Consider a classical system of N noninteracting diatomic molecules enclosed in a
ID: 3307511 • Letter: C
Question
Consider a classical system of N noninteracting diatomic molecules enclosed in a box of volume V at temperature T. The Hamiltonian of the system is taken to be 2 2 where M = 2m the total mass of the molecule, -m/2 the reduced mass of the molecule. P is the momentum of the center of mass, p1 p2, and p the reduced momentum p = (pi-P2)/2, and 1-1r-ral. a Calculate the Helmholtz free energy, F, of the system. b Calculate the internal energy U from the relation U-(SF)/83. Does this agree with the result you would get from the equipartition theorem? c What is the specific heat at constant volume? d Calculate the mean square molecular diameter 2 >. You can get this in two ways: from = 20(F/N)/K or from the equipartition theorem. Do they agree? e Calculate the pressure p =-8F/OV.Explanation / Answer
In this solution, the partial derivative will be denoted by d/dt or d/dB so on with respect to the independent variable:
(a) Since the Hamiltonian is unperturbed, using Bogoliubov inequality we get,
F =< H - TS , where H is the given Hamiltonian, T is the temperature, S is the entropy.
F =< P2/2M + p2/2ur + 1/2 Kr2 - TS , which is the required free energy expression.
(b) B will be denoted by beta = 1/kT , where k is the boltzmann constant, therefore,
U = d(BF)/ dB , where d/dB denotes the partial derivative with respect to B,
BF = F/kT = B[(P2/2M + p2/2ur + 1/2 Kr2) - TS]
U = d(BF)/dB = P2/2M + p2/2ur + 1/2 Kr2 - TS = F , therefore the internal energy of the system is equal to the Helmohltz free energy.
(c) CV = (dU/dT)V = (dF/dT)V = - S.
(d) < r2 > = 2 d(F/N)/dK = 2 d( P2/2M + p2/2ur + 1/2 Kr2 - TS)/N /dK = 2 x 1/2 x r2 /N = r2/N .
(e) In a system of ideal gas, p`v = NRT which implies T = p`v/NR , where p` is the pressure and v is the volume of the gas and R is the gas constant. Therefore,
F = P2/2M + p2/2ur + 1/2Kr2 - (p`v/NR) S
p = -dF/dv = p`S/NR.
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