A rigid vessel of volume is held at by a reservoir

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Statistical mechanics of ideal classical systems

Thermal Physics, Ch The classical ideal gas from the partition function

Problem

A rigid vessel of volume

V = 2.00\times10^{-3}\,\mathrm{m^3}

is held at

T = 350\,\mathrm{K}

by a reservoir. It contains

N_{\rm He} = 3.00\times10^{22}

helium atoms (

M = 4.003\,\mathrm{g\,mol^{-1}}

) and

N_{\rm Ar} = 1.00\times10^{22}

argon atoms (

M = 39.95\,\mathrm{g\,mol^{-1}}

). Both gases are monatomic and ideal, the atoms are independent, and the classical regime holds. For such a mixture the partition function factorises,

Z = Z_{\rm He}Z_{\rm Ar}

, each species being divided by its own

N_i!

, so the free energies add:

F = F_{\rm He} + F_{\rm Ar}

with

F_i = -N_ik_BT[\ln(V/N_i\lambda_{T,i}^3) + 1]

. ((a)) Chapter 1 obtained the pressure from wall collisions as

p_i = \tfrac13 n_i m_i \overline{v_i^2}

. Using this chapter's internal energy of an ideal monatomic gas to supply

\overline{v_i^2}

, find the root-mean-square speed of a helium atom and of an argon atom, and hence the total pressure in the vessel. ((b)) Obtain the total pressure again from

p = -(\partial F/\partial V)_{T,N_{\rm He},N_{\rm Ar}}

, and say in one sentence why the atomic masses cancel out of

p

although they appear in

\lambda_{T,i}

. ((c)) Using

\mu_i = (\partial F/\partial N_i)_{T,V,N_{j\ne i}}

, evaluate the chemical potential of each species in units of

k_BT

, and say which species has the more negative

\mu

and why. Take

k_B = 1.381\times10^{-23}\,\mathrm{J\,K^{-1}}

h = 6.626\times10^{-34}\,\mathrm{J\,s}

N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}

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