Assignment 8

Problem 1:

Inside a blackbody cavity, the energy density per unit frequency interval, ρ(ν), is given by Planck's formula
ρ(ν) = (8πν2/c3) hν/(exp(hν/kT) - 1).
The intensity per unit frequency interval, I(ν), of the radiation emitted by the blackbody is given by I(ν) = ρ(ν) c.
Commonly, Wien's displacement law is written as λmax (m) = (2.9*10-3 m K)/T.
Derive the Wien's displacement law for the frequency νmax (s-1), and show that νmax is NOT equal to c/ λmax.  Why?

Problem 2:

Consider a system of one mole of an ideal gas A and three moles of an ideal gas B at the same pressure P and temperature T, in volumes of VA and VB respectively.  The two gases are separated by a partition so they are each sequestered in their respective volumes.  If the partition is removed, calculate the change in entropy of the system.

Problem 3:

A pipe of length 180 m, open on one end and closed on the other, lies at the bottom of a 200 m deep lake.  A light movable piston is placed inside the pipe.  The space between the closed end of the pipe and the piston is filled with air.  The piston is in equilibrium 20 m away from the closed end of the pipe.
The open end of the pipe is very slowly raised until the pipe is brought into the vertical position, its closed end resting at the bottom of the lake.
What is the height of the air column inside the vertical pipe?
Neglect the atmospheric pressure and assume the water temperature is the same throughout the lake.

Problem 4:

Consider a neutron star, a macroscopic body composed of neutrons, at a density ρ = 1017 kg/m3. The temperature of the star's interior is approximately 107 K.  For this problem, consider the star to be a non-interacting Fermi gas of neutrons.
For an ideal non-relativistic 3D Fermi gas comprising N non-interacting fermions, the Fermi energy (the energy difference between the highest and lowest occupied single-particle states at T = 0) is given by EF = [ħ2/(2m)](3Nπ2/V)2/3, and the average energy per fermion at absolute zero <E> = (3/5) EF.
(a)  Determine the Fermi energy of the neutrons in the neutron star.  Are the neutrons relativistic or nonrelativistic?
(b)  Determine whether or not the neutrons are reasonably well considered to be a zero temperature (EF >> kT) Fermi gas.
(c)  Estimate the pressure in the neutron star P = -∂U/∂V.  (quasi-static, adiabatic, fixed # of particles)  In equilibrium this pressure balances the pressure due to gravity.
(d)  Use (c) to estimate the mass of this neutron star.

Problem 5:

The operation of a gasoline engine is (roughly) similar to the Otto cycle.  A S-V diagram is shown.

A → B:  Gas compressed adiabatically
B → C:  Gas heated isochorically (constant volume; corresponds to combustion of gasoline)
C → D:  Gas expanded adiabatically (power stroke)
D → A:  Gas cooled isochorically.
Compute the efficiency of the Otto cycle for an ideal gas (with temperature-independent heat capacities) as a function of the compression ratio VA/VB, and the heat capacity of the gas, CV .