Boltzman: Boltzman








Physics 418: Statistical Mechanics I
Prof. S. Teitel ----- Spring 2002

Problem Set 3

Due Thursday, March 21, 11 am. Hand in to the grader.

  • Problem 1

    Consider a classical gas of very weakly interacting molecules of different species. There are Ni molecules of species number i, i = 1, 2, ..., m. The molecules of different species may have different masses and different internal degrees of freedom (such as vibrational or rotational modes), however you may assume that the molecules do not interact with each other. The molecules of a given species are indistiguishable from each other.

    (a) Show that the canonical partition function for the gas has the form

    Q = Q(1)Q(2)....Q(m)      where      Q(i) = [Q1(i)]Ni/Ni!
    and Q1(i) is the single particle partition function for molecules of species i. [4 points]

    (b) Using the result in (a), show that the total pressure of the gas is the sum of the pressures that each of the species of molecule would have on its own (total pressure is the sum of the "partial pressures"). Similarly show that the total entropy of the gas is the sum of the entropies that each species would have on its own. [2 points]

    (c) By taking the appropriate derivative of the total Helmholtz free energy of the gas, compute the chemical potential µi of species i. Express your answer in terms of Q1(i). [2 points]

    (d) Assume that the molecules are free (i.e. not in any external potential, except for their confinement to a box of volume V). Show that Q1(i) can be written as Q1(i) = Vq1(i)(T) where q1(i) depends only on temperature T. [2 points]

    (e) Suppose that the species of molecules undergo the chemical reaction

    a1A1 + ... + ajAj <--> aj+1Aj+1 + ... + amAm
    where ai is the number of molecules of species Ai (i = 1,...,j) that combine to create ak molecules of species Ak (k = j+1,...,m). As discussed in lecture, the equilibrium number of molecules of each species will be determined by the condition

    a1µ1 + ... + ajµj = aj+1µj+1 + ... + amµm
    (make sure you understand where this result comes from!). Use the above, and the results of the previous parts, to show that the equilibrium concentrations of the species are given by

    ([n1]a1 [n2]a2...[nj]aj )/([nj+1]aj+1...[nm]am ) = K(T)
    where ni is the concentration of species i, ni = Ni/V, and K(T) is a function of temperature only. Derive an expression for K(T) in terms of the q1(i)(T). The above result is known as the "law of mass action". [4 points]

    (f) Consider the reaction

    A + B <--> C
    Suppose that an energy Eo is released when A and B combine to form C, i.e. Eo is the binding energy of the molecule C when compared to its separated constituents A and B. Assume that A, B, and C are free point particles (i.e. no internal degrees of freedom are excited). By explicitly evaluating the single particle partition function of the three species, determine the function K(T). Remember, you must properly include the binding energy Eo of molecule C when you compute its partition function. [4 points]

    (g) Suppose that initially there are equal concentrations of A and B, nA = nB = no, while the concentration of C is initially nC = 0. Find the resulting equilibrium concentrations of A, B and C, in terms of no and the function K(T). [2 points]

  • Problem 2

    In the grand canonical ensemble, the probability to have a given state with total energy Ea and total number of particles Na is,

    Pa = [e-(Ea-µNa)/kBT]/L

    L = Suma [e-(Ea-µNa)/kBT]       is the grand canonical partition function.

    (a) For a quantum ideal gas, with single particle states i of energy ei, many particle states are specified by the occupation numbers {ni} and have energy E = Sumi [eini]. Show that the probability for the state with occupations {ni} is given by

    P({ni}) = Producti [pi(ni)]
    where pi(ni) is the probability that single particle state i has occupation ni

    pi(ni) = [e-(ei-µ)ni/kBT]/wi

    wi = Sumni [e-(ei-µ)ni/kBT]
    can be thought of as the partition function for the single particle state i. The above factorization says that the number of particles ni in state i, is independent of the number of particles nj in state j. [6 points]

    (b) Using the above result, show that the Shannon definition of entropy can be written as

    S = -kB Sum{ni} [P({ni}) ln P({ni})] = -kB Sumi Sumni [pi(ni) ln pi(ni)]
    [6 points]

    (c) Using the above result, show that the following expressions apply for the entropies of an ideal gas of bosons and fermions, respectively

    bosons: S = kB Sumi [(1+<ni>) ln (1+<ni>) - <ni> ln <ni>]
    fermions: S = kB Sumi [-(1-<ni>) ln (1-<ni>) - <ni> ln <ni>]
    where <ni> = Sumni [ni pi(ni)] is the average occupation number of state i. [8 points]

  • Problem 3

    Consider photons of a given energy e = hbar omega.

    (a) If <n> is the average number of such photons in equilibrium at temperature T, show that the fluctuation in the number of photons is

    <n2> - <n>2 = - (1/e) (d<n>/dbeta)     where beta = 1/kBT
    [5 points]

    (b) Using the forumula for the equilibrium value of <n>, apply the above result to determine the relative fluctuation in the number of photons

    [<n2> - <n>2]/<n>2
    Is this large or small? [5 points]

Last update: Monday, August 20, 2007 at 12:09:29 PM.