270
6 The Thermodynamics of Solutions
which is the same as
d ln(γ 2 ) dφ +
φ − 1
m 2
dm 2
(6.4-16)
If data for the osmotic coefficient are available over this molality range, Eq. (6.4-16)
can be integrated from m 2 0 to m 2 m
2 , a particular value of m 2 :
m 2 m
2
m 2 0
d ln(γ 2 )
m 2 m
2
m 2 0
dφ +
m
2
0
φ − 1
m 2
dm 2
(6.4-17)
The integral on the left-hand side of this equation yields zero at its lower limit, since
the activity coefficient approaches unity as m 2 approaches 0. It can be shown that φ
approaches unity as m 2 approaches zero, so that the integral converges and we can
write
ln(γ 2 (m
2 )) φ(m
2 ) − 1 +
m
2
0
φ − 1
m 2
dm 2
(6.4-18)
The integral is approximated numerically from the data, giving the value of γ 2 at m
2 .
Exercise 6.16
Using the relation x 1 1 − x 2 1 − n 2 /n 1 at high dilution, show that φ approaches 1 as m 2
approaches 0.
The Debye–Hückel Theory
This is a theory for the activity coefficients of dilute electrolyte solutes. Ions exert large
forces on each other, and these forces are called long-range forces because they act
over large distances compared with other intermolecular forces. The electrostatic force
on a charge Q 1 due to a charge Q 2 is given by Coulomb’s law,
Coulomb’s law is named for Charles
Augustin de Coulomb, 1736–1806, the
French physicist who discovered the
law.
F 12 e r
Q 1 Q 2
4πεr 2
12
(6.4-19)
where r 12 is the distance between the charges and where the unit vector e r points from
object 1 toward object 2. If the two charges have the same sign the force is a repulsion,
and if they have opposite signs the force is an attraction. The quantity ε is the permittivity
of the medium between the charges. The permittivity of a vacuum is denoted by ε 0
and is equal to 8.854519 × 10 −12 C 2 N −1 m −2 . The ratio of the permittivity of a given
material to that of a vacuum is called the dielectric constant and is denoted by ε rel :
ε rel
ε(material)
ε 0
(6.4-20)
The dielectric constant of water is larger than that of most substances and at 25 ◦ C is
equal to 78.54. Even though the electrostatic forces between ions in an aqueous solution
are weaker than those in a vacuum by the reciprocal of this factor, the electrostatic forces
between ions in a solution are strong compared with other intermolecular forces, and
they act over a large distance compared to other intermolecular forces.
6 The Thermodynamics of Solutions
which is the same as
d ln(γ 2 ) dφ +
φ − 1
m 2
dm 2
(6.4-16)
If data for the osmotic coefficient are available over this molality range, Eq. (6.4-16)
can be integrated from m 2 0 to m 2 m
2 , a particular value of m 2 :
m 2 m
2
m 2 0
d ln(γ 2 )
m 2 m
2
m 2 0
dφ +
m
2
0
φ − 1
m 2
dm 2
(6.4-17)
The integral on the left-hand side of this equation yields zero at its lower limit, since
the activity coefficient approaches unity as m 2 approaches 0. It can be shown that φ
approaches unity as m 2 approaches zero, so that the integral converges and we can
write
ln(γ 2 (m
2 )) φ(m
2 ) − 1 +
m
2
0
φ − 1
m 2
dm 2
(6.4-18)
The integral is approximated numerically from the data, giving the value of γ 2 at m
2 .
Exercise 6.16
Using the relation x 1 1 − x 2 1 − n 2 /n 1 at high dilution, show that φ approaches 1 as m 2
approaches 0.
The Debye–Hückel Theory
This is a theory for the activity coefficients of dilute electrolyte solutes. Ions exert large
forces on each other, and these forces are called long-range forces because they act
over large distances compared with other intermolecular forces. The electrostatic force
on a charge Q 1 due to a charge Q 2 is given by Coulomb’s law,
Coulomb’s law is named for Charles
Augustin de Coulomb, 1736–1806, the
French physicist who discovered the
law.
F 12 e r
Q 1 Q 2
4πεr 2
12
(6.4-19)
where r 12 is the distance between the charges and where the unit vector e r points from
object 1 toward object 2. If the two charges have the same sign the force is a repulsion,
and if they have opposite signs the force is an attraction. The quantity ε is the permittivity
of the medium between the charges. The permittivity of a vacuum is denoted by ε 0
and is equal to 8.854519 × 10 −12 C 2 N −1 m −2 . The ratio of the permittivity of a given
material to that of a vacuum is called the dielectric constant and is denoted by ε rel :
ε rel
ε(material)
ε 0
(6.4-20)
The dielectric constant of water is larger than that of most substances and at 25 ◦ C is
equal to 78.54. Even though the electrostatic forces between ions in an aqueous solution
are weaker than those in a vacuum by the reciprocal of this factor, the electrostatic forces
between ions in a solution are strong compared with other intermolecular forces, and
they act over a large distance compared to other intermolecular forces.