If the piston head is pushed into the container very slowly, and the container remains in contact with the large water bath,
will the temperature of the gas in the container increase, decrease, or stay the same?

The temperature will decrease.
The temperature will stay the same.

The piston does work on the gas, which, if the container were isolated, would increase the internal energy and therefore
the temperature of the gas. But this energy flows as heat out of the container into the large water bath, since the
container and the bath are in thermal equilibrium with each other. Since the water bath is large, its temperature does not
change significantly due to due the heat.

Part C
Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Lock the piston head in place and plunge the piston into water that is colder than the gas."

Part D
Starting from equilibrium at point 0, what point on the pV diagram will describe the ideal gas after the following process?

"Wrap the piston in insulation. Pull the piston head further out of the container."

Part D.1     Find the change in volume
If the piston head is pulled further out of the container, will the volume of the gas increase, decrease, or stay the same?

The volume will decrease.
The volume will stay the same.

Part D.2     Find the change in temperature
If the piston head is pulled further out of the container, and the piston is wrapped in insulation, will the temperature of the
gas increase, decrease, or stay the same?

http://session.masteringphysics.com/myct                                                                                                 Page 18 of 27
MasteringPhysics                                                                                                                          11/19/08 6:13 PM

The temperature will decrease.
The temperature will stay the same.

Simple Ways of Expanding
Description: Identify isothermal and isobaric curves on a p-V diagram. For an arbitrary curve, determine change in pressure,
temperature, and internal energy between endpoints.

A plot of pressure as a function of volume is known as a pV diagram. pV diagrams are often used in analyzing thermodynamic
processes. Consider an ideal gas that starts in state O, as indicated in the diagram. Your task is to describe how the gas proceeds
to one of four different states, along the different curves indicated.

Although there are an infinite number of such curves, several are
particularly simple because one quantity or another does not
change:

Isothermal:                   . The temperature does not change.
(The prefix "iso" means equal or alike.)
Isobaric:         . The pressure does not change. (A
barometer measures the pressure.)
Isochoric:         . The volume does not change. (This
process is infrequently used.)

The key idea in determining which of these processes is occurring from a pV plot is to recall that an ideal gas must obey the
ideal gas equation of state:             (where the constant      is the Boltzmann constant, which has the value
in SI units). Generally       , the number of gas particles, is held constant, so you can determine what
happens to       at various points along the curve on the pV diagram.

Note that in this problem, as is usually assumed, the processes happen slowly enough that the gas remains in equilibrium without
hot spots, without propagating pressure waves from a rapid change in volume, or without involving similar nonequilibrium
phenomena. Indeed, the word "adiabatic" is often used by scientists to describe a process that happens slowly and smoothly
without irreversible changes in the system.

Part A
What type of process does curve OA represent?

Hint A.1       What quanitity remains constant?
Curve OA is horizontal; the pressure of the system remains constant throughout this process. What is the name of a process
in which pressure remains constant?

isobaric
isochoric
isothermal

Part B
What type of process does curve OC represent?

Hint B.1       Relationship between pressure and volume
When                ,                   ; when             ,             . Evidently, pressure is proportional to the inverse of the
volume. What is the name of a process in which             is proportional to     ?

Hint B.2       Use the ideal gas law
Recall the ideal gas equation of state for a fixed amount of gas:                 , where    is some constant. Solving this equation
for   yields                 . If     is to be proportional to     , the temperature    must remain constant throughout the process.
What is the name of a process in which temperature remains constant?

isobaric
isochoric

http://session.masteringphysics.com/myct                                                                                                      Page 19 of 27
MasteringPhysics                                                                                                                    11/19/08 6:13 PM

isothermal

Detailed analysis of curve OB

The following questions refer to the process represented by curve OB, in which an ideal gas proceeds from state O to state B.

Part C
The pressure of the system in state B is __________ the pressure of the system in state O.

Part D
The work done by the system is __________.

Hint D.1     How to find the work done using the pV diagram
On a pV diagram, the work done during a particular process is represented by the area under the curve describing that
process. Is the area under curve OB greater than, less than, or equal to zero?

Part E
The temperature of the system in state B is __________ the temperature of the system in state O.

Hint E.1     Compare curve OB to an isothermal process
Does curve OB lie above or below a curve representing an isothermal (constant-temperature) process? Use this, along with
the ideal gas equation of state, to figure out how changes as the system proceeds from state O to state B.

Hint E.2     Computing the change in temperature mathematically
The ideal gas equation of state for a fixed amount of gas is           , where   is some constant. To find the sign of the
change in temperature, you could find                  at point B, then subtract the value of                at point O.
Although this method will not yield the actual change in temperature, it will give a number proportional to the change in
temperature (with the proper sign).

Part F
The internal energy of the system in state B is __________ the internal energy of the system in state O.

Hint F.1     Relationship between internal energy and temperature
The change in internal energy of an ideal gas is proportional to its change in temperature; the constant of proportionality is
the heat capacity at constant volume.

The First Law of Thermodynamics Reviewed
Description: Know meaning of first law and its terms
The first law of thermodynamics is usually written as

.

You need to look at the signs carefully; some other disciplines (chemistry comes to mind) may use other signs, which implies
other definitions of these quantities.

Part A
Which physical law underlies the first law of thermodynamics?

entropy always increases
conservation of energy
conservation of work

Part B

http://session.masteringphysics.com/myct                                                                                                Page 20 of 27
MasteringPhysics                                                                                                                     11/19/08 6:13 PM

The quantity      is the __________ that is added to the system.

work
entropy
internal energy

Part C
The quantity       in the first law refers to the __________.

Hint C.1      A wording convention
If the system expands, then it is said that work was done by the system. Conversely, if the system is compressed, then it is
said that work was done on the system.

ANSWER:            work done on the system
work done by the system
change of energy in the system
change of volume in the system

Part D
The quantity      in the first law of thermodynamics is the change in the __________ of the system.

Choose the correct option.

internal energy
heat energy
temperature

Part E
If heat is added to a system (        ) and no change in internal energy of the system occurs (          ), then the system must
do __________ work on the outside world.

positive

Two Closed Thermodynamic Cycles Conceptual Question
Description: Two closed thermodynamic cycles for an ideal gas are depicted on a pressure versus volume graph. Describe
change in internal energy, work done, and heat transferred to the gas. (conceptual)
Two closed thermodynamic cycles are illustrated in the figure.
The ideal gas sample can be processed clockwise or
counterclockwise through either cycle.

Part A
Imagine processing the gas clockwise through Cycle 1. Determine whether the change in internal energy of the gas in the entire
cycle is positive, negative, or zero.

Hint A.1      Closed thermodynamic cycles
A series of actions undertaken with a gas sample that result in the sample returning to its initial state form a closed
thermodynamic cycle. From steam engines to air conditioners, many machines employ closed thermodynamic cycles. Since
the cycle can be endlessly repeated, the precise starting point of the cycle is not necessary for its analysis.

Hint A.2      Change in internal energy in a closed cycle

http://session.masteringphysics.com/myct                                                                                                 Page 21 of 27
MasteringPhysics                                                                                                                       11/19/08 6:13 PM

The change in internal energy of a gas depends only on the initial and final states of the gas, not on the process that takes the
gas from one state to another. This fact allows you to determine the change in internal energy for any closed cycle.

Hint A.3     Initial and final state in a closed cycle
In any closed cycle, the initial and final states of the gas are the same. This is what allows the cycle to be repeated.

Choose the correct description of                 for Cycle 1.

zero
negative
cannot be determined

Part B
Imagine processing the gas clockwise through Cycle 1. Determine whether the work done by the gas in the entire cycle is
positive, negative, or zero.

Hint B.1     Work in a closed cycle
Work depends on the precise path from initial state to final state. To determine the work done in a closed cycle, you must
sum the work done in each portion of the cycle.

Part B.2     Compare the work done in different parts of a closed cycle
The figure shows a        plot depicting a closed thermodynamic cycle. Is the magnitude         of the work done by the gas in
portion 2 greater than, less than, or the same as the magnitude       of the work done in portion 4 of the cycle? The parts of
the cycle are labeled clockwise beginning from the left side of the cycle.

Hint B.2.a Magnitude of work done
The magnitude of the work done by the gas during a portion of the cycle is equal to the area under the curve on the pV-
diagram for that portion.

Choose the correct comparison between the value of              in portion 2 and that of      in portion 4 of the cycle.

Work done during expansion of the gas is positive, while
work done during contraction of the gas is negative. When
Cycle 1 is followed clockwise, the gas expands during
portion 2 and contracts during portion 4. Thus, the net
work done by the gas in following cycle 1 clockwise is
.

The magnitude of the work done in following a closed
cycle will always be the area enclosed by the cycle on a
pV-diagram. The sign of the work will be determined by
whether the larger magnitude of work is done during
expansion or contraction.

Choose the correct description of                for Cycle 1.

zero

http://session.masteringphysics.com/myct                                                                                                   Page 22 of 27
MasteringPhysics                                                                                                                  11/19/08 6:13 PM

negative
cannot be determined

Part C
Imagine processing the gas clockwise through Cycle 1. Determine whether the heat energy transferred to the gas in the entire
cycle is positive, negative, or zero.

Hint C.1     Heat transfer in a closed cycle
The easiest way to determine the heat transferred in a closed cycle is by applying the first law of thermodynamics to the
cycle.

Hint C.2     Simplifying the first law of thermodynamics
All closed cycles involve no change in internal energy. Therefore,

becomes

.

Based on the work done in each cycle, you should be able to correctly determine the heat transfer.

Choose the correct description of                 for Cycle 1.

zero
negative
cannot be determined

Part D
Imagine processing the gas clockwise through Cycle 1 and then counterclockwise through Cycle 1. Compare these two
processes on the basis of the work done by the gas in the entire cycle.

Part D.1     Find the magnitude of the work done
How does the magnitude of the work done in going around the cycle clockwise compare to the magnitude in traversing the
cycle counterclockwise?
Compare               to                     .

You know that the magnitude of the work done by traversing the cycle is the same regardless of direction of traversal.
Consider which direction has the gas doing positive work and which direction has the gas doing negative work.

Choose the correct comparison symbol.

ANSWER:                     for Cycle 1   >                        for Cycle 1.

Part E
Imagine processing the gas clockwise through Cycle 1 and then counterclockwise through Cycle 1. Compare these two
processes on the basis of the heat energy transferred to the gas in the entire cycle.

Hint E.1     Heat transfer in a closed cycle
The easiest way to determine the heat transferred in a closed cycle is by applying the first law of thermodynamics to the
cycle. Recall that                , where       is the change in internal energy,    is the work done by the gas, and is the
heat transferred into the gas.

Choose the correct comparison symbol.

ANSWER:                     for Cycle 1   >                       for Cycle 1.

Part F
Imagine processing the gas clockwise through Cycle 1 and then clockwise through Cycle 2. Compare these two processes on
the basis of the work done by the gas in the entire cycle.
Choose the correct comparison symbol.

ANSWER:                     for Cycle 1   =                  for Cycle 2

http://session.masteringphysics.com/myct                                                                                              Page 23 of 27
MasteringPhysics                                                                                                                   11/19/08 6:13 PM

Part G
Imagine processing the gas clockwise through Cycle 1 and then clockwise through Cycle 2. Compare these two processes on
the basis of the heat energy transferred to the gas in the entire cycle.

Part G.1     Find the magnitude of work done
How does the magnitude of the work done in Cycle 1 clockwise compare to the magnitude in Cycle 2 clockwise?
Compare                in Cycle 1 to                in Cycle 2.

ANSWER:                     in Cycle 1    =                   in Cycle 2.

Choose the correct comparison symbol.

ANSWER:                     for Cycle 1   =                   for Cycle 2

Work Integral in the pV Plane
Description: Given a rectangular pV curve for an ideal gas, find work done for each leg of the curve, then net work for one
cycle. Find the temperature at one corner of the cycle in terms of temperature at the opposite corner.
The diagram shows the pressure and volume of an ideal gas during one cycle of an engine.
As the gas proceeds from state 1 to state 2, it is heated at constant
pressure. It is then cooled at constant volume, until it reaches state
3. The gas is then cooled at constant pressure to state 4. Finally, the
gas is heated at constant volume until it returns to state 1.

Part A
Find      , the work done by the gas as it expands from state 1 to state 2.

Part A.1     Relating work, pressure, and volume
If the pressure of a gas is , and its infinitesimal change in volume is      , what is   , the infinitesimal work done by the
gas?

Part A.2     Doing the integration
To find the work done by the gas as it expands from state 1 to state 2, multiply the gas pressure by the change in volume
between states 1 and 2. When you do this, what value should you use for the pressure ?

Express the work done in terms of          and       .

Part B
Find      , the work done by the gas as it cools from state 2 to state 3.

Hint B.1     Volume of the gas
Note that the volume of the gas remains constant during this part of the cycle.

http://session.masteringphysics.com/myct                                                                                               Page 24 of 27
MasteringPhysics                                                                                                                              11/19/08 6:13 PM

Part C
Find        , the work done by the gas as it is compressed from state 3 to state 4.

Part D
[ Print ]
Find        , the work done by the gas as it is heated from state 4 to state 1.

Part E
What is         , the total work done by the gas during one cycle?

Notice that the net work done by the gas during this cycle is equal to the area of the rectangle that appears in the pV
(pressure-volume) diagram. (The width of this rectangle is      , and its height is     .) In other words, the net work done
by this gas is equal to the net area under its pV curve. For a cycle, this is equivalent to the (signed) area enclosed by the
cycle.

Part F
When the gas is in state 1, its temperature is         . Find the temperature     of the gas when it is in state 3. (Remember, this is an
ideal gas.)

Part F.1       Equation of state in terms of          and

If an ideal gas is held at a fixed number of moles, then its equation of state is                  , where   is some constant. For the
gas given, find an expression for        in terms of given quantities.

Express      in terms of     ,   , and      .

=

Express       in terms of     .

=

Isobaric Expansion
Description: For an isobaric expansion, given initial and final volume along with initial pressure and temperature, find final
temperature, total work done, and total heat absorbed. (uses applet)

This applet shows a gas undergoing isobaric compression and expansion. It should help you to see the qualitative behavior of

A monatomic ideal gas composed of             atoms, originally with volume        and at pressure      and temperature     , expands at
constant pressure to five times its initial volume (                  ).

Use       for Boltzmann's constant.

Part A
What is the final temperature          of the gas after it has expanded to five times its initial volume?

Part A.1       Identify the ideal gas law
Which of the following equations expresses the ideal gas equation of state?

http://session.masteringphysics.com/myct                                                                                                          Page 25 of 27
MasteringPhysics                                                                                                                                   11/19/08 6:13 PM

Set up the ideal gas law for the initial and final situations. If you then divide the two equations, all of the terms should
cancel so that you have just       , , and some numerical constants.

Part B
How much work            is done by the gas during the expansion?

Hint B.1       General expression for work done by a gas
The amount of work done by a gas when its volume changes by an infinitesimal amount                             is given by            .
Integrate         to find the total work done, recalling that, in this problem, the pressure                is constant.

Part B.2       Evaluate the integral
What is the result of integrating           from            to             ?

=

Since you are dealing with an isobaric (constant pressure) process, the pressure can be pulled out of the work integral:

.

Express the work in terms of             and             , the initial pressure and volume of the gas.

Part C
How much heat         is absorbed by the gas during the expansion?

Part C.1       Relate         ,       and

If     is the heat absorbed by the gas,                 is the change in internal energy of the gas, and        is the amount of work done by
the gas, which of the following equations expresses the first law of thermodynamics?

You know       from Part B, so all you have to do is find                 and add it to       to get   .

Part C.2       Determine the initial internal energy
What is           , the initial internal energy of the gas?

Part C.2.a Specify the initial internal energy in terms of temperature
What is the initial internal energy of the gas in terms of the initial temperature                 ?

=

http://session.masteringphysics.com/myct                                                                                                               Page 26 of 27
MasteringPhysics                                                                                                                      11/19/08 6:13 PM

Express the initial internal energy in terms of          and          .

=

Part C.3    Determine the final internal energy
What is       , the final internal energy of the gas?

Part C.3.a Specify the final internal energy in terms of temperature
What is the final internal energy of the gas in terms of the temperature     ?

=

Express the final internal energy in terms of           and       .

=

Part C.4    Determine the change in internal energy
Find                           , the net change in the gas's internal energy during the isobaric expansion. Use the previous two
hints if you need help finding        or         .

Express the total heat absorbed in terms of             and    .

Summary      1 of 13 items complete (7.69% avg. score)
4 of 44 points

http://session.masteringphysics.com/myct                                                                                                  Page 27 of 27

```
To top