Atkins’ Physical ChemistryEighth Edition

Chapter 3 – Lecture 3The Second Law

Copyright © 2006 by Peter Atkins and Julio de Paula

Peter Atkins • Julio de Paula

Concentrating on the System

• Consider a system in thermal equilibrium with itssurroundings at constant T

• When heat is transferred between system and surroundings,the Clausius inequality reads:

• We can use this to develop expressions for sponteneityunder constant volume and under constant pressure

0T

dqdS

(a) Constant volume conditions

• dqV = dU

• so becomes

• which rearranges to

(b) Constant pressure conditions

• dqP = dH

• so becomes

• which rearranges to

0T

dqdS V 0

T

dUdS

dUTdS

0T

dqdS P 0

T

dHdS

dHTdS

Concentrating on the System

• We introduce two new thermo functions:

• Helmholtz energy (constant V)

• A ≡ U – TS → dA = dU - TdS → ΔAT,V = ΔU - TΔS

• Gibbs energy (constant P)

• G ≡ H – TS → dG = dH - TdS → ΔGT,P = ΔH - TΔS

• Criterion for a spontaneous process:Criterion for a spontaneous process:

• dGdGT,PT,P ≤ O ≤ O based on the system alonebased on the system alone

• “ “Free energy”: Free energy”:

• max additional (non-PV) work obtainable max additional (non-PV) work obtainable

dwdwadd,maxadd,max = dG = dG or

wwadd,maxadd,max = = ΔΔGG

The Gibbs Energy

Standard Molar Gibbs EnergiesStandard Molar Gibbs Energies

rxnrxnrxn STHG

where: )tstanreac(Hm)products(HnΔH ffrxn

and: )tstanreac(mS)products(nSΔS mmrxn

Or:

)tstanreac(Gm)products(GnΔG ffrxn

Combining First and Second LawsCombining First and Second Laws

Major objectives: Major objectives:

• Find relationships between various thermo propertiesFind relationships between various thermo properties

• Derive expressions for G(T,P)Derive expressions for G(T,P)

The Fundamental Equation

• From 1From 1stst Law: Law: dU = dq + dw

• From 2nd Law:

• Substituting:Substituting: dw = -PΔV and dq = TdS

• Gives:Gives:

dU = TdS - PdV

T

dqdS

• dU = TdS – PdV indicates that U = f (S,T)

• So the full differential of U:

• Indicates that:

• The first partial is a pure thermo definition of temperature

Properties of the Internal Energy

dVV

UdS

S

UdU

SV

TS

U

V

PV

U

S

A Math Lesson

• An infinitesimal change in the function f(x,y) is:

df = M dx + N dy

where M and N are functions of x and y

• Criterion for df to be exact is:

• Euler Reciprocity relationship (p 968)

yxx

N

y

M

A Math Lesson (cont’d)

• dU = TdS – PdV must pass this test if it indeed is exact

• Applying the Euler Reciprocity:

• This equation is one of the four Maxwell relations

yxx

N

y

M

df = M dx + N dy

VS S

P

V

T

The Maxwell RelationsTable 3.5

VS SP

VT

PS SV

PT

TV VS

TP

TP PS

TV

Natural Variables

U(S,V)

H(S,P)

A(V,T)

G(P,T)

Relation

Internal pressure

Beginning with dU = TdS – PdV

Dividing both sides by dV and imposing constant T

TT V

U

π

Variation of Internal Energy with Volume

STVT V

U

V

S

S

U

V

U

TS

U

V

PV

U

S

P

V

ST

TT

π

A thermo equation of state

Maxwell #3 here...

Properties of Gibbs Energy

• dG = dH – TdS

• So: G(H,T,S) and expanding:

• dG = dH – d(TS)

= dH – TdS – SdT

= dU + d(PV) – TdS – SdT

= dU + PdV + VdP – TdS – SdT

= TdS - PdV + PdV + VdP – TdS – SdT

• dG = VdP - SdT

dG = VdP - SdT

Fig 3.18 Variation of Gibbs energy with T and with P

dTT

GdP

P

GdG

PT

VP

G

T

ST

G

P

Fig 3.19 Variation of Gibbs energy with T depends on S

ST

G

P

T

HG

T

G

P

2P T

H

T

G

T

Fig 3.20 Variation of Gibbs energy with P depends on V

VP

G

T

oomm

P

Pln RTG(P)G