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