MASENO UNIVERSITY
DEPARTMENT OF PHYSICS & MATERIAL SCIENCE
SPH 302 : THERMODYNAMICS
(1) Consider a gas in a container of volume V at a temperature T . Use the fact that the velocity distribution
is isotropic to show that the number of particles dN striking a surface of area dA during a time dt is
1
dN = nv̄ dAdt where n = N/V and v̄ is the average velocity of the gas particles. If the velocity distribution
4
is of the Maxwell form,
m 3/2
2
f (v) = 4π
v 2 e−mv /2kT
2πkT
where m is the mass of the particles and k is Boltzmann’s constant, show that the flux J = dN/dAdt is
given by
1/2
1
8kT
P
J= n
=√
4
πm
2πmkT
where the last expression is obtained by regarding the gas as ideal with pressure P.
(2) The equation of state for a mole of a van der Waals fluid is given by
α P + 2 (V − β) = RT
V
where P is the pressure, V is the volume per mole, T is the temperature, while α and β are arbitrary
constants. Using this information, obtain the following quantities and the verify that they satisfy the
results for an ideal gas in the limit α = β = 0:
1
∂V
a) Isothermal compressibility, κT = −
V
∂P T
1
∂V
b) Isobaric coefficient of thermal expansion, α =
V
∂T P
c) Molar heat capacity difference, (Cp − CV )/N
α
∂P
=
. Is this true? Explain.
(3) In problem (2) above, show that
∂T V
κT
(4) Given that after a combustion process in a cylinder, the pressure is 1200 kPa and the temperature is 350°C.
The gases are expanded to 140 kPa with a reversible adiabatic process. Calculate the work done by the
gases, assuming they can be approximated as air with constant specific heats.
(5) Given that the heat capacity at constant volume of a substance can be represented empirically by an
equation of the form CV = α + βT + γT 2 , where α, β, and γ are constants, find the change in internal
energy (∆U ) and the change in entropy (∆S) when the temperature changes from Ta to Tb at constant
volume.
(6) Establish thermodynamically the formulae
∂P
V
=S
∂T µ
and
V
∂P
∂µ
=N
T
Express the pressure P of an ideal classical gas in terms of the variables µ and T , and verify the above
formulae.
(7) Show that
Cv = −T
∂V
∂T
S
∂P
∂T
and
Cp = T
V
∂P
∂T
S
∂V
∂T
P
(8) Starting with the relation dH = T dS + V dP , show that the slope of a constant-pressure line on an H − S
diagram
a) is constant in the saturation region, and
b) increases with temperature in the superheated region.
(9) Using the cyclic relation and the first Maxwell relation, derive the other three Maxwell relations.
(10) Show that the difference between the reversible steady-flow work and reversible moving boundary work is
equal to the flow energy.
SPH 302 : Assignment Set One (20/05/2022)
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