Physical Chemistry Third Edition

1204 28 The Structure of Solids, Liquids, and Polymers

Since all microstates are assumed equally probable,Pis proportional toΩand we can

write a formula for the entropy change:

∆SS(stretched)−S(equilibrium)k[ln(P′)−ln(P)]

kB

∑

i

Niln

(

p′i

pi

)

NkB

∑

i

piln

(

p′i

pi

)

(28.9-18)

where we have used the fact thatpiNi/N.

We pretend thatx,y, andzrange continuously and replace the sum by an integral:

∆SNkB

∫

pln

(

p′

p

)

dxdydz (28.9-19)

where the integral is over all values ofx,y, andz. From Eq. (28.7-18) and Eq. (28.9-13),

ln

(

p′

p

)



3

2 na^2

[

−x^2 (α^2 −1)−(y^2 +z^2 )

(

1

α

− 1

)]

(28.9-20)

When Eq. (28.9-20) is substituted into Eq. (28.9-19), we obtain

∆S

3 NkB

2 na^2

[

−〈x^2 〉n(α^2 −1)−

(

〈y^2 〉n+〈z^2 〉n

)( 1

α

− 1

)]

−

NkB

2

(

α^2 +

2

α

− 3

)

(28.9-21)

where we have used Eq. (28.7-11) for the equilibrium value of〈x^2 〉n, which is also

equal to〈y^2 〉nand〈z^2 〉n. Using Eqs. (28.9-21) and (28.9-12), we can write an equation

of state for ideal rubber:

f−T

(

∂S

∂L

)

T

−

T

L 0

(

∂S

∂α

)

T



NkBT

L 0

(

α−

1

α^2

)

(28.9-22)

This equation of state agrees fairly well with experiment for values ofαno larger

than 3.^44

EXAMPLE28.16

Derive an expression for the reversible work done in stretching a piece of ideal rubber at

constant temperature.

Solution

Letα′be the final value of the extent of elongation.

dwfdL

NkBT

L

(

α−

1

α^2

)

dLNkBT

(

α−

1

α^2

)

dα

wNkBT

∫α′

1

(

α−

1

α^2

)

dα

NkBT

2

(

α′^2 +

2

α′

− 3

)

(^44) F. T. Wall,op. cit., Ch. 16 (note 37).