April 19, 2006
Name:_____________________________
PHY3513/Kumar
Midterm Examination III
1. Show that during a first order phase transition:
(a) The change of entropy of the system undergoing transition is a linear
function of the volume change
(Problem 9.10) The Clausius-Clapeyron equation describes the phase boundary and also
the relationship between the entropy discontinuity and the one in molar volume.
dP Δs
|=
⇒ Δs ∝ Δv
dT Δv
(b) The change of the internal energy is given by
⎡ d ln T ⎤
ΔU = L ⎢1 −
⎣ d ln P ⎥⎦
Where L is the latent heat of transformation.
Notice that at a first order phase transition ΔH = L. Since H = U+PV, and the transition
takes place at constant pressure, ΔU = ΔH - PΔV = L-PΔV. Substituting for ΔV from
part a, we have
PL dT
ΔU = L −
T dP
⎡ d ln T ⎤
= L ⎢1 −
⎣ d ln P ⎥⎦
2. (Problem 10.5 and 10.7, part c is new)
(a) Show that if Cv = bTα at low temperatures then the third law of
thermodynamic requires that α>0.
dT
C v and the third law stipulates that S(T=0) = 0. Here S =(b/α)Tα
T
which in the limit of TÆ 0 is finite only if α ≥ 0.
Recall that S = ∫
(b) If Cv = a(P)T + b(P)T3 at low temperatures (as is the case with most
metals, ask Stewart or Andraka), calculate the temperature variation of the
entropy.
b
S = aT + T 3
3
Note that this is not a classical ideal gas. A classical ideal gas has a constant specific
heat. The term linear in T can be understood as arising from a quantum ideal gas of
fermions but then its coefficient would have a specific value. In general, the coefficient
varies from its quantum ideal gas value. The term cubic in temperature also can be
derived from a quantum ideal gas of bosons.
(c) Calculate the temperature dependence of the coefficient of volume thermal
expansion (vβ = dv/dT|P) from the entropy in part (b). If β>o, what does
that say about the coefficients a(P) and b (P).
Define ap(P) = da(P)/dP. Then vβ = ∂v/∂T = - ∂S/∂P = -[apT + (bp/3)T3]. Since β ≥0,
that means that ap ≤ 0, i.e. at low temperatures the specific heat decreases with pressure
at a fixed temperature. There is no restriction on bp though at low T.
(d) A low temperature physicist wishes to publish his experimental result that
the heat capacity of a non-magnetic dielectric material between 0.05 K and
0.5 K varies as AT1/2 + BT3. As editor of the journal, should you accept
this paper for publication? Why?
As discussed in part (a), the specific heat does not violate the third law of
thermodynamics.
3. Compute the mean energy in electron volts and the rms speed in m/s of an
electron at 1000K. At 10,000K, what fraction of the speed of light is the rms speed?
( 1 eV = 1.6×10-19J)
(Problem 11.5) Mean energy is (3/2) kT where k is the Boltzmann constant
=1.5×1.4×10-23×1000
= 2.1×10-20 J
= 0.131 eV
E = (1/2) mvrms2⇒ vrms = √(2×2.1×10-20/9.11×10-31)
=2.13×105m/s
At T = 10,000 K, vrms = 6.74×105m/s and vrms/c = 0.22 %