Physical Chemistry Assignment 1 (25 points) (

Chemistry 360
Fall 2014
Dr. Jean M. Standard
September 3, 2014
Physical Chemistry Assignment 1
(25 points)
This assignment is due on Friday, September 12, 2014.
Part 1 (15 points): Ideal and Real Gas Isotherms
This problem demonstrates the use of equations of state for both ideal and real gases to understand isotherms,
critical behavior, and molar volumes. Recall that the equation of state for an ideal gas is given by
PV m = RT ,
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where V m is the molar volume. The most common equation of state that represents the behavior of real gases is the
van der Waals equation,
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"
$
P +
#
a
V m
2
%
' V m − b
&
( ) = RT .
where a and b are the van der Waals constants.
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a. Use a spreadsheet program such as Microsoft Excel to generate and plot isotherms of the ideal gas equation.
Remember that an isotherm is a plot of P (on the y-axis) vs. V m (on the x-axis) for a specific fixed temperature.
Plot five isotherms on the same graph, for T = 100, 300, 500, 700, and 900 K. For convenience, you might
want to set the x- and y-axis ranges in your calculation to run from 0 to 1 L/mol for the molar volume and 0 to
200 atm for the pressure. Choose reasonable increments in the x and y variables that enable you to generate
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fairly smooth curves. Include the plot when you turn in your assignment. Discuss whether or not critical
behavior is observed for the ideal gas.
b. Use a spreadsheet program such as Microsoft Excel to generate and plot isotherms of the van der Waals
equation for CO 2 . You will have to obtain the van der Waals constants a and b for CO 2 . Plot the same five
isotherms as you did in part (a). Make sure to use the same range for the x and y axes as given in part (a).
Include the plot when you turn in your assignment. Compare and contrast the van der Waals isotherms to the
ideal gas isotherms. In what regions are they similar In what regions are differences observed Discuss
whether or not critical behavior is observed in this case, and if so, indicate the region of the plot where critical
behavior occurs.
c. The final step is to calculate the critical point for CO 2 , assuming the gas obeys the van der Waals equation of
state. As discussed in class, the critical point appears as an inflection point on the pressure vs. molar volume
plot. At an inflection point, both the first and second derivatives are zero, which yields two equations,
# ∂P &
% (
$ ∂V m '
T
= 0 ;
# ∂ 2 P &
% (
2
$ ∂V m '
T
= 0 .
To find the pressure, volume, and temperature of the critical point (labeled P c , V c , and T c ), solve the van der
Waals equation for P. Then, evaluate the first and second partial derivatives given in the equations above.
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Setting those partial derivatives equal to zero yields two equations that can be solved for the temperature and
volume of the critical point. Substitution of these two values back into the van der Waals equation gives the
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critical pressure. Report the values of the critical constants that you obtained and show your work.

2
Part 2 (10 points): Rubber Band Thermodynamics
This problem deals with an extension of ideas that you learned in class about equations of state and work. It has to
do with the tension in a rubber band and the work required to stretch or compress the rubber band. The variables
involved are
F – the tension of the rubber band measured in Newtons
L – the length of the rubber band in meters
T – the temperature in degrees Kelvin.
The infinitesimal work dw for stretching a rubber band from L to L+dL is defined as
dw = F dL .
Note the similarity of this expression (with the exception of the sign) to the pressure-volume work discussed in
class. As a result of this similarity, it should not be too surprising that equations of state may be defined to describe
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the tension F of the rubber band in terms of its length and the temperature, F( L, T ) , just as we can write an
equation of state for the pressure P of a gas in terms of volume and temperature, P( V, T ).
A typical equation of state, known as the Guth-James equation, for
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an elastic substance such as a rubber band is
given by
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#
F( L,T ) = CT L − L 2 &
o
%
$ L o L 2 ( ,
'
where C and
L o are constants. The constant
L o is defined to be the length of the rubber band at zero tension.
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a. Derive an expression for the work done in stretching a rubber band from L 1 to L 2 under isothermal reversible
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conditions. Assume that the tension F is given by the equation of state shown above.
b. A certain sample of rubber has been determined to have C = 1.33×
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10
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−2 N/K and L o = 0.10 m. Using your
result from part (a), calculate the work in Joules for stretching the rubber sample from 0.2 to 1.0 m at 300 K.
c. Use a spreadsheet program such as Microsoft Excel to generate and plot isotherms of the Guth-James equation.
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In this case, you should create a plot of F (on the y-axis) vs. L (on the x-axis) for a series of fixed temperatures.
Plot five isotherms on the same graph, for T = 100, 300, 500, 700, and 900 K. Use the values of C and L o
given in part (b). Set the x- and y-axis ranges in your calculation to run from 0.1 to 0.5 m for the length and 0 to
70 N for the force, respectively. Choose reasonable increments in the x and y variables that enable you to
generate fairly smooth curves. Include the plot when you turn in your assignment.
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d. Using the values of C and L o given in part (b), calculate the length of a rubber band for T = 100, 300, 500, 700,
and 900 K under a constant tension of 10 Newtons. To obtain the length L, you will have to solve a cubic
equation - you may do this using a calculator. Based on your results, discuss whether or not the length of a
rubber band increases or decreases when the rubber band is heated while held under constant tension conditions.
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