A sealed cell of volume at contains atoms of ( ) and atoms of ( )

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 sealed cell of volume

V = 1.00\times10^{-3}\,\mathrm{m^3}

T = 320\,\mathrm{K}

contains

4.00\times10^{21}

atoms of

^{20}\mathrm{Ne}

(

M = 19.99\,\mathrm{g\,mol^{-1}}

) and

4.00\times10^{21}

atoms of

^{22}\mathrm{Ne}

(

M = 21.99\,\mathrm{g\,mol^{-1}}

). Both are monatomic ideal gases in the classical regime; atoms of different mass are distinguishable from each other, so the partition function is

Z = Z_{20}Z_{22}

with each species divided by its own

N_i!

, and

F = F_{20} + F_{22}

with

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

. ((a)) Chapter 1's wall-collision result is

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

. Find the root-mean-square speed of each isotope (using this chapter's internal energy to supply

\overline{v_i^2}

) and show that the two partial pressures are equal; give the total pressure in the cell. ((b)) From

U = -(\partial \ln Z/\partial\beta)_{V,N}

, find the internal energy of the gas in the cell, in joules. ((c)) Evaluate

\mu

for

^{20}\mathrm{Ne}

in units of

k_BT

, and the difference

\mu(^{22}\mathrm{Ne}) - \mu(^{20}\mathrm{Ne})

in units of

k_BT

. Show that this difference equals

-\tfrac32k_BT\ln(m_{22}/m_{20})

and say which isotope is the more strongly bound to the gas, in the sense of having the lower

\mu

. 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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