Work Done During Reversible and Irreversible Expansions of an Ideal Gas
by Arthur Ferguson 2002Dept. of ChemistryWorcester State CollegeWorcester, MA 01602Translated and modified from the MathCAD document by Alex Grushow 2004©
Copyright 2004 by the Division of Chemical Education, Inc., American Chemical Society. All rights reserved.For classroom use by teachers, one copy per student in the class may be made free of charge. Write to JCEOnline, jceonline@chem.wisc.edu, for permission to place a document, free of charge, on a class Intranet.
Abstract
This exercise examines how the work associated with stepped irreversible isothermal expansions and compressions of an ideal gas changes as the number of steps is increased. It uses the graphing power of
Mathematica
tolead the user to discover that the work approaches an asymptote as the number of steps becomes very large andthat asymptote is the work for the reversible expansion or compression. It then challenges the user to recognizethat the work for the reversible expansion or compression represents a limiting value for the work for the corresponding irreversible processes.
Goal
The goal of this exercise is to help the user to discover in a clear, visual, concrete way1. that as the number of steps in an isothermal expansion or compression of an ideal gas increases and the size of each step decreases, the work for the expansion or compression approaches that for the corresponding reversible process, and by implication that the irreversible process approaches reversibility;2. that the work for the reversible expansion or compression represents a limiting value of the work for thecorresponding irreversible processes.
Performance Objectives
Isothermal_Expansion.nb
1
After completing this exercise the user should1. be able to describe how the work for stepped isothermal expansions and compressions of an ideal gas changeas the number of steps increases, including sketching graphs of the work vs. the number of steps and relatingthat graph to the work for the corresponding reversible expansion or compression;2. recognize that the work for the reversible expansion or compression is the limiting value of the work for thecorresponding irreversible expansions or compressions
.
Introduction
The work resulting from the expansion or compression of a pas is PV work, for which the equation is
Ÿ
V
initial
V
final

P
ext
„
V
(1)where
P
ext
is the pressure exerted by the surroundings on the system. By the definition of a reversible process, a reversibleexpansion is one that occurs so slowly that the system always remains in equilibrium, both internally and with the surroundings. When the system and surrounding are in equilibrium,
P
ext
= P of the system. Therefore, for a reversible expansion
Ÿ
V
initial
V
final

P
„
V
(2)What does this equation become for one mole of an ideal gas?For an irreversible expansion
P
ext
has to be used in the calculation of w. The irreversible expansions we will look at willinvolve a series of steps in which the gas starts out at equilibrium,
P
ext
is dropped instantaneously from one value to anew, lower one and is held constant at the new value while the gas expands until a new equilibrium is reached. Then
P
ext
is instantaneously dropped again to start the next step. For example, the expansion of a gas from a pressure of 10 atm to a pressure of 1 atm might happen in a series of 1atm steps, with
P
ext
first dropping instantaneously from 10 atm to 9 atmand holding there until the gas has expanded enough to come into equilibrium with it, then dropping to 8 atm, etc. (Thesudden drop in pressure at the beginning of each step destroys the equilibrium attained at the end of the last step, and it isthis destruction of equilibrium that makes the expansion irreversible.) Since
P
ext
is constant during eachof these steps, the value of w for each step as calculated from Equation 1 is simply 
P
ext
D
V, where
P
ext
is the value of
P
ext
for that step and
D
V is the change in volume during that step. The total work for the expansion is the sum of thevalues of w for the individual steps. In this exercise we will examine how the work for an irreversible stepped expansion of 1 mole of an ideal gas at 300 K from a pressure of 10 atm to a pressure of 1 atm changes as the number of steps is increased and will compare the work for the irreversible expansions to the work for a reversible expansion of the same gas between the same two pressures.
Setting up the initial parameters:
Isothermal_Expansion.nb
2
<<
Miscellaneous`Units`
<<
Graphics`Graphics`
<<
Graphics`MultipleListPlot`P
initial
=
10 Atmosphere;P
final
=
1 Atmosphere;T
=
300 Kelvin; R
=
0.08205Liter Atmosphere
Kelvin; V
initial
=
R T
P
initial
í
Liter; V
final
=
R T
P
final
Liter;
Establishing Arrays for P
ext,
V and w
Our approach will be to set up arrays for P
ext
, V and w. Each element of P
ext
will be the external pressure during a step inthe expansion and will be smaller than the previous one by an amount
int
gotten by dividing the range between P
initial
andP
final
by the number of steps the expansion will take. Each element of V will be the volume of the gas at the end of a step,when it is in equilibrium with the corresponding value of P
ext
. Each value of w will be the work for a step, which is equal toP
ext
for the step times the change in volume during the step, calculated as the difference between the volume at the end of the step and the volume at the end of the previous step. The total work will be the sum of the elements of the
w
array. Note that the values of the 0th elements of the P
ext
and V arrays are set manually and that the range value
i
starts at 1 rather than 0.
Steps
=
1;Pex
@
0
D
=
P
initial
; V
@
0
D
=
V
initial
Liter;int
=
P
initial
−
P
final
Steps;Do
@
Pex
@
i
D
=
Pex
@
i
−
1
D
−
int,
8
i, Steps
<D
;Do
A
V
@
i
D
=
R T
Pex
@
i
D
,
8
i, Steps
<E
;Do
@
w
@
i
D
= −
Pex
@
i
D
H
V
@
i
D
−
V
@
i
−
1
DL
,
8
i, Steps
<D
;totalwork
=
Convert
A ‚
i
=
1Steps
H
w
@
i
DL
, Joule
E
−
2244.7Joule
Isothermal_Expansion.nb
3
Exercise 1
In this exercise you will want to fill in the
X
and
Y
matrix below with at least 10 elements each (the first two are given for you).
X
=
8
1, 5
<
;Y
=
8
−
2244.7,
−
4277.22
<
;
Fill up the matrix elements by repeating the calculation above for
totalwork
with different values for
Steps
. Add thenumber of steps into the
X
matrix and
totalwork
into the
Y
matrix. Verify that the first two entries are correct. Repeat atleast 8 more times.
Make sure that your values of Steps cover a large range, with the largest value being at least 5000.
(The values 1, 2, 10, 25, 50, 100, 500, 1000, 5000, 10000 work well, but you may want to experiment with other values aswell.)
wlist
=
Table
@8
X
@@
i
DD
, Y
@@
i
DD<
,
8
i, Length
@
X
D<D
;zplot
=
ListPlot
@
wlist, FrameLabel
→
8
Number of Steps , Work
<
,RotateLabel
→
False, PlotStyle
→
8
Hue
@
0
D
, PointSize
@
0.010
D<
,Frame
→
True, TextStyle
→
8
FontFamily
→
Times , FontSize
→
12
<D
1234 NumberofSteps

4000

3500

3000

2500Work
a. What does the full graph above look like? Does it look useful?
Isothermal_Expansion.nb
4