Statistical Molecular Thermodynamicspollux.chem.umn.edu/4501/Lectures/ThermoVid_8_03.pdfThermodynamics Christopher J. Cramer Video 8.3 Maxwell Relations from A Interrelated Thermodynamic
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Statistical Molecular Thermodynamics
Christopher J. Cramer
Video 8.3
Maxwell Relations from A
Interrelated Thermodynamic Quantities When you are not able to directly measure a given thermo-dynamic property, it is very useful to express it in terms of other properties.
for a reversible process PdVTdSdU −=
SdTPdVdA −−=
compare with the formal derivative of A=A(V,T): dT
TAdV
VAdA
VT⎟⎠
⎞⎜⎝
⎛∂
∂+⎟
⎠
⎞⎜⎝
⎛∂
∂=
Thus PVA
T
−=⎟⎠
⎞⎜⎝
⎛∂
∂ STA
V
−=⎟⎠
⎞⎜⎝
⎛∂
∂and
SdTTdSdUdA −−= (general)
Equating Key Cross Derivatives
James Clerk Maxwell
PVA
T
−=⎟⎠
⎞⎜⎝
⎛∂
∂ STA
V
−=⎟⎠
⎞⎜⎝
⎛∂
∂and
€
∂A∂V⎛
⎝ ⎜
⎞
⎠ ⎟ T
= −P
€
∂∂T
∂A∂V⎛
⎝ ⎜
⎞
⎠ ⎟ = −
∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟ V
€
∂A∂T⎛
⎝ ⎜
⎞
⎠ ⎟ V
= −S
€
∂∂V
∂A∂T⎛
⎝ ⎜
⎞
⎠ ⎟ = −
∂S∂V⎛
⎝ ⎜
⎞
⎠ ⎟ T
One of many Maxwell relations
€
∂∂T
∂A∂V⎛
⎝ ⎜
⎞
⎠ ⎟ =
∂∂V
∂A∂T⎛
⎝ ⎜
⎞
⎠ ⎟ As equality of mixed
partial derivatives
€
∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟ V
=∂S∂V⎛
⎝ ⎜
⎞
⎠ ⎟ T
Utility of a Maxwell Relation
€
∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟ V
=∂S∂V⎛
⎝ ⎜
⎞
⎠ ⎟ T
From this Maxwell relation we can determine how S changes with V given an equation of state
Integrate at constant T:
€
ΔS =∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟
V1
V2∫V
dVNote that T is held
constant during integration over V
Get V (or ρ) dependence of S from P-V-T data.
Example: Ideal gas
VnR
TP
V
=⎟⎠
⎞⎜⎝
⎛∂
∂
€
ΔS = nR dVVV1
V2∫ = nR lnV2V1
(isothermal)
(a previous result derived another way, cf. Video 6.2)
Entropy of Ethane
€
ΔS = S T ,V2( ) − S(ρ→0)id =
∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟ V id
V2∫V
dV (constant T )
If V1 is chosen to be so large that a gas behaves ideally (=Vid),
Ethane at 400 K €
S (P→ 0)id 246.45 J • mol−1 • K−1 at 1 bar from Q!( )[ ]
€
S T,V( ) = S (ρ→ 0)id +
∂P∂T⎛
⎝ ⎜
⎞
⎠ ⎟
V id
V2∫V
dV
For real gases, i.e., those having no readily available, analytical equation of state, this requires data for how pressure varies with temperature over a full range of volumes (or densities, since density is equal to V –1)
Internal Energy of Ethane Differentiating A = U – TS wrt V :
TTT VST
VU
VA
⎟⎠
⎞⎜⎝
⎛∂
∂−⎟
⎠
⎞⎜⎝
⎛∂
∂=⎟
⎠
⎞⎜⎝
⎛∂
∂(isothermal)
using TV V
STP
⎟⎠
⎞⎜⎝
⎛∂
∂=⎟
⎠
⎞⎜⎝
⎛∂
∂
Maxwell relation
and PVA
T
−=⎟⎠
⎞⎜⎝
⎛∂
∂
previously derived VT T
PTPVU
⎟⎠
⎞⎜⎝
⎛∂
∂+−=⎟
⎠
⎞⎜⎝
⎛∂
∂
Ethane at 400 K €
U (P→ 0)id 14.55 kJ • mol−1 from Q!( )[ ]
€
U T,V( ) = U (ρ→ 0)id + T ∂P
∂T⎛
⎝ ⎜
⎞
⎠ ⎟
V
− P⎡
⎣ ⎢
⎤
⎦ ⎥ V id
V2∫ dV
For real gases, i.e., those having no readily available, analytical equation of state, this again requires data for how pressure varies with temperature over a full range of volumes (although the plot here is over pressures, which are obviously readily measured for each volume)
Volume Dependence of A
Ideal gas example, P = nRT/V :
PVA
T
−=⎟⎠
⎞⎜⎝
⎛∂
∂
€
ΔA = − PdVV1
V2∫integrate (constant T)
€
ΔA = −nRT 1VdV
V1
V2∫ = −nRT lnV2V1
(constant T)
Compare this to a previous result for an ideal gas at constant T:
1
2lnVVnRS =Δ
As expected, ΔΑ = ΔU –TΔS is equal simply to –TΔS since ΔU = 0 at constant T for an ideal gas
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