Statistical Molecular Thermodynamics

Christopher J. Cramer

Video 12.6

Temperature Dependence of K

Dependence of K on T

We can apply the Gibbs-Helmholtz equation (cf. video 8.7)

∂ΔG! /T∂T

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((P

= −ΔH!

T 2

( ) ( )TKRTTG Plnr −=Δ !And substitute

∂ lnKP T( )∂T

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=d lnKP T( )

dT=ΔrH!

RT 2van’t Hoff Equation

Another manifestation of Le Châtelier’s principle

Endothermic: 0r >Δ !H KP(T) increases with T.

Exothermic: 0r <Δ !H KP(T) decreases with T.

Integrating the van’t Hoff Equation d lnKP T( )

dT=ΔrH!

RT 2 lnKP T2( )KP T1( )

=ΔrH! T( )

RT 2T1

T2

∫ dTintegrate

Over small T ranges we can consider ΔrHo constant (T independent): ln

KP T2( )KP T1( )

= −ΔrH!

R1T2−1T1

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600oC

900oC slope =

RH !rΔ−

( ) ( ) ( ) ( )gOHgCOgCOgH 222 ++

Clausius-Clapeyron analog

ln P2P1= −

ΔvapHR

1T2−1T1

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(cf. video 9.6)

( ) ( )lXgX

Self-assessment Is the water gas shift reaction as written below

exothermic or endothermic?

600oC

900oC slope =

RH !rΔ−

( ) ( ) ( ) ( )gOHgCOgCOgH 222 ++

Self-assessment Explained Is the water gas shift reaction as written below

exothermic or endothermic?

600oC

900oC slope =

RH !rΔ−

( ) ( ) ( ) ( )gOHgCOgCOgH 222 ++

Endothermic. One can see this either by noting that KP is larger at higher temperature, or by noting that the slope being negative implies ΔrHo must be positive.

Integrating the van’t Hoff Equation

lnKP T2( )KP T1( )

=ΔrH! T( )

RT 2T1

T2

∫ dT lnKP T( ) = lnKP T1( )+ΔrH! "T( )

R "T 2T1

T

∫ d "T

If we cannot consider ΔrH(T) constant over the T range,

ΔrH! T2( ) = ΔrH! T1( )+ CP

! T( )T1

T2∫ dT

We do know how to calculate the T dependence of ΔH,

where CP(T) is often reported as a series in T from an experimental fit,

CP! T( ) = A+BT +CT 2 +"( ) ( ) ( )gNHgN

21gH

23

322 +

the van’t Hoff plot is no longer linear

Next: Determining K from Q (the other Q)