Page 18 - Partition & Adsorption of Organic Contaminants in Environmental Systems

ACTIVITY OF A SUBSTANCE 9

and at fixed T,

i Ú
Ú dm = V dP (1.30)
i
m i - m i ∞ = P P∞ Ú i VdP (1.31)

where m i is the chemical potential at P and m i ° is the chemical potential at a
reference pressure P°. For solid or liquid substances, V i does not vary much
with P and may be treated as constant; hence,

(
m i = m i ∞ + i VP - P∞) (1.32)
For ideal gases, V i = RT/P i , where R is the gas constant (8.31J/mol·K); one gets

)
m i Ú
m i -∞ = P ( RT i P dP i (1.33)
P∞
or
ln
m i =∞ + RT ( i P P∞) (1.34)
m i
i
In our treatment of gases, it is common and convenient to set P i ° = 1 atmos-
phere (atm) as the reference state of a gas at temperature T. In this case, m i °
is the reference chemical potential of gas i at 1atm pressure and temperature
T, and m i is the chemical potential of gas i at P i (atm) and T. With P i ° = 1atm,
Eq. (1.34) is thus reduced to

m i =∞ + RT ln i P (1.35)
m i
If P i ° π 1atm, P i in Eq. (1.35) is simply the dimensionless ratio of P i to P i °. If
the gas behaves nonideally (when under high pressure), m i =m i ° + RTln f i is
used instead, where f i is the fugacity of vapor i (i.e., the vapor pressure cor-
rected for deviation from the ideal-gas law). From Eq. (1.35), the differential
change in m i with P i at constant temperature is therefore

dm= RTdln P i = RTdln f i (1.36)
i

1.8 ACTIVITY OF A SUBSTANCE
By Raoult’s law convention, the activity of a substance at temperature T is the
ratio of its fugacity or vapor pressure to that of the fugacity or vapor pressure
of the substance at some reference state at T, that is,

a i = f f i ∞ P P i ∞ (1.37)
i
i