Imagine a piston with an external pressure of 1 bar that contains liquid water,
ID: 903620 • Letter: I
Question
Imagine a piston with an external pressure of 1 bar that contains liquid water, water vapor, and nitrogen gas. The piston is in the thermal contact with a large reservoir held at 270.15 Kelvin. Initially the partial pressure of water vapor in the piston is 489 Pa. At some point the water crystallizes and the system comes to a new equilibrium. Now the partial pressure of water vapor in the piston is 475 Pa. Calculate the difference in the chemical potential of the liquid water and the crystalline water.
(Hint: The partial pressure of the gas can be thought of as a reporter that provides easy access to the chemical potential of the liquid or the crystal, assume that water vapor is an ideal gas.)
Explanation / Answer
In the first equilibrium(a), pure liquid water and water vapor was at equilibrium. So chemical potentials of pure liquid water (l0) and chemical potential of water vapor (va) were equal.
We can write,
l0 = va = v0 + RTlnpva (pva = partial pressure of vapor in system a)
In the second equilibrium(b), ice and water vapor was at equilibrium. So chemical potentials of ice (ice0) and chemical potential of water vapor (vb) were equal.
We can write,
ice0 = vb = v0 + RTlnpvb (pvb = partial pressure of vapor in system b)
The difference in the chemical potential of the liquid water and the crystalline water(ice)
= (v0 + RTlnpva) - (v0 + RTlnpvb)
= RT (lnpva - lnpvb)
= RT ln (pva/pva)
= 8.314 J.K-1.mol-1 × 270.15 K × ln489/475
= 65.24 J/mol