Post on 08-Jul-2020
Statistical Molecular Thermodynamics
Christopher J. Cramer
Video 8.4
Maxwell Relations from G
Working With Gibbs Free Energy
PVTSUG +−=
using dU = TdS – PdV
compare with the formal derivative of G = G(P,T): dP
PGdT
TGdG
TP⎟⎠
⎞⎜⎝
⎛∂
∂+⎟
⎠
⎞⎜⎝
⎛∂
∂=
VdPPdVSdTTdSdUdG ++−−=differentiate
VdPSdTdG +−=
STG
P
−=⎟⎠
⎞⎜⎝
⎛∂
∂ VPG
T
=⎟⎠
⎞⎜⎝
⎛∂
∂and Thus
Equating Key Cross Derivatives
James Clerk Maxwell
€
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=V
€
∂G∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
= −Sand
€
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=V
€
∂∂T
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ =
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
€
∂G∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
= −S
€
∂∂P
∂G∂T⎛
⎝ ⎜
⎞
⎠ ⎟ = −
∂S∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
Another of many Maxwell relations
€
∂∂T
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ =
∂∂P
∂G∂T⎛
⎝ ⎜
⎞
⎠ ⎟ As equality of mixed
partial derivatives
€
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
= −∂S∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
Utility of The Maxwell Relation
€
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
= −∂S∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
From this Maxwell relation we can see how S changes with P given an equation of state
Integrate at constant T:
€
ΔS = −∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟
P1
P2∫P
dPNote that T is held
constant during integration over P
Get P dependence of S from P-V-T data.
Example: Ideal gas
€
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
=nRP
€
ΔS = −nR dPPP1
P2∫ = −nR ln P2P1
(isothermal)
Entropy of Ethane Again
€
ΔS = S T,P2( ) − S(P→ 0)id = −
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟
P id
P2∫P
dP (constant T)
If P1 is chosen to be so small that a gas behaves ideally (P 0),
Ethane at 400 K
€
S (P→ 0)id 246.45 J • mol−1 • K−1 at 1 bar from Q!( )[ ]
€
S T,P( ) = S (P→ 0)id −
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟
P id
P2∫P
dP
For real gases, i.e., those having no readily available, analytical equation of state, this requires data for how volume varies with temperature over a full range of pressures
Enthalpy Dependence on P Differentiating G = H – TS wrt P :
€
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=∂H∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
−T ∂S∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
(isothermal)
using
€
∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
= −∂S∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
Maxwell relation
and
€
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=V
previously derived
€
∂H∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=V −T ∂V∂T⎛
⎝ ⎜
⎞
⎠ ⎟ P
Ethane at 400 K
€
H (P→ 0)id 17.87 kJ • mol−1 from Q!( )[ ]
€
H T,P( ) = H (P→ 0)id + V −T ∂V
∂T⎛
⎝ ⎜
⎞
⎠ ⎟
P
⎡
⎣ ⎢
⎤
⎦ ⎥ P id
P2∫ dPFor real gases, i.e., those having no readily available, analytical equation of state, this again requires data for how volume varies with temperature over a full range of pressures
Pressure Dependence of G
Ideal gas example, V = nRT/P :
€
∂G∂P⎛
⎝ ⎜
⎞
⎠ ⎟ T
=V
€
ΔG = VdPP1
P2∫integrate (constant T)
€
ΔG = nRT 1PdP
P1
P2∫ = nRT ln P2P1
(constant T)
Compare this to a previous result for an ideal gas at constant T:
€
ΔS = nR lnV2V1
= −nR ln P2P1
As expected, ΔG = ΔH –TΔS is equal simply to –TΔS since ΔH = 0 at constant T for an ideal gas