Here is a first attempt at the entropy of argon atoms (molar mass ) held at in a…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Statistical foundations and the equilibrium ensembles · Statistical mechanics of ideal classical systems

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

Problem

Here is a first attempt at the entropy of

N = 2.00\times10^{23}

argon atoms (molar mass

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

) held at

T = 250\,\mathrm{K}

in a rigid box of volume

V = 5.00\,\mathrm{L} = 5.00\times10^{-3}\,\mathrm{m^3}

. Argon is monatomic, the gas is dilute enough to be treated as an ideal gas in the classical regime, and only translational states are counted. The attempt writes

Z_1 = V/\lambda_T^3

with

\lambda_T = h/\sqrt{2\pi m k_B T}

, then

Z_N = Z_1^{\,N}

, so that

F = -Nk_BT\ln(V/\lambda_T^3)

and

S_{\rm trial} = Nk_B\left[\ln\!\left(\frac{V}{\lambda_T^3}\right) + \frac{3}{2}\right].

One factor has been left out of

Z_N

. Name it, then work through the consequences. Use

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 Stirling's approximation

\ln N! \approx N\ln N - N

. ((a)) Evaluate

\lambda_T

and the trial entropy

S_{\rm trial}

. ((b)) Evaluate the correct entropy from the Sackur-Tetrode formula. ((c)) Show that the difference

S_{\rm trial} - S

k_B\ln N!

, and evaluate it. ((d)) The system is now doubled at the same temperature (

N \to 2N

V \to 2V

). By how much does the trial formula exceed twice its original value?

Browse the problem library