Ph641 - Statistical Thermophysics - Spring 2016
Problem Set 1 - due Wednesday, Apr 13th
1. Internal energy: I.
The internal energy of system is given by U (S, V, N ) = S α V β N γ .
a) Using the fact that U is extensive, find a relation between α, β, and γ.
b) Calculate the pressure p, temperature T , and chemical potential µ as a
function of S, V , and N .
c) Calculate the heat capacity at constant volume V (and constant N ).
d) Calculate the adiabatic compressibility (at constant N ).
e) Based on the sign of these response functions, find inequalities for α and
β.
2. Idealized combustion engine
Consider an ideal gas consisting of N particles (molecules) with f degrees of
freedom per molecule. The equation of state and the internal energy U for
this gas are given by
pV
= N kT
f
N kT
U =
2
(1)
(2)
(f = 3 for a mono atomic gas, f = 5 for a diatomic gas, etc.)
a) Consider an adiabatic process with constant particle number N . Starting
from the above equations show that
pV γ = const.
and
T V γ−1 = const.
(3)
and express γ in terms of f : γ(f ).
b) Consider an (idealized) four stroke engine (Otto motor) working with an
ideal gas as a working medium. The (idealized) cycle is given in the figure
below:
1 → 2 : expansion
2 → 3 : exhaust
3 : intake
3 → 4 : compression
4 → 1 : combustion
Determine the efficiency η = ∆W/∆Q of this cycle, where ∆W is the performed work, and ∆Q is the generated heat during combustion. Express
the efficiency in terms of the compression ratio V3 /V4 of the engine and the
degrees of freedom f of the working gas.
3. Entropy of mixing
Calculate the entropy of mixing between ideal gases A and B, which are
initially separated in volumes xV and (1 − x)V (equal pressure p). After
removal of the barrier the two gases will mix.
a) Calculate the entropy increase upon mixing using thermodynamic relations (express your result in terms of N = NA + NB , the total number of
atoms and x).
4. Internal energy: II. Old Comp Exam Question (Graded for (any) attempt
only feel free to look on the exam archive or simply wait for the solution)