PHYSICS 140A : STATISTICAL PHYSICS
HW ASSIGNMENT #1
(1) To measure the heat capacity of an object, all you usually have to do is put it in thermal
contact with another object whose heat capacity is known. Suppose that a 100 g chunk of
metal is immersed in boiling water (100◦ C). After a time, the metal is removed and quickly
transferred to a Styrofoam cup containing 250 g of water at 20◦ C. After a little while, the
temperature of the contents of the cup is found to be 24◦ C. You may assume that the heat
transferred from contents of the cup to its surroundings, and the heat capacity of the cup
itself are both negligible.
(a) How much heat is gained by the water in the Styrofoam cup?
(b) How much heat is lost by the metal?
(c) What is the heat capacity of the metal?
(d) What is the specific heat (in J/g K) of the metal?
(2) The heat capacity of Albertson’s Rotini Tricolore is approximately 1.8 J/g ◦ C. Suppose
you toss 340 g of pasta at 15◦ C into 1.5 liters of boiling water. What effect does this have
on the temperature of the water, before the stove has time to provide any more heat?
Hint: You will need to look up some constants in order to solve this problem. Part of the
problem is in understanding what constants you need. Once you know, try Google.
(3) Consider the van der Waals equation of state,
aν 2
p + 2 V − νb = νRT
V
where ν = N/NA is the number of moles. Find expressions for each of the following quantities
in terms of the specific volume v = V /ν, the temperature T , and constants. Don’t confuse
the symbols ν and v!
(a)
−V
∂p
∂V
(b)
T
1
∂p
∂T
(c)
V
1
V
∂V
∂T
p
(4) A peculiar molecule discovered in Santa Barbara, consisting of uranium, carbon, and
antimony (UCSb) is found in its gas phase to obey the modified van der Waals equation,
p (V − νb) = νRT .
From calorimetry, the energy is determined to be
E(T, V, ν) = ν ε0 eT /T0 ,
which is volume-independent, with ε0 = 1000 J/mol and T0 = 300 K.
(a) Find an expression for the molar specific heat cp .
(b) How much heat energy Q is required to raise ν = 5 moles of the UCSb gas from
TA = 300 K to TB = 600 K and at constant pressure?
(c) (Challenging!) Find an expression for the adiabatic speed of sound cS (v, T ) in terms
of the molar volume v = V /ν, the temperature T , and other constants (including the
molecular mass M ). For a discussion of sound in fluids (and gases), see §14.5.5 of my
lecture notes from Physics 110B (still available online).
(5) Which of the following quantities is a ‘state function’ for driving? Explain your answers.
(a) travel time
(b) elevation
(c) aggravation
(6) For each of the following differentials, identify whether it is exact or inexact. If you
can, find the integrating factor L for which dU = e−L dW
¯ is exact.
(a) dW
¯ = 2x2 y dx + x3 dy
(b) dW
¯ =
y2 z
(x+y)2
dx +
x2 z
(x+y)2
(c) dW
¯ = exy dx + 2x +
x
y
dy +
exy
xy
x+y
dz
dy
2