Problem Set 4

Problem Set 4

Due Wednesday, March 16, in lecture

Problem 1 [15 points total]
Consider a classical ideal gas of indistinguishable, non-interacting, particles confined to a region of three dimensional space by a harmonic potential V(r) rather than the walls of a box. This might be a model for a gas of atoms in a magnetic trap (we will talk more about this later in the semester). The single particle Hamiltonian is then,

H⁽¹⁾(r, p) = |p|²
2m + 1
2 m&omega_o²|r|²

Working in the canonical ensemble for a gas of N particles,
a) Compute the average energy E of the gas as a function of temperature T.
b) Compute the density of particles n(r) as a function of the radial distance r=|r| from the origin. n(r) should be normalized so that ∫d³r n(r) = N.
c) What is the average radial distance <r> of particles from the origin?
d) What is the pressure of the gas p(r) as a function of the radial distance r from the orign? To do this part, it might help to think of only the fraction of the gas that is within a spherical shell between radii r and r+Δr, for small Δr. Treat this shell as your system of interest, and find the pressure as a function of the density of the gas in this shell.
Problem 2 [15 points total]
Consider a system of particles in the grand canonical ensemble, with μ the chemical potential and z=exp(βμ) the fugacity. If < N> is the average number of particles, and <E> is the average total energy,
a) show that,

1
T ( ∂<E>
∂μ )_T,V = 1
k_BT² [ <EN> - <E><N> ]

b) show that,

( ∂<N>
∂T )_z,V = 1
T ( ∂<E>
∂μ )_T,V

Note, on the left hand side the derivative is taken at constant fugacity z, rather than constant chemical potential μ.
Problem 3 [35 points total]
Consider a classical gas of very weakly interacting molecules of different species. There are N_i 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⁾]^N_i/N_i!
and Q₁⁽ⁱ⁾ is the single particle partition function for molecules of species i. [5 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. [5 points]
(c) By taking the appropriate derivative of the total Helmholtz free energy of the gas, compute the chemical potential µ_i of species i. Express your answer in terms of Q₁⁽ⁱ⁾. [5 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. [5 points]
(e) Suppose that the species of molecules undergo the chemical reaction
a₁A₁ + ... + a_jA_j ↔ a_j+1A_j+1 + ... + a_mA_m
where a_i is the number of molecules of species A_i (i = 1,...,j) that combine to create a_k molecules of species A_k (k = j+1,...,m). As discussed in lecture, the equilibrium number of molecules of each species will be determined by the condition
a₁µ₁ + ... + a_jµ_j = a_j+1µ_j+1 + ... + a_mµ_m
(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...[n_j]^a_j)/([n_j+1]^a_j+1...[n_m]^a_m) = K(T)
where n_i is the concentration of species i, n_i = N_i/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". [5 points]
(f) Consider the reaction
A + B ↔ C
Suppose that an energy E_o is released when A and B combine to form C, i.e. E_o 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 E_o of molecule C when you compute its partition function. [5 points]
(g) Suppose that initially there are equal concentrations of A and B, n_A = n_B = n_o, while the concentration of C is initially n_C = 0. Find the resulting equilibrium concentrations of A, B and C, in terms of n_o and the function K(T). [5 points]