PHYS-4420 THERMODYNAMICS & STATISTICAL MECHANICS SPRING 2002

Quiz 2 Friday, April 19, 2002

NAME: ______

To receive credit for a problem, you must show your work, or explain how you arrived at your answer.

1. (30%) Enthalpy is defined as H = E + pV, where E is internal energy, p is pressure, and V is volume.

a) Show that: dH = TdS + Vdp + mdN

b) Use the equation given in part a) to show that:

c) Use the equation given in part a) to show that:

(This is one of the Maxwell relations.)


2. (25%) Consider a collection of 3N identical, distinguishable harmonic oscillators, all of frequency n. The energies that one these oscillators can take on (measured relative to the ground state) are en = nhn. Where n is an integer that can take on values from 0 to ¥.

a) Find the partition function z, for one of these oscillators.

(Hint: for x < 1)

z = ______

b) Find the partition function Z, for the collection of oscillators. (Since the oscillators are distinguishable, no N! is needed in the denominator.)

Z = ______

c) Find the Helmholtz function F, for this collection of oscillators.

(Hint: F = – kT ln Z)

F = ______

d) Find the entropy S, for this collection of oscillators. Hint: . This will not give a particularly neat expression.

S = ______

e) Show that the expression for the entropy that you obtained in part d) goes to zero when T goes to zero, in agreement with the third law of thermodynamics.

3. (15%) The number density of the gas, mainly hydrogen, that fills interstellar space is about one molecule per cubic centimeter (r = 1 ×106 m-3). The diameter of the molecules is about

d = 1 ×10-10 m, and the temperature of interstellar space is about 10K.

a) Find the mean free path of the molecules in interstellar space.

l = ______

units

b) Find the average speed of the molecules in interstellar space. (The mass of molecular hydrogen is m = 2 amu, and 1 amu = 1.66 ×10-27 kg.)

= ______

units

c) Find the average time between collisions for molecules in interstellar space. Express your answer in centuries. (1 year = 3.16 ×107 s, and 1 century = 100 years.)

t = ______

units


4. (30%) One mole of an ideal monatomic gas traverses the cycle shown in the figure. Process 1®2 takes place at constant volume, process 2®3 is adiabatic, and process 3®1 takes place at constant pressure. In the work that follows, express all answers in terms of the gas constant R. (Hint: for one mole of an ideal monatomic gas, the internal energy is .)

a) Compute the heat added, and the work done for the process 1®2.

Q1®2 = ______

units

W1®2 = ______

units

b) Compute the heat added, and the work done for the process 2®3.

Q2®3 = ______

units

W2®3 = ______

units


c) Compute the heat added, and the work done for the process 3®1.

Q3®1 = ______

units

W3®1 = ______

units

d) Compute the net heat added, and the net work done for the entire cycle.

Qnet = ______

units

Wnet = ______

units

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