Sodium vapour (monatomic, ) is treated as a classical ideal gas at

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

Sodium vapour (monatomic,

M = 22.99\,\mathrm{g\,mol^{-1}}

) is treated as a classical ideal gas at

T = 1000\,\mathrm{K}

. Chapter 8 wrote the chemical potential of an ideal gas as

\mu = k_BT\ln(p/p_0) + \mu_0(T)

with an undetermined function

\mu_0(T)

and a chosen reference pressure, here

p_0 = 1.00\times10^{5}\,\mathrm{Pa}

. The partition function gives

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)) From

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

and

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

, show that

\mu = k_BT\ln(p/p_0) + \mu_0(T)

with

\mu_0(T) = k_BT\ln\!\left(p_0\lambda_T^{3}/k_BT\right)

, and check that the argument of the logarithm is dimensionless. ((b)) Evaluate

\mu_0

1000\,\mathrm{K}

, in J and in eV. ((c)) Evaluate

\mu

for the vapour at

p = 0.0500\,\mathrm{bar} = 5.00\times10^{3}\,\mathrm{Pa}

and

1000\,\mathrm{K}

, in eV. 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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