Lecture 18
• The solvent activity
• The solute activity
Ch. 5 Simple Mixtures
where the fugacity (f) is an effective pressure of the real gas and is the dimensionless fugacity coefficient .
• Now we consider how to take into account deviations of real solutions from ideal behavior.
• In chapter 3, the imperfections of real gases are taken into account by introducing a quantity called fugacity.
• Here we introduce the concept of activity to express the real solutions.
• For a perfect gas, m mo o
p RT p
G p
G ( ) ln
• For real gases,
o o
m
m p
RT f G
p
G ( ) ln f p
Chapter 3
• The general form of the chemical potential of a real or ideal solvent is given by:
*
ln *A A A
A p
RT p l
l
where pA* is the vapor pressure of pure A and pA is the vapor pressure of A when it is a component of a solution.
• For an ideal solution (of which the solvent and solute obey the Raoult’s law),
A
AA l * l RT ln x
For the ideal solvent:
* A A
A p
x p
Raoult’s law
• For a real solution (of which the solvent and solute does not obey the Raoult’s law),
A
AA l * l RT ln a For the real solvent:
where the quantity aA is the activity, a kind of effective mole fraction of the real solvent A.
By comparing two equations,
* ln *
A A A
A p
RT p l
l
* A A
A p
a p
The activity can be determined by measuring pA and pA*.
• Because all solvents obey Raoult’s law (pA/pA* = xA), when xB0 (that is, xA1),
1 as
A A
A x x
a
where is the activity coefficient.
• By definition, at all T and p.
1 as
A A
A x x
a
A A
A x
a
1 as
1
A
A x
• The chemical potential of the solvent is:
A
A AA l l RT x RT
* ln ln
A
A A
A AA l l RT a l RT x RT
* ln * ln ln Ideal solvent
Correction for real solvent and
• The solute in an ideal-dilute solution (the solute and solvent obeying the Henry’s law and Raoult’s law, respectively),
• The chemical potential of B in the ideal-dilute solution,
B B
B K x
p
BB B B
B B B B
B B B
B
x p RT
RT K l
p x RT K
p l RT p
l l
ln ln
ln ln
*
*
*
*
*
*
The 2nd term may be combined with the 1st to give a new standard chemical potential ( ) of B:
*
ln *B B B
o
B p
RT K l
l
lo
B
* ln *
B B B
B p
RT p l
l
Bo
BB l l RT ln x
*
ln *B B B
o
B p
RT K l
l
• For ideal-dilute solutes,
If the solution is ideal, (the solute also obeys the Raoult’s law, KB = pB* and . Bo
l B*
l• The standard state ( ) can be regarded as one in which the pure solute is giving
rise to a vapor pressure of magnitude of KB.
• The standard state is a hypothetical state in which the solute is pure, but behaving as though it still obeyed Henry’s law (the
extrapolation of behavior at lower concentrations).
o
*
ln *B B B
o
B p
RT K l
l
• Now we permit the solute to deviate from ideal-dilute behavior (Henry’s law .
• For real solutes, we introduce aB in place of xB,
Bo
BB l l RT ln a
*
ln *B B B
o
B p
RT K l
l
The standard state remains unchanged in this equation, and all the deviations from ideality are captured in the activity aB. The activity of solute B is: cf) *
A A
A p
a p
B B
B K
a p
As for the solvent, the activity coefficient is introduced as:
B B
B K x
p
and at all T and p.
• Now all the deviations from ideality (Henry’s law) are captured in the activity coefficient (B).
• Because the solute obeys Henry’s law as its concentration goes to zero,
*
ln *B B B
o
B p
RT K l
l
BB B B
B B
B B B
B B
B B B
B B
o B B
p RT RT p
l
p RT x RT K
l
RT x
p RT RT K
l
RT x
RT l
l
ln ln
ln ln
ln ln
ln
ln ln
*
*
*
*
*
*
0 as
x x 0 a
as
1
x
Bo
BB l l RT lna
• The selection of a standard state is entirely arbitrary.
• In chemistry, compositions are often expressed as molalities (b) in place of mole fractions.
Bo
B oB l l RT lnb b
Note that here has a different value from the standard values introduced earlier.
lo
B
• According to this definition, the chemical potential of the solute (B) has its standard value when the molality (b) is equal to 1 mol/kg ( )
• Note that as bB 0, B ; that is, as the solution becomes diluted, so the solute becomes increasingly stabilized. difficult to remove the last trace of a solute from a solution.
lo
B
bo
• Now, as before, we introduce deviations from ideality by
introducing a dimensionless activity (aB) and its coefficient (B).
0 as
1
where
B Bo B B
B b
b
a b at all T and p.
Bo
BB l l RT ln a
• The standard state remains unchanged in this stage, and all deviations from ideality are captured in the activity coefficient (B).
Bo
B o BB l l RT b b RT
ln ln
• In dilute solutions, . Therefore nB nA
A B B
B n
n n
x n
A m A
B
m A
m A
B B
B n
M n
M M n
b n ,
, ,
kg kg 1
1
B A m
A B B
B b M
n n n
x n ,
m A B
B b M
x ,
• Because the Henry’s law is obeyed in dilute solutions,
Bo
BB l l RT ln x
*
ln *B B B
o
B p
RT K l
l
o B
o
m A o o o B
B
m A B o
B B
RT b M
b RT l
M b b
RT b l
M b RT l
l
ln ln
ln
ln
, ,
l o
l RT ln bB
New standard state
• The conventional standard state of hydrogen ions (unit activity, corresponding to pH = 0, ) is not appropriate to normal biological conditions.
• In biochemistry, it is common to adopt the biological standard state, in which pH = 7 (an activity of 10-7, neutral solution).
• The corresponding standard thermodynamic functions are denoted as , , , .
• The relation between the thermodynamic and biological standard values of the chemical potential of hydrogen ions:
aH O
3
log pH
G H S
ln10
pHln
Ho RT aH Ho RT
H
when pH = 7, H Ho 7RT ln10
• At 298 K, 7RT ln10 ~ 40 kJ/mol
• Reading: page 162 ~ 169
• Problem set (Ch. 4): Discussion 4.4 4.2a, 4.8a, 4.10a Due dates: 3215 (May 6)
3996 (May 7)
• The 2nd exam: May 23 (Fri), 19:00, B566, Ch. 3~5
