Download Work Done by an Expanding Gas

Work Done by an Expanding Gas
Learning Goal: To derive the expression for the work done by an expanding gas,
to understand how it follows from the expression
, and
for mechanical work.
Especially from the historically important perspective of making engines to convert heat energy into
work, the work in thermodynamics is defined as the work done by the system on the exterior world,
and not vice versa as is done in the rest of classical mechanics. In classical mechanics, one always
considers the work done on a system by the outside world. Rarely does one think about the work done
by the system. Suppose you push a large block with a certain force of magnitude over some distance.
You have done work on the block; hence the energy of the block should increase. According to
Newton's 3rd law, the block exerts the same magnitude of force , but in the opposite direction (i.e.,
directed back at you). Hence, the work done by the block (on you) is negative, since the direction of
motion opposes the direction of the force. In summary, you have to be careful about the sign of the
work: the same situation gives opposite signs of the work depending on whether our perspective is
classical mechanics or thermodynamics.
In thermodynamics, one often deals with liquids and gases that exert forces on their containers (i.e., the
fluids exert pressure over an area). If the container changes volume, then this force acts through a
distance and hence does work.
For a steam engine, the example pictured
here, the "container" is a cylinder whose
volume changes as the piston slides in or
out. Suppose a gas is confined within the
cylinder. The pressure of the gas is , and
the area of the cylinder is . Consider the
work done as the gas expands, pushing the
piston to the right. Call the infinitesimal
distance the piston moves
.
A. What force does the gas exert
on the piston? (Note that the
positive x axis is to the right in the
figure.)
Express the force in terms of ,
, and any constants,
= p*A
B. If the piston moves a distance
, what is
, the work done by the gas?
Express the work done by the gas in terms of given quantities.
= p*A*dx
C. What is
, the increase in volume of the gas?
Express the differential increase in terms of
and other given quantities.
= A*dx
D.
Now find the work done by the gas in terms of the thermodynamic variables.
Express the differential work
pressure , temperature
in terms of thermodynamic variables such as the gas's
, volume
, and its change in volume
.
= p*dV
E. Suppose that the gas expands from
done by the gas?
Express the work in terms of
,
to
at constant pressure
, and
. How much work
is
.
= p_0*(V_1-V_0)
F.
Is the work you just computed positive or negative?
positive
negative
G. Assume now that the diameter of the piston is reduced by a factor of 2. What is the amount of
work
done by a gas of pressure
Express your answer in terms of
in expanding from
,
,
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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.
A. Find
, the work done by the
gas as it expands from state 1 to
state 2.
Express the work done in terms
?
, and simple numerical factors.
= p_0*(V_1-V_0)
Work Integral in the pV Plane
to
of
and
.
= 9*p_0*V_0
B. Find
, the work done by the gas as it cools from state 2 to state 3.
Express your answer in terms of
and
.
= 0
C. Find
, the work done by the gas as it is compressed from state 3 to state 4.
Express your answer in terms of
and
.
= -3*p_0*V_0
D. Find
, the work done by the gas as it is heated from state 4 to state 1.
Express your answer in terms of
and
.
= 0
E. What is
, the total work done by the gas during one cycle?
Express your answer in terms of
and
.
= 6*p_0*V_0
F.
When the gas is in state 1, its temperature is
in state 3. (Remember, this is an ideal gas.)
Express
in terms of
. Find the temperature
of the gas when it is
.
= (4/3)*T_1
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Various Gas Expansions: pV Plots and Work
An ideal monatomic gas is contained in a cylinder with a movable piston so that the gas can do work
on the outside world, and heat can be added or removed as necessary. The
figure
shows various paths that the gas
might take in expanding from an initial state whose pressure, volume, and temperature are
respectively. The gas expands to a state with final volume
,
, and
. For some answers it will be
convenient to generalize your results by using the variable
final to initial volumes (equal to 4 for the expansions shown in the figure.)
, which is the ratio of
The figure shows several possible paths of the system in the pV plane. Although there are an infinite
number of paths possible, several of those shown are special because one of their state variables
remains constant during the expansion. These have the following names:



Adiabiatic: No heat is added or removed during the expansion.
Isobaric: The pressure remains constant during the expansion.
Isothermal: The temperature remains constant during the expansion.
A. Which of the curves in the figure represents an isobaric process?
A
B
C
D
B. What happens to the temperature of the gas during an isobaric expansion?
Temperature increases.
Temperature remains constant.
Temperature decreases.
C. Which of the curves in the figure represents an isothermal process?
A
B
C
D
D. Graphically, the work along any path in the pV plot ____________.
is the area to the left of the curve from
is the area under the curve from
to
requires knowledge of the temperature
to
E. Calculate
Express
, the work done by the gas as it expands along path A from
in terms of
,
, and
to
.
.
= p_0* V_0 * (R_v - 1)
F.
Calculate the work
Express
in terms of
done by the gas during the isothermal expansion.
,
, and
.
= p_0*V_0*ln(R_v)
G. Which of the curves shown represents an adiabatic expansion?
A
B
C
D
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Fast vs. Slow Tire Pumping
Imagine the following design for a simple tire pump. The pump is filled with a volume
atmospheric pressure
and ambient temperature
of air at
. 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
.
A. First imagine that you push the pump handle quickly, so that the compression of air in the
pump occurs adiabatically. Find the absolute temperature
rapid compression from volume
Express your answer in terms of
T_a*(V_i/V_f)^(gamma-1)
or
=
T_a*(V_i/V_f)^(0.40)
to volume
,
,
of the air inside the pump after a
, assuming an ambient temperature of
, and
.
B.
Once the tire and pump pressures have equilibrated at , what fraction
particles initially in the pump will have ended up in the tire?
Express the fraction in terms of ,
should not appear in your answer.
,
,
of the gas
, and . The temperatures
and
1-(p_t/p_a)*(V_f/V_i)^gamma
or
=
1-(p_t/p_a)*(V_f/V_i)^(1.40)
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
assuming an ambient temperature of
of the air inside the pump
.
= T_a
D.
Once the tire and pump pressures have equilibrated at , what fraction
particles initially in the pump will have ended up in the tire?
Express the fraction in terms of ,
should not appear in your answer.
,
,
of the gas
, and . The temperatures
and
= 1-(p_t/p_a)*(V_f/V_i)
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?
fast pumping
slow pumping
Carnot Cycle
After Count Rumford (Benjamin Thompson) and James Prescott Joule had shown the equivalence of
mechanical energy and heat, it was natural that engineers believed it possible to make a "heat engine"
(e.g., a steam engine) that would convert heat completely into mechanical energy. Sadi Carnot
considered a hypothetical piston engine that contained moles of an ideal gas, showing first that it was
reversible, and most importantly that--regardless of the specific heat of the gas--it had limited
efficiency, defined as
, where
of heat put into the engine at a (high)
temperature . Furthermore, he showed
that the engine must necessarily put an
amount of heat
back into a heat
reservoir at a lower temperature
.
The cycle associated with a Carnot engine
is known as a Carnot cycle. A pV plot of
the Carnot cycle is shown in the figure. The
is the net work done by the engine and
is the quantity
working gas first expands isothermally from state A to state B, absorbing heat
temperature
. The gas then expands adiabatically until it reaches a temperature
gas is compressed isothermally to state D, giving off heat
compressed to state A, its original state.
from a reservoir at
, in state C. The
. Finally, the gas is adiabatically
A. Indicate whether the following statements are true or false:
1. For the gas to do positive work, the cycle must be traversed in a clockwise manner.
2. Positive heat is added to the gas as it proceeds from state C to state D.
3. The net work done by the gas is proportional to the area inside the closed curve.
4. The heat transferred as the gas proceeds from state B to state C is greater than the
heat transferred as the gas proceeds from state D to state A.
Use t for true or f for false. Separate your answers with commas.
t , f , t , f
B. Find the total work
done by the gas after it completes a single Carnot cycle.
Express the work in terms of any or all of the quantities
,
,
, and
.
Q_h-Q_c
or
=
Q_h*(1-T_c/T_h)
C. Suppose there are
respectively,
state B.
moles of the ideal gas, and the volumes of the gas in states A and B are,
and
. Find
, the heat absorbed by the gas as it expands from state A to
Express the heat absorbed by the gas in terms of
reservoir
, and the gas constant
,
,
, the termperature of the hot
.
= n*R*T_h*ln(V_B/V_A)
D. The volume of the gas in state C is
, and its volume in state D is
. Find
, the
magnitude of the heat that flows out of the gas as it proceeds from state C to state D.
Express your answer in terms of
and
,
,
,
(the temperature of the cold reservoir),
.
= n*R*T_c*ln(V_C/V_D)
E. Now, by considering the adiabatic processes (from B to C and from D to A), find the ratio
in terms of
and
.
= V_B/V_A
F.
Using your expressions for
and
find a simplified expression for
(found in Parts C and D), and your result from Part E,
.
No volume variables should appear in your expression, nor should any constants (e.g.,
or
).
= T_c/T_h
G. The efficiency of any engine is, by definition,
. Carnot proved that no engine can
have an efficiency greater than that of a Carnot engine. Find the efficiency
of a Carnot
engine.
Express the efficiency in terms of
and
.
= 1-T_c/T_h
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Isothermal Engine
Consider a thermal engine filled with
cyclic transformation, as
molecules of a monatomic ideal gas. The gas undergoes a
shown.
Processes AB
and CD are isothermal (constant temperature), and processes BC and DA are isochoric (constant
volume).
The quantities
has pressure
given.
,
,
, and
and volume
are defined in the figure. For example, when the gas is in state A, it
; when the gas is in state B, it has volume
, but the pressure is not
Work done by the gas
A. Find
, the work done by the gas during process AB.
Express the work in terms of
,
,
, and/or
.
= p_2*V_1*ln(V_2/V_1)
B. Find
, the work done by the gas during process BC.
Express your answer in terms of
,
,
, and/or
.
= 0
C. Find
, the work done by the gas during process CD.
Express your answer in terms of
,
,
, and/or
.
= -p_1*V_2*ln(V_2/V_1)
D. Find
, the work done by the gas during process DA.
Express your answer in terms of
,
,
, and/or
.
= 0
The internal energy
Now, you will investigate the change in the gas's internal energy during each of the four segments of its
cycle. You may find it helpful to recall that, for a monatomic ideal gas,
the heat capacity of the gas at constant volume.
A. What is the change in internal energy
during the process AB?
Express the change in internal energy in terms of
= 0
, where
,
,
, and/or
.
is
B. What is the change in internal energy
during the process BC?
Express the change in internal energy in terms of
,
,
, and/or
.
= (3/2)*(p_1*V_2-p_2*V_1)
C. What is the change in internal energy
Express your answer in terms of
during the process CD?
,
,
, and/or
.
= 0
D. What is the change in internal energy
Express your answer in terms of
during the process DA?
,
,
, and/or
.
= -(3/2)*(p_1*V_2-p_2*V_1)
E. What is the total change in the gas's internal energy after it has completed a cycle?
= 0
Heat
Finally, consider the heat absorbed by the gas during its cycle. Define
to be heat absorbed by the
gas in process AB, etc. If a process is exothermic (i.e., heat is given off by the gas during the process),
then the quantity of heat absorbed will be negative.
A. Indicate the processes in this cycle during which heat is flowing into the gas. That is, indicate
the segments where
is positive.
DA only
CD and DA
DA and AB
AB and BC
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Heat Pumps and Refrigerators
Learning Goal: To understand that a heat engine run backward is a heat pump that can be used as a
refrigerator.
By now you should be familiar with heat engines--devices, theoretical or actual, designed to convert
heat into work. You should understand the following:
1.
Heat engines must be cyclical; that is, they must return to their original state some time after
having absorbed some heat and done some work).
2.
Heat engines cannot convert heat into work without generating some waste heat in the
process.
The second characteristic is a rigorous result even for a perfect engine and follows from
thermodynamics. A perfect heat engine is reversible, another result of the laws of thermodynamics.
If a heat engine is run backward (i.e., with every input and output reversed), it becomes a heat pump (as
pictured schematically
). Work
must be
put into a heat pump, and it then pumps heat from a colder temperature to a hotter temperature ,
that is, against the usual direction of heat flow (which explains why it is called a "heat pump"). The big
surprise is that the laws of thermodynamics seem to allow more than 100% efficiency: The heat
coming out the hot side
of a heat pump or the heat going in to the cold side
of a refrigerator is
more than the work put in; in fact it can be many times as large. For this reason, the ratio of the heat to
the work in heat pumps and refrigerators is called the coefficient of performance. In a refrigerator, this
is the ratio of heat removed from the cold side
to work put in:
.
In a heat pump the coefficient of performance is the ratio of heat exiting the hot side
in:
to the work put
.
A. What is the relationship of
Express
in terms of
to the usual thermodynamic definition of work
and other quantities given in the introduction.
= -W
B. Find
Express
, the heat pumped out by the ideal heat pump.
in terms of
= W_in + Q_c
and
.
?
C. A heat pump is used to heat a house in winter; the inside radiators are at
and the outside
heat exchanger is at . If it is a perfect (e.g., Carnot cycle) heat pump, what is
coefficient of performance?
Give your answer in terms of
and
, its
.
= T_h/(T_h - T_c)
D. The heat pump is designed to move heat. This is only possible if certain relationships between
the heats and temperatures at the hot and cold sides hold true. Indicate the statement that must
apply for the heat pump to work.
and
.
and
.
and
.
and
.
E. Assume that you heat your home with a heat pump whose heat exchanger is at
, and
which maintains the baseboard radiators at
. If it would cost $1000 to heat the
house for one winter with ideal electric heaters (which have a coefficient of performance of 1),
how much would it cost if the actual coefficient of performance of the heat pump were 75% of
that allowed by thermodynamics?
Express the cost in dollars.
Cost = 187.5 dollars
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