Physical Chemistry week 4 Monday January 28, 2013 page 1
∆Smix =-nA Rln
VA*
VB*
- nB Rln
V
V
ideal solution mixture
Since A and B form an ideal solution:
*
*
Vm,A
= Vm,B
*
VA* = nA Vm,A
∆Smix = -nA Rln
∆Smix = -R(nA lnχA +nB lnχB )
*
VB = nB Vm,B
*
nA Vm,A
*
(nA + nB )Vm,A
- nB Rln
V= VA* + VB*
*
nB Vm,B
*
(nA +nB )Vm,B
ideal solution, const T and P, similar to mixing ideal gasses
ΔHmix = 0 for ideal solutions
ΔGmix = ΔHmix – TΔSmix = 0 – T(-nARlnχA – nBRlnχB) = RT(nAlnχA + nBRlnχB)
∆Gmix =RT ∑ ni lnχi
i
R = N * k = Avagadro’s number * Boltzmann’s constant
ΔGmix = G – G*
G= ∑ ni G̅ i
G̅ i is partial molar Gibbs energy
i
*
G* = ∑ ni Gi,m
μi = G̅ i
*
μ*i = Gi,m
i
∆Gmix =G- G* = ∑ ni μi - ∑ ni μ*i =RT ∑ ni lnχi
i
RT ∑ ni lnχi = ∑ ni μi - ∑ ni μ*i
i
∑ ni μi = ∑ ni μ*i + RT ∑ ni lnχi = ∑ ni (μ*i +RTlnχi )
i
i
i
The only solution is: μi = μ*i + RTlnχi
μi = μ*i (T,P)+ RTlnχi ideal solution law, thermodynamic definition of ideal solution
PV = nRT was ideal gas law, better as P→0 or ρ→0
RTlnχi < 0 normally negative
so μi < μ*i
μ*i is constant
As χi →0, µi →-∞ at constant T,P
Thermodynamic properties of ideal solutions
a. standard states
For gas: pure ideal gas, P°=1 bar, T = Tinterest
For liquid solutions: pure liquid(before mixing), T = Tsoln, P=Psoln
For solid solutions: pure solid, T=Tsoln, P=Psoln
μ°i is chemical potential of i at standard state
1
∴ μ°i = μ*i (T,P)
° means standard state *means pure substance
2
μi = μ*i + RTlnχi
Statements 1 and 2 are definition of ideal solution
∆Gmix =RT ∑ ni lnχi
pressure independent
i
Since 0 < χi < 1 then lnχi <0 and ΔGmix < 0 so mixing is irreversible(spontaneous)
dG = VdP – SdT dT = 0
dG = VdP
∂G
V= ( )
∂P T
∆Vmix = (
∂∆Gmix
)
∂P T,ni
Since ∆Gmix is independent of P:
∴ ∆Vmix =0
constant T and P
∂∆Gmix
∂
∆Smix = - (
) = - (RT ∑ ni lnχi )
∂T P,ni
∂T
i
∂∆Gmix
(
) =0
∂P T,ni
= -R ∑ ni lnχi
ideal solution
P,ni
Since lnχi < 0, ΔSmix > 0 usually
Exception: diethyl amine + water: ΔSmix<0, not really ideal since water is much smaller molecule
ΔGmix = ΔHmix – TΔSmix so ΔHmix = ΔGmix + TΔSmix
∆Hmix =RT ∑ ni lnχi -RT ∑ ni ln χi =0
ΔHmix = 0 for ideal solutions
When TΔS is at max, ΔG is at min
On graph of mixing quantities versions mole fraction of substance B, the slope is steepest at the edges,
which is why it’s hard to purify a substance out to 99.9999% purity. As an example, it’s not worth the
trouble to extract gold that’s dissolved in the ocean.
For ideal solutions: ΔHmix = 0 ΔVmix = 0
If a solution has ΔHmix = 0 and ΔVmix = 0 that’s not enough to conclude that it’s an ideal solution.
Immiscible doesn’t count as a solution.
For a “real” solution, ΔGmix <0 (always!)