CW01-1
Exercise 1.1 Kinetic Theory of Gases
A gas is considered to be isotropic (equal in all directions) as part of the assumptions
intrinsic to the KTG. It can be shown that for an isotropic distribution, the flux of particles
(number of particles passing through a surface per unit area per unit time) in one direction
is
F =
1
4
ρhci
where ρ is the number density (number of particles per unit volume) and hci is the mean
speed of the particles. (This seems like a pretty trivial relation, except for that pesky factor
of 1/4, which requires a bit of thought). This relation can be used to calculate the frequency
of collisions that an ideal gas makes with the walls of its container.
Consider n moles of ideal gas of molecular mass m at a temperature T confined in a cubical
box of volume V . Define f as the average number of collisions per second with the gas
molecules with the walls of the container.
i How does this frequency scale with temperature for a given gas if the volume and moles are
constant?
ii How does this frequency scale with volume for a given gas if the temperature and moles are
constant?
iii How does this frequency scale with molecular mass if the temperature, volume, and moles
are fixed?
iv Derive an expression for f in terms of n, m, T, and V . Is this consistent with the above
observations?
v What would be different about your expression in item iv if the container were not cubical?
vi Calculate the collision frequency f in seconds−1
for Argon gas at STP in a 22.41397 L
cubical container.
Exercise 1.2 Intensive Properties of a Gas
Intensive properties are typically directly related to the microscopic behavior (molecular
motion) of the system. For a gas, the pressure, mass density, and root mean square speed
of the molecules are all intensive properties, and thus could be related. This problem asks
you to prove that they are.
i Derive a symbolic relationship between the pressure, p, the mass density ρm and the root
mean square speed, crms for an Ideal Gas
ii Consider an Ideal Gas with a mass density of 17.86 g/L at a pressure of 10.00 atm. What
is the root mean square speed of the molecules in the gas, in m/s?
Exercise 1.3 The Maxwell-Boltzmann Distribution
In the notes, a discussion of the Maxwell-Boltzmann distribution for molecular speeds was
made, and this was used to create an energy distribution. In many ways the energy of a
molecule is a much more ‘chemically significant’ quantity than speed. Use the expression
for the MB energy distribution, eq. 1.6.8, to derive the following quantities in analogy to
what was done in eq. 1.6.6 for molecular speeds.
i hi
ii p
h
2i ≡ rms
iii mp
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CW01-2
Exercise 2.1 The van der Waals Equation of State (vdW EOS)
Steam (gaseous water) has been used as the working substance in industrial-scale engines
for well over a century, and is still used in that capacity in modern nuclear power plants
(amazing!). Steam is not a very ideal gas, as one might expect. If steam is treated as a van
der Waals gas, its ‘best fit’ parameters are:
a = 5.464 L2
·atm·mol−2
b = 0.03049 L·mol−1
Measurement of the mass density of steam at 776.4 K and 327.6 atm places it at 133.2 g/L.
i What is the measured compression factor, Z, for steam using the experimental mass density
at 776.4 K and 327.6 atm?
ii What is the compression factor, Z, for steam computed using the vdW EOS and the above
a and b parameters at 776.4 K and 327.6 atm?
iii What is the percent error of the vdW EOS prediction of Z?
Exercise 2.2 Critical Behavior
Consider a gas that obeys the following Equation of State (EOS)
p =
RT
V
−
B
V
2 +
C
V
3
where B and C are positive empirical constants peculiar to a given gas.
i Explain and justify the condensability of this gas.
ii Determine the critical parameters, Tc, pc and V c of the gas in terms of B and C.
iii Determine the critical compression factor, Zc, for this gas.
Exercise 2.3 Isobaric Thermal Expansivity
The thermal expansion of a material may be characterized by the following intensive quantity
α ≡
1
V
∂V
∂T
p
This quantity is related, naturally, to the equation of state of the material. Consider a gas
that obeys the following EOS
p =
RT
V
+
(a + bT)
V
2
where a and b are empirical constants peculiar to a given gas.
i Determine α for this gas. (Hint: The EOS involves state variables, which must have exact
differentials.)
ii Is it possible for the quantity, α, to be negative? Justify your answer.
— End of CW01 —
CHM4411 pjbrucat 2022 page 2
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