A small system is weakly coupled to a block of aluminium of mass which acts as its…

PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Statistical foundations and the equilibrium ensembles

Thermal Physics, Ch The grand canonical ensemble, ensemble equivalence and fluctuations

Problem

A small system is weakly coupled to a block of aluminium of mass

2.0\,\mathrm{g}

which acts as its reservoir, exchanging both energy and particles with it. The total energy

U_0

and total particle number

N_0

of system plus block are fixed, and the block is so large that its temperature

T = 300\,\mathrm{K}

and its chemical potential

\mu

do not change in the exchange. The block's entropy is

S_R(U,N)

, and its heat capacity may be taken as the Dulong-Petit value

3Nk_B

with molar mass

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

. Take

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}

. (a) By expanding

S_R(U_0 - E_s, N_0 - N_s)

to first order in both arguments, show that the probability of a system state

s

with energy

E_s

and particle number

N_s

is proportional to

e^{-(E_s - \mu N_s)/k_BT}

. Name the two reservoir derivatives you use. (b) Explain in words, from the entropy the block loses when it gives up a particle, why the exponent carries

+\mu N_s

, and why this does not mean the system gains an energy

\mu

per particle. (c) The dropped term of the expansion is of order

E_s^2/(2k_BT^2C_R)

, with

C_R

the block's heat capacity. Evaluate this term and the first-order term

E_s/k_BT

for

E_s = 1.5\,\mathrm{eV}

, and say whether the expansion is safe. (d) Find the root-mean-square fluctuation

\sigma_E

of the block's own energy in a bath at

300\,\mathrm{K}

, using

\sigma_E^2 = k_BT^2C_V

, and compare it with the block's internal energy

U \approx 3Nk_BT

by giving the ratio

\sigma_E/U

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