Physical Chemistry week 5 Friday February 8, 2013 page 1
µi = µ+ν+ + µ-νμA = μ*A (T,P)+ RTln(γX,A χA )
γX,A is the mole fraction activity coefficient of the solvent
ai = γi𝜒i
want to express in terms of molality
m+ = molality of cation Mz+
m- = molality of anion Mzγ+ = molality scale activity coefficient
γ- = molality scale activity coefficient
γ+ m+
μ+ = μ°+ + RTln (
)
m°
m° =
γ- mμ- = μ°- + RTln ( ° )
m
γ∞
+
∞
γ∞
+ = γ- =1
1 mol
kg (solvent)
γ has no unit
m is molality
is gamma plus at infinite dilution
(the infinity means infinite dilution)
µi = µ+ν+ + µ-νγ+ m+
γ- mμi = ν+ μ°+ + ν+ RTln (
) + ν- μ°- + ν- RTln ( ° )
°
m
m
m+ ν+ m- νμi = ν+ μ°+ + ν- μ°- + RTln (γ+ ν+ γ- ν- ( ° ) ( ° ) )
m
m
Those individual quantities can’t be measured.
(γ+ )ν+ (γ- )ν- is measurable
Define: γ± : molality scale mean activity coefficient
(γ± )ν++ν- = (γ+ )ν+ (γ- )ν- for 1 or more electrolytes
Ex: BaCl2
1
(γ± )3 = (γ+ )1 (γ- )2 ⟹ γ± ((γ+ )1 (γ- )2 )3
m+ ν+ m- νμi =μ°i + RTln ((γ± )ν ( ° ) ( ° ) )
m
m
ν = ν+ + νmi =
ni
wA
i is solute w is mass of solvent in kilograms
Assume:
a. no ion pairing
Mν+𝜒ν- → ν+ (cations) + ν- (anions)
m+ = ν+mi
m- = ν-mi
m+ and m- are ionic molality
(m+ )ν+ (m- )ν- = (ν+ mi )ν+ (ν- mi )ν- = (ν+ )ν+ (ν- )ν- (mi )ν
Just like γ± ,
since ν= ν+ +ν-
(ν± )ν = (ν+ )ν+ (ν- )ν-
Ex: MgCl2
(ν± )3 = (1)1 (2)2
3
ν± = √4=1.587
m+ ν+ m- νmi ν
(γ± )ν ( ° ) ( ° ) = (ν± γ± ° )
m
m
m
μi = μ°i = νRTln (ν± γ±
mi
)
m°
any electrolyte, no ion pairing, strong electrolytes
Non ideal gas mixtures
Fugacity (fi) = effective pressure in a mixture of gasses
fi
= ai
P°
fi has unit of pressure
fi is intensive P°=1bar
For ideal gas mixture:
Pi
°
μid
i = μi + RTln ( ° )
P
For nonideal gas mixture:
fi
μi = μ°i + RTln ( ° )
P
fi plays the same role in real mixture that partial pressure did in ideal mixture
As P→0 in nonideal gas mixture fi →Pi
fugacity approaches partial pressure
φ is fugacity coefficient, letter phi
ϕi =
fi = φi𝜒iP
𝜒 is mole fraction
fi
⟹ fi = ϕi Pi
Pi
Pi = χi P (always valid)
P is total pressure
Consider a graph with chemical potential of solute i(µi) on the vertical axis and pressure on the
horizontal axis. On the graph is a curve that starts near the lower left and curves upward and right. This
is the ideal gas mixture curve. A second curve starts near the lower left close to the first curve but
doesn’t curve upward as much as first, then crosses higher. This second curve is for a nonideal gas
mixture. Where the nonideal gas mixture has lower chemical potential than the ideal gas mixture is
where attractions are dominant and f<P. Where the nonideal gas mixture has higher chemical potential
than the ideal gas mixture is where repulsions are dominant and f>P.
end of chapter 10/starting chapter 12
A. colligative properties
Colligative means bound together.
1. vapor pressure lowering
2. freezing point depression
3. boiling point elevation
4. osmotic pressure
Consider the phase diagram for pure H2O. If some solute is added, the curve representing liquid-gas
equilibrium will go down a little, causing the boiling point elevation and vapor pressure lowering. Also,
the curve representing solid-liquid equilibrium will go down a little, causing freezing point depression.
Osmotic pressure can’t be shown on this type of diagram.
Adding a solute to a solvent makes the chemical potential of the solvent drop.
∂μA
(
)
∂χA T,P,n
>0
i≠A
μA = μ*A (T,P)+ RTlnχA
𝜒A = 1 for pure solvent 𝜒A < 1 after adding solute
I vapor pressure lowering
Assume nonvolatile solute
Psoln = PA = γA χA PA* for any solution
γA is solvent activity coefficient
a = γA𝜒A
γA = 1 for nonelectrolyte
∆P= Psoln - PA* = γA χA PA* -PA* =(γA χA -1)PA*
∆P=(γA χA -1)PA*
γA =1 for ideal or ideally dilute solution
For ideally dilute:
∆P=(γA χA -1)PA* = -χB PA* B is solute
Consider a graph with ΔP on the vertical axis and mole fraction of the solute on the horizontal axis. A
dotted line from the top left to the bottom right with slope = PA* is for an ideal or ideally dilute solution.
A curve from the top left but come down faster than the first line represents a real solution. The
departure from ideal is based on the number of solute particles, not their identity, just like with
electrolytes.