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5.60 Thermodynamics & Kinetics

Spring 2008

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5.60 Spring 2008 Lecture #10 page 1

Entropy and Disorder

Mixing of ideal gases at constant Tand pnA A (g, VA, T) + nB A (g, VB, T) = n(A + B) (g, V, T)

nVA

nVB

nVB

spontaneous +n =nA B A Bmixing V V += A

To calculate Smix , we need to find a reversible path between

the two states.

constant T A B

A + B

piston piston back to initial statepermeable permeable

to A only to B only

Sdemix = Smix function of stateFor demixing process

U =0 q = w =pdV +pdVrev rev A A B Bwork of compression of each gas

S = dqrev = VA pAdVA + VB pBdVB =nRlnVA +n RlnVBdemix T V T V T A V B VPut in terms of mole fractions XA =nA XB =nBn n

Ideal gas XA =VA XB =VBV V

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5.60 Spring 2008 Lecture #10 page 2

Sdemix = nR XAlnXA XB ln B[ + X ] Smix = nR XAlnXA XB ln B [ + X ]

Since A, < 1 Smix > 0 mixing is always spontaneousX XB

The mixed state is more disordered or random than the

demixed state.Smixed > Sdemixed

This is a general result

Entropy is a measure of the disorder of a system

For an isolated system (or the universe)

>S 0 Spontaneous, increased randomness

=S 0 Reversible, no change in disorder

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5.60 Spring 2008 Lecture #10 page 3

(b) Heating (or cooling) at constant VA (T1, V) = A (T2, V)

S T2 C dT if CV is = qrev = T V = C lnT2T 1 T T-independent V T1[Note S > 0 if T2 >T1 ]

(c) Reversible phase change at constant Tand pe.g. H2O (l, 100C, 1 bar) = H2O (g, 100C, 1 bar)

qp = Hvap

vap ( ) qpvap = Hvap (Tb = boiling Temp at 1 bar)S 100 C =Tb Tb

(d) Irreversible phase change at constant Tand pe.g. H

2O (l, -10C, 1 bar) = H

2O (s, -10C, 1 bar)

This is spontaneous and irreversible. We need to find a reversible path between the two states

to calculate S.

irreversible

H2O (l, -10C, 1 bar) = H2O (s, -10C, 1 bar)qrev = Cp( )A dT qrev = Cp (s)dT

reversible

H2O (l, 0C, 1 bar) H2O (s, 0C, 1 bar)=revqp = Hfus

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5.60 Spring 2008 Lecture #10 page 4

S S + = Sheating + fus Scoolingfus p( )dT Hfus T1 Cp( )s dTT C A= T1 T + Tfus + Tfus T

= T

+ T1 C A C s T SHfus Tfus

p( ) p( ) dT

=Hfus C ( ) ( ) lnTfus if Cp values are T-independentS

T+ p A Cp s T1