Two glass bulbs are joined by a narrow vertical pipe of negligible volume fitted with…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Thermodynamic potentials and the formal structure of thermodynamics · Thermodynamic potentials and the formal structure of thermodynamics

Thermal Physics, Ch Chemical potential, open systems and chemical equilibrium

Problem

Two glass bulbs are joined by a narrow vertical pipe of negligible volume fitted with a closed valve. The lower bulb sits at

z = 0

on a mountainside, the upper bulb at

z = 1200\,\mathrm{m}

. Both contain nitrogen, treated as an ideal gas, and both are held at

T = 290\,\mathrm{K}

throughout. The number densities are

n_{\rm low} = 2.50\times10^{25}\,\mathrm{m^{-3}}

in the lower bulb and

n_{\rm up} = 2.30\times10^{25}\,\mathrm{m^{-3}}

in the upper one. At the common temperature the chemical potential of the gas in a bulb at height

z

may be written

\mu = k_BT\ln(n/n_0) + \mu_0'(T) + mgz

, with

n

the number density and

n_0

any fixed reference density. Molar mass of nitrogen

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

;

g = 9.81\,\mathrm{m\,s^{-2}}

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

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

. ((a)) Compute

\Delta\mu = \mu_{\rm up} - \mu_{\rm low}

per molecule for the gas as it stands. From its sign, say which way molecules move when the valve is opened (a positive value means the upper bulb has the higher chemical potential). ((b)) Find the ratio

n_{\rm up}/n_{\rm low}

at which the valve could be opened with no net flow at all.

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