For a one-component system open to particle exchange, Chapter 7's differential of the…

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

For a one-component system open to particle exchange, Chapter 7's differential of the Gibbs energy gains the chemical-potential term:

\mathrm{d}G = -S\,\mathrm{d}T + V\,\mathrm{d}p + \mu\,\mathrm{d}N

. Hydrogen may be treated as an ideal gas throughout. ((a)) By equating the mixed second derivatives of

G

with respect to

p

and

N

, obtain a Maxwell relation between

(\partial\mu/\partial p)_{T,N}

and a derivative of

V

, and evaluate it for hydrogen at

350\,\mathrm{K}

and

2.00\,\mathrm{bar} = 2.00\times10^{5}\,\mathrm{Pa}

. ((b)) Integrate that derivative at fixed temperature to find the change in

\mu

per molecule when the hydrogen is compressed from

2.00\,\mathrm{bar}

5.00\,\mathrm{bar}

350\,\mathrm{K}

. ((c)) Express the same change per mole, in

\mathrm{J\,mol^{-1}}

. ((d))

0.500\,\mathrm{mol}

of that hydrogen at

350\,\mathrm{K}

and

5.00\,\mathrm{bar}

is then mixed, at fixed temperature and pressure, with

0.500\,\mathrm{mol}

of argon at the same temperature and pressure. Find the entropy of mixing.

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