P URE S UBSTANCES IN S INGLE P HASES
7.8 CHEMICAL POTENTIAL AND FUGACITY
Thechemical potential,, of a pure substance has as one of its definitions (page 141) defD GmD G
n (7.8.1)
(pure substance) That is, is equal to the molar Gibbs energy of the substance at a given temperature and pressure. (Section 9.2.6 will introduce a more general definition of chemical potential that applies also to a constituent of a mixture.) The chemical potential is an intensive state function.
The total differential of the Gibbs energy of a fixed amount of a pure substance in a single phase, withT andpas independent variables, is dG D SdT CV dp (Eq. 5.4.4).
8The Plimsoll mark is named after the British merchant Samuel Plimsoll, at whose instigation Parliament passed an act in 1875 requiring the symbol to be placed on the hulls of cargo ships to indicate the maximum depth for safe loading.
9Ref. [36], p. 61–62.
CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES
7.8 CHEMICALPOTENTIAL ANDFUGACITY 182
Dividing both sides of this equation byngives the total differential of the chemical potential with these same independent variables:
dD SmdT CVmdp (7.8.2)
(pure substance,PD1) (Since all quantities in this equation are intensive, it is not necessary to specify a closed system; the amount of the substance in the system is irrelevant.)
We identify the coefficients of the terms on the right side of Eq. 7.8.2 as the partial derivatives
@
@T
p D Sm (7.8.3)
(pure substance,PD1)
and
@
@p
T
DVm (7.8.4)
(pure substance,PD1) SinceVm is positive, Eq. 7.8.4 shows that the chemical potential increases with increasing pressure in an isothermal process.
The standard chemical potential,ı, of a pure substance in a given phase and at a given temperature is the chemical potential of the substance when it is in the standard state of the phase at this temperature and the standard pressurepı.
There is no way we can evaluate the absolute value of at a given temperature and pressure, or ofıat the same temperature,10but we can measure or calculate thedifference ı. The general procedure is to integrate dDVmdp(Eq. 7.8.2 with dT set equal to zero) from the standard state at pressurepıto the experimental state at pressurep0:
.p0/ ıD Z p0
pı
Vmdp (7.8.5)
(constantT)
7.8.1 Gases
For the standard chemical potential of a gas, this book will usually use the notationı(g) to emphasize the choice of agasstandard state.
An ideal gas is in its standard state at a given temperature when its pressure is the standard pressure. We find the relation of the chemical potential of an ideal gas to its pressure and its standard chemical potential at the same temperature by settingVmequal to RT =pin Eq. 7.8.5:.p0/ ıDRp0
pı.RT =p/dp DRTln.p0=pı/. The general relation foras a function ofp, then, is
Dı(g)CRT ln p
pı (7.8.6)
(pure ideal gas, constantT) This function is shown as the dashed curve in Fig. 7.6 on the next page.
10At least not to any useful degree of precision. The values ofandıinclude the molar internal energy whose absolute value can only be calculated from the Einstein relation; see Sec. 2.6.2.
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version:www.chem.umd.edu/thermobook
CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES
7.8 CHEMICALPOTENTIAL ANDFUGACITY 183
p
0
p ặ
f.p 0
/ p 0
ặ(g) b
b
b
A
B C
Figure 7.6 Chemical potential as a function of pressure at constant temperature, for a real gas (solid curve) and the same gas behaving ideally (dashed curve). Point A is the gas standard state. Point B is a state of the real gas at pressurep0. The fugacity f .p0/of the real gas at pressurep0is equal to the pressure of the ideal gas having the same chemical potential as the real gas (point C).
If a gas isnotan ideal gas, its standard state is a hypothetical state. Thefugacity,f, of a real gas (a gas that is not necessarily an ideal gas) is defined by an equation with the same form as Eq. 7.8.6:
Dı(g)CRT ln f
pı (7.8.7)
(pure gas) or
f defD pıexp
ı(g) RT
(7.8.8) (pure gas) Note that fugacity has the dimensions of pressure. Fugacity is a kind of effective pressure.
Specifically, it is the pressure that the hypothetical ideal gas (the gas with intermolecular forces “turned off”) would need to have in order for its chemical potential at the given temperature to be the same as the chemical potential of the real gas (see point C in Fig. 7.6).
If the gas is an ideal gas, its fugacity is equal to its pressure.
To evaluate the fugacity of a real gas at a givenT andp, we must relate the chemical potential to the pressure–volume behavior. Let 0 be the chemical potential and f0 be the fugacity at the pressurep0 of interest; let00 be the chemical potential andf00 be the fugacity of the same gas at some low pressurep00(all at the same temperature). Then we use Eq. 7.8.5 to write0 ı(g)DRT ln.f0=pı/and00 ı(g)DRTln.f00=pı/, from which we obtain
0 00 DRTln f0
f00 (7.8.9)
CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES
7.8 CHEMICALPOTENTIAL ANDFUGACITY 184
By integrating dDVmdpfrom pressurep00to pressurep0, we obtain 0 00 D
Z 0 00
dD
Z p0 p00
Vmdp (7.8.10)
Equating the two expressions for0 00and dividing byRT gives ln f0
f00 D Z p0
p00
Vm
RT dp (7.8.11)
In principle, we could use the integral on the right side of Eq. 7.8.11 to evaluatef0by choosing the lower integration limit p00 to be such a low pressure that the gas behaves as an ideal gas and replacing f00 by p00. However, because the integrandVm=RT becomes very large at low pressure, the integral is difficult to evaluate. We avoid this difficulty by subtracting from the preceding equation the identity
ln p0 p00 D
Z p0 p00
dp
p (7.8.12)
which is simply the result of integrating the function1=pfromp00top0. The result is lnf0p00
f00p0 D Z p0
p00
Vm RT
1 p
dp (7.8.13)
Now we take the limit of both sides of Eq. 7.8.13 asp00approaches zero. In this limit, the gas at pressurep00approaches ideal-gas behavior,f00approachesp00, and the ratiof0p00=f00p0 approachesf0=p0:
lnf0 p0 D
Z p0 0
Vm RT
1 p
dp (7.8.14)
The integrand.Vm=RT 1=p/of this integral approaches zero at low pressure, making it feasible to evaluate the integral from experimental data.
Thefugacity coefficientof a gas is defined by defD f
p or f Dp (7.8.15)
(pure gas) The fugacity coefficient at pressurep0is then given by Eq. 7.8.14:
ln.p0/D Z p0
0
Vm RT
1 p
dp (7.8.16)
(pure gas, constantT) The isothermal behavior of real gases at low to moderate pressures (up to at least1bar) is usually adequately described by a two-term equation of state of the form given in Eq.
2.2.8:
Vm RT
p CB (7.8.17)
Thermodynamics and Chemistry, 2nd edition, version 7a © 2015 by Howard DeVoe. Latest version:www.chem.umd.edu/thermobook
CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES