6.3 Activity and Activity Coefficients
261
where we attach the superscript (I) to specify that we are using convention I.
Since µ
◦(I)
i
is the chemical potential of the pure liquid, µ ∗
i , it is equal to the chemical potential of the gaseous substance at a partial pressure equal to the equilibrium
vapor pressure P ∗
i :
µ
∗
i µ
◦(I)
i
µ
◦(g)
i
+ RT ln
P ∗
i
P ◦
(6.3-20)
We are ignoring a small correction in µ
◦(I)
i
due to the fact that the standard state is
at P P ◦ 1 bar, whereas µ ∗
i pertains to pressure P ∗
i . The relation of Eq. (6.3-19)
becomes
µ
◦(g)
i
+ RT ln
P ∗
i
P ◦
+ RT lna
(I)
i µ
◦(g)
i
+ RT ln
P i
P ◦
(6.3-21)
Canceling equal terms and taking antilogarithms in Eq. (6.3-21), we obtain
P i P
∗
i a
(I)
i
(6.3-22)
which resembles Raoult’s law except that the activity a
(I)
i occurs instead of the mole
reaction x i . The activity acts as an “effective” mole fraction in determining the partial
vapor pressure of the substance. Equation (6.3-22) is equivalent to
a
(I)
i
P i
P ∗
i
(6.3-23)
The activity coefficient in convention I is defined as the ratio of the activity to the
mole fraction:
γ
(I)
i
a
(I)
i
x i
(definition of the activity coefficient)
(6.3-24)
It is equal to the actual vapor pressure divided by the value of the vapor pressure that
is predicted by Raoult’s law:
γ
(I)
i
P i /P ◦
P i,ideal /P ◦
P i
P ∗
i x i
(6.3-25)
The activity coefficient specifies how the substance deviates from Raoult’s law. If
γ i > 1, the partial vapor pressure of substance i is higher than predicted by Raoult’s
law, and if γ i < 1, it is lower than predicted by Raoult’s law.
E X A M P L E 6.11
Find the value of the activity and the activity coefficient of 2,2,4-trimethyl pentane (component 2) in ethanol at 25 ◦ C at a mole fraction of 0.2748, according to convention I. The
partial vapor pressure is equal to 48.31 torr and the vapor pressure of the pure liquid is equal
to 59.03 torr.
261
where we attach the superscript (I) to specify that we are using convention I.
Since µ
◦(I)
i
is the chemical potential of the pure liquid, µ ∗
i , it is equal to the chemical potential of the gaseous substance at a partial pressure equal to the equilibrium
vapor pressure P ∗
i :
µ
∗
i µ
◦(I)
i
µ
◦(g)
i
+ RT ln
P ∗
i
P ◦
(6.3-20)
We are ignoring a small correction in µ
◦(I)
i
due to the fact that the standard state is
at P P ◦ 1 bar, whereas µ ∗
i pertains to pressure P ∗
i . The relation of Eq. (6.3-19)
becomes
µ
◦(g)
i
+ RT ln
P ∗
i
P ◦
+ RT lna
(I)
i µ
◦(g)
i
+ RT ln
P i
P ◦
(6.3-21)
Canceling equal terms and taking antilogarithms in Eq. (6.3-21), we obtain
P i P
∗
i a
(I)
i
(6.3-22)
which resembles Raoult’s law except that the activity a
(I)
i occurs instead of the mole
reaction x i . The activity acts as an “effective” mole fraction in determining the partial
vapor pressure of the substance. Equation (6.3-22) is equivalent to
a
(I)
i
P i
P ∗
i
(6.3-23)
The activity coefficient in convention I is defined as the ratio of the activity to the
mole fraction:
γ
(I)
i
a
(I)
i
x i
(definition of the activity coefficient)
(6.3-24)
It is equal to the actual vapor pressure divided by the value of the vapor pressure that
is predicted by Raoult’s law:
γ
(I)
i
P i /P ◦
P i,ideal /P ◦
P i
P ∗
i x i
(6.3-25)
The activity coefficient specifies how the substance deviates from Raoult’s law. If
γ i > 1, the partial vapor pressure of substance i is higher than predicted by Raoult’s
law, and if γ i < 1, it is lower than predicted by Raoult’s law.
E X A M P L E 6.11
Find the value of the activity and the activity coefficient of 2,2,4-trimethyl pentane (component 2) in ethanol at 25 ◦ C at a mole fraction of 0.2748, according to convention I. The
partial vapor pressure is equal to 48.31 torr and the vapor pressure of the pure liquid is equal
to 59.03 torr.