1.6 Real Gases
1.20.Liquid nitrogen comes in large cylinders that require
special tank carts and hold 120 L of liquid at 77 K. Given the
density of liquid nitrogen of 0.840 g/cm^3 , use the van der
Waals equation to estimate the volume of the nitrogen gas af-
ter it evaporates at 77 K. (Hint:because Vshows up in two
places in the van der Waals equation, you will have to do an
iteration procedure to estimate V. Neglect the an^2 /V^2 term ini-
tially and calculate V; then substitute this into the an^2 /V^2 term,
evaluate the pressure term, resolve for V, and repeat until the
number doesn’t change. A programmable calculator or spread-
sheet program might be useful.)
1.21.Calculate the Boyle temperatures for carbon dioxide,
oxygen, and nitrogen using the van der Waals constants in
Table 1.6. How close do they come to the experimental val-
ues from Table 1.5?
1.22.Determine the expression for ( p/ V)Tfor a van der
Waals gas and for the virial equation in terms of volume.
1.23.What are the units of the virial coefficient C? of C?
1.24.Table 1.4 shows that the second virial coefficient Bfor
He is negative at low temperature, seems to maximize at a
little over 12.0 cm^3 /mol, and then decreases. Do you think it
will become negative again at higher temperatures? Why is it
decreasing?
1.25.Use Table 1.5 to list the gases from most ideal to least
ideal. What trend or trends are obvious from this list?
1.26.What is the van der Waals constant afor Ne in units of
barcm^6 /mol^2?
1.27.By definition, the compressibility of an ideal gas is 1. By
approximately what percentage does this change for hydro-
gen upon inclusion of the second virial coefficient term? How
about for water vapor? Give the conditions under which you
make this estimate.
1.28.The second virial coefficient Band the third virial coef-
ficient Cfor Ar are 0.021 L/mol and 0.0012 L^2 /mol^2 at 273
K, respectively. By what percentage does the compressibility
change when you include the third virial term?
1.29.Use the approximation (1 x)^1 1 xx^2
to determine an expression for Cin terms of the van der Waals
constants.
1.30.Why is nitrogen a good choice for the study of ideal gas
behavior around room temperature?
1.7 & 1.8 Partial Derivatives and Definitions
1.31.Write two other forms of the cyclic rule in equation
1.26, using the mnemonic in Figure 1.11.
1.32.Use Figure 1.11 to construct the cyclic rule equivalent
of ( p/ p)T. Does the answer make sense in light of the origi-
nal partial derivative?
1.33.What are the units for and ?
1.34.Why is it difficult to determine an analytic expression
for and for a van der Waals gas?
1.35.Show that (T/p) for an ideal gas.
1.36.Determine an expression for ( V/ T)p,nin terms of
and . Does the sign on the expression make sense in terms
of what you know happens to volume as temperature changes?
1.37.Density is defined as molar mass, M, divided by molar
volume:dM
VEvaluate ( d/ T)p,nfor an ideal gas in terms of M, V, and p.
1.38.Write the fraction /in a different form using the
cyclic rule of partial derivatives.(Note: The Symbolic Math Exercise problems at the end of
each chapter are more complex, and typically require addi-
tional tools like a symbolic math program—MathCad, Maple,
Mathematica—or a programmable calculator.)
1.39.Table 1.4 gives different values of the second virial co-
efficient Bfor different temperatures. Assuming standard pres-
sure of 1 bar, determine the molar volumes of He, Ne, and Ar
for the different temperatures. What does a graph of Vversus
Tlook like?
1.40.Using the van der Waals constants given in Table 1.6,
predict the molar volumes of (a)krypton, Kr; (b)ethane,
C 2 H 6 ; and (c)mercury, Hg, at 25°C and 1 bar pressure.
1.41.Use the ideal gas law to symbolically prove the cyclic
rule of partial derivatives.
1.42.Using your results from exercise 1.39, can you set up
the expressions to evaluate and for Ar?Exercises for Chapter 1 23Symbolic Math Exercises