Thermodynamics and Chemistry

CHAPTER 8 PHASE TRANSITIONS AND EQUILIBRIA OF PURE SUBSTANCES

8.1 PHASEEQUILIBRIA 196

Imagine a small sample of gas of massmthat is initially at elevationhD 0. The vertical
extent of this sample should be small enough for the variation of the gravitational force
field within the sample to be negligible. The gravitational work needed to raise the gas to
an arbitrary elevationhisw^0 Dmgh(page 81 ). We assume this process is carried out
reversibly at constant volume and without heat, so that there is no change inT,p,V,S, or
f. The internal energyUof the gas must increase bymghDnMgh, whereMis the molar
mass. Then, because the Gibbs energyGdepends onUaccording toGDUTSCpV,
Gmust also increase bynMgh.
The chemical potentialis the molar Gibbs energyG=n. During the elevation process,
fremains the same andincreases byMgh:

.h/D.0/CMgh (8.1.9)

(f .h/Df .0/)

From Eqs.8.1.8and8.1.9, we can deduce the following general relation between chemical
potential, fugacity, and elevation:

.h/D(g)CRTln

f .h/

p

CMgh (8.1.10)

(pure gas in

gravitational field)

Compare this relation with the equation that defines the fugacity when the effect of a grav-
itational field is negligible: D (g)CRTln.f=p/(Eq.7.8.7on page 183 ). The
additional termMghis needed when the vertical extent of the gas is considerable.

Some thermodynamicists call the expression on the right side of Eq.8.1.10the “to-

tal chemical potential” or “gravitochemical potential” and reserve the term “chemical

potential” for the function(g)CRTln.f=p/. With these definitions, in an equilib-

rium state the “total chemical potential” is the same at all elevations and the “chemical

potential” decreases with increasing elevation.

This book instead defines the chemical potentialof a pure substance at any ele-

vation as the molar Gibbs energy at that elevation, as recommended in a 2001 IUPAC

technical report.^1 When the chemical potential is defined in this way, it has the same

value at all elevations in an equilibrium state.

We know that in the equilibrium state of the gas column, the chemical potential.h/
has the same value at each elevationh. Equation8.1.10shows that in order for this to be
possible, the fugacity must decrease with increasing elevation. By equating expressions
from Eq.8.1.10for.h/at an arbitrary elevationh, and for.0/at the reference elevation,
we obtain

(g)CRTln

f .h/

p

CMghD(g)CRTln

f .0/

p

(8.1.11)

Solving forf .h/gives

f .h/Df .0/eMgh=RT (8.1.12)

(pure gas at equilibrium

in gravitational field)

(^1) Ref. [ 2 ].