Helium gas (monatomic, ) is held at and in a container in contact with 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

Helium gas (monatomic,

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

) is held at

T = 290\,\mathrm{K}

and

p = 1.20\times10^{5}\,\mathrm{Pa}

in a container in contact with a reservoir. Treat it as a classical ideal gas, for which

F(T,V,N) = -Nk_BT\left[\ln\!\left(\frac{V}{N\lambda_T^{3}}\right) + 1\right]

\lambda_T = \frac{h}{\sqrt{2\pi m k_B T}}, \qquad m = \frac{M}{N_A}.

((a)) Derive

\mu = (\partial F/\partial N)_{T,V}

in terms of the number density

n = N/V

and

\lambda_T

. ((b)) Using the gas law

p = nk_BT

(itself obtained from

-(\partial F/\partial V)_{T,N}

) to get

n

, evaluate

\lambda_T

and then

\mu

, as a multiple of

k_BT

and in eV. ((c)) The mean translational energy per atom is

U/N = \tfrac32 k_BT

. By what factor does

|\mu|

exceed it? Say in one sentence why

\mu

is nevertheless negative. Take

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

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

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

and

1\,\mathrm{eV} = 1.602\times10^{-19}\,\mathrm{J}

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