Problem Set 1
Physics 7230
Spring 2007
Due: Friday February 9, 2007
1.
c=
The compressibility K, and mass density ρ of a fluid determine the speed of sound c in the fluid by
the relation
1
=
ρK
1
mnK
where n=N/V is the number density of the molecules in the fluid, m is the mass of one molecule and K
is the compressibility. The isothermal compressibility is defined by
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−1  ∂V 
1  ∂n   ∂n   ∂ 2 p 
KT =   =   =   =  2 
V  ∂p T n  ∂p T  ∂µ T  ∂µ T
Isaac Newton first calculated the speed of sound in air using the isothermal value of the
compressibility from Boyle’s law. Use the ideal gas law to show that the isothermal speed of sound in
a gas does not depend on the density of the gas and give a numerical value for the prediction of the
speed of sound in air at STP. Does this correspond to the actual speed of sound? The key point that
Newton did not understand was that for audible frequencies sound waves are not isothermal but rather
are nearly adiabatic. This led to a famous 20% “fudge” by Newton to give agreement between his
theory and the experimentally measured speed of sound. The appropriate compressibility is the one at
constant entropy since there is negligible heat flow within the wave there is not enough time per period
for heat to flow across a wavelength to keep the gas at one temperature. The temperature of the gas
fluctuates in phase with the pressure variations due to adiabatic compression and expansion.
KS =
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−1  ∂V 
 
V  ∂p  S
Show that the ratio of the adiabatic compressibility to the isothermal compressibility is related to the
ratio of the constant pressure and constant volume heat capacities. Use this to show that KS is always
less that the isothermal compressibility. The constant volume specific heat (heat capacity per particle)
for air is (5/2)kB. Use this to predict the experimentally observed speed of sound at STP.
2. The Helmholtz free energy of a model of a gas that includes molecular repulsion is
 V − Nv 
F(T,V,N) = −Nk B T ln 3 1  − Nk B T
 NΛ (T) 
where v1 is a microscopic parameter representing the volume of one particle and
Λ(T) =
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h
2πmk B T
is the thermal deBroglie wavelength that depends only on temperature.
a) Calculate the pressure, isothermal compressibility, entropy, chemical potential, and constant
volume and constant pressure heat capacities of the gas. Show the properties of the model
approach that of an ideal gas in the limit of low density.
b) Calculate the Gibbs free energy G=F+pV and show that G can be written explicitly as a function
of T, p, and N. Show that G=N µ(T,p) and
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 ∂G 
  =V
 ∂p T ,N
c)
Show that the entropy, chemical potential and heat capacities are the same as for the Helmholtz
free energy.
d) Show the Maxwell relations
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 ∂S 
 ∂p 
  =   and
 ∂V T ,N  ∂T V ,N
 ∂S 
 ∂V 
  = − 
 ∂T  p,N
 ∂p T ,N
are satisfied for this particular model system.
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Finally use this model to approximate the properties of a typical gas. What pressure would be required
to give a 10% increase in the speed of sound in air at room temperature?
3. The Helmholtz free energy of a gas of N atoms of helium in container of volume V is
 V 
Fbulk (T,V,N) = −Nk B T ln 3  − Nk B T
 NΛ (T) 
h
where Λ(T) =
is the thermal deBroglie wavelength. Helium atoms are attracted to the walls
2πmk B T
of the container by van der Waals forces. Some of them stick to the walls with binding energy is ε>0 but
are free to move on the walls like a two-dimensional ideal gas. The Helmholtz free energy of the helium
bound to the walls is
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 A 
Fwalls (T, A,N) = −Nε − Nk B T ln 2  − Nk B T
 NΛ (T) 
Show that if the walls and bulk are in equilibrium then the chemical potentials for those two systems must
be the same. Determine the equilibrium area density of helium on walls as a function of temperature and
pressure of the bulk gas. Use some estimate of the van der Waals binding energy and analyze the problem
of creating an ultra high vacuum (10-9 torr and below) in a vacuum chamber starting from STP. How does
the amount of gas bound to the walls affect the ability of the vacuum pumps to maintain an ultra high
vacuum? What experimental methods might you use to minimize the vapor pressure that results from
outgassing of the walls?
4. Use the Clausius-Clapyron equation
dpσ s2 − s1
L
=
=
dT v 2 − v1 T (v 2 − v1 )
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to derive an analytical expression for the equilibrium vapor pressure pσ of water between 0°C and 200°C.
Assume the following:
1. The equilibrium vapor pressure of water at 100°C is 1.00 atmospheres.
2. The latent heat of vaporization of water LV is independent of temperature and has the value 540
kcal/kg.
3. The volume per mole of the liquid phase is negligible compared to that of the vapor phase.
4. Water vapor can be treated as an ideal gas for the range of temperatures and pressures needed in
this problem.
Plot pσ vs. T between 0°C and 200°C. Compare your results with experimental values for pσ in this
temperature range.
5. The Helmholtz Free Energy of a magnet is given by the following Landau expansion
F(T, M) = a(T) +
b(t) 2 c(t) 4
M +
M
2
4
a) Derive expressions for H, S, U, χT , and CH as functions of T and M.
b) Assume that the coefficients are of the form a(T)=a0, b(T)=b1(T-Tc)/Tc, and c(T)=c0>0 where a0, b1, and
c0 are constants. Show that the Landau Free Energy is unphysical for a range of values of T and M. Fix this
using a convex cover. Determine analytical forms for the spontaneous magnetization, the zero field
isothermal susceptability and the zero field specific heat as functions of temperature and sketch.