Partition function problem (Reif)

[Tom1]

Hi, I am trying to work through some of the problems in Reif, not for homework, but to prepare for a final. I was hoping someone could show me how to work through the first problem of chapter seven in Reif.

7.1 - Consider a homogeneous mixture of intert monatomic ideal gases at absolute temperature T in a container of volume V. Let there be v1 moles of gas 1, and v2 moles of gas 2, ..., and vk moles of gas k.

(a) By considering the classical partition function of this system, derive its equation of state, i.e., find an expression for the total mean pressure p.

(b) How is this total pressure p of the gas related to the pressure p_i, which the ith gas would produce if it alone occupied the entire volume at this temperature?

 

[BigJDubs]

Answer: (a) The classical partition function of the system is given by Z = (V/h^3)^(N/2) ∏_i (2πm_i kT/h^2)^(v_i/2). Here, N is the total number of moles of the gas mixture, h is Planck's constant, m_i is the mass of the ith gas, and v_i is the number of moles of the ith gas. Taking the natural logarithm of both sides, we get ln Z = (N/2)ln(V/h^3) + ∑_i (v_i/2)ln(2πm_i kT/h^2). Using the Gibbs-Helmholtz equation, we can rewrite this expression as ln Z = (N/2)lnV - (3/2)NkTln(h/2π) + (3/2)NkT ∑_i (v_i/2)ln(m_i/M), where M is the molar mass of the gas mixture. Differentiating with respect to V and setting the result equal to the total mean pressure p, we get: p = (1/V) * (∂/∂V) ln Z = (N/V)kT - (3/2)NkTln(h/2π) + (3/2)NkT ∑_i (v_i/2)ln(m_i/M).(b) The pressure p_i that the ith gas would produce if it alone occupied the entire volume V at temperature T can be found by rewriting the above equation as p_i = (v_i/V)kT - (3/2)v_ikTln(h/2π) + (3/2)v_ikTln(m_i/M). This expression shows that the total pressure p is equal to the sum of the partial pressures of the individual gases.