Equilibrium and Kinetics

Chapter 2

In the last lecture we used the mechanicalAnalogy to understand the concept ofStability and metastability

Recap

metastable

unstable

stable

Activation barrier

Fig. 2.2Recap

P.E

Configuration

Mechanical push to overcome activation barrier

System automaticallyattains the stable state

Recap

If we want to transform the Local Minimum - METASTABLE to Global Minimum - Most STABLE then we have to overcome the activation barrier (could be by mechanical push, thermal activation)

Thermodynamic functions

U = internal energy

Courtsey: H. Bhadhesia

At constant pressure

Courtsey: H. Bhadhesia

This expression can also be expressed as: U = Uo + dtCt

o

v

Sum of internal energy and external energy

For solids and liquid the PV term is negligible

Courtsey: H. Bhadhesia

dtCt

o

pThe above expression can also be expressed as: H = Ho +

Courtsey: H. Bhadhesia

P

Courtsey: H. Bhadhesia

Entropy

Courtsey: H. Bhadhesia

Courtsey: H. Bhadhesia

How do you measure the entropy?

Gibbs Free Energy

TSHG

Condition for equilibrium

≡ minimization of G

Local minimum ≡ metastable equilibrium

Global minimum ≡ stable equilibrium

(2.6)

G = GfinalGinitial

G = 0 reversible change

G < 0 irreversible or spontaneous change

G > 0 impossible

(2.7)

(2.8)

The variation of G with temperature

Atomic

or

statistical

interpretation of entropy

The entropy of a system can be defined by two components:

Thermal:

Configurational: WkS ln

Entropy

Boltzmann’s Epitaph

WkS lnW is the number of microstates corresponding to a given macrostate

(2.5)

N=16, n=8, W=12,870

)!(!

!

nNn

NCW n

N

(2.9)

Stirling’s Approximation

nnnn ln!ln(2.11)

If n>>>1

WkS ln

)!(!

!ln

nNn

Nk

)]ln()(lnln[ nNnNnnNNk

(2.10)

(2.12)

KINETICS: Arrhenius equationSvante Augustus

Arrhenius

1859-1927

Nobel 1903

RT

QArate exp

(2.15)

Rate of a chemical reaction varies with temperature

R

Qslope

RT

QArate exp

ln (rate)

T

1

Fig. 2.4

Arrhenius plot

Thermal energy

Average thermal energy per atom per mode of oscillation is kT

Average thermal energy per mole of atoms per mode of oscillation is NkT=RT

(2.13)

Maxwell-Boltzmann Distribution

kT

E

N

nexp

Fraction of atoms having an energy E

at temperature T

(2.14)