Physical Chemistry Third Edition

25.3 The Probability Distribution and the Molecular Partition Function 1059

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Curve representing the

function exp( 2 h^2 nx^2 /8ma^2 )

Value of term

nx (quantum number)

Figure 25.2 A Graphical Representation of the Translational Partition Function

(Schematic).

in Exercise 25.14. The factorszyandzzare similar tozxexcept for the replacement

ofabyborc, so the translational partition function can be written as

ztr

(

2 πm

h^2 β

) 3 / 2

abc

(

2 πm

h^2 β

) 3 / 2

V (25.3-18)

whereVabc, the volume of the box containing the gas. The thermodynamic proper-

ties of a dilute gas are independent of the shape of the container in which it is confined,

so we will use the second version of Eq. (25.3-18).

The parameterβcan now be evaluated. We substitute Eq. (25.3-18) into Eq. (25.3-7),

assuming thatzelis equal to a constant:

U

N



(

∂

∂β

ln(ztrzel)

)

V



(

∂

∂β

ln(ztr)

)

V

+

(

∂

∂β

ln(zel)

)

V



(

∂

∂β

ln(ztr)

)

V

+ 0



(

∂

∂β

ln

((

2 πm

h^2 β

) 3 / 2

V

))

V

−

3

2

dln(1/β)

dβ



3

2

dln(β)

dβ



3

2 β

(25.3-19)

To make Eq. (25.3-19) agree with Eq. (25.3-8) we set

β

1

kBT

(25.3-20)

The translational partition function is now

ztr

(

2 πmkBT

h^2

) 3 / 2

V (any gaseous substance) (25.3-21)