Physical Chemistry Third Edition

644 14 Classical Mechanics and the Old Quantum Theory

EXAMPLE14.2

Show that the wavelength of maximum spectral radiant emittance is inversely proportional

to the absolute temperature and find the proportionality constant.

Solution

To find the maximum, we set the derivative of the function of Eq. (14.4-5) equal to zero:

dη

dλ

 2 πhc^2

(hc/λkB)Tehc/λkBT−5(ehc/λkBT−1)

λ^6 [ehc/λkBT−1]

 0

This expression can vanish only if the numerator vanishes, which is equivalent to

hc

λmaxkBT

 5

(

1 −e−hc/λmaxkBT

)

or,ifweletxhc/λmaxkBT, then

x 5

(

1 −e−x

)

This equation must be solved by numerical approximation. The result is thatx 4 .965, so

that

λmax

1

4. 965

hc

kBT



2. 898 × 10 −^3 mK

T

(14.4-6)

This result agrees with Wien’s law and with the experimental value of the proportionality

constant.

EXAMPLE14.3

Use the definite integral

∫∞

0

x^3

ex− 1

dx

π^4

15

(14.4-7)

to derive the Stefan–Boltzmann law, Eq. (14.4-1). Calculate the theoretical value of the

Stefan–Boltzmann constant.

Solution

The total energy flux is

∫∞

0

η(λ)dλ

∫∞

0

2 πhc^2

λ^5 (ehc/λkBT−1)

dλ

We make the variable change

x

hc

λkBT

so that

∫∞

0

2 πhc^2

λ^5 (ehc/λkBT−1)

dλ 2 πhc^2

(

kBT

hc

) 5 (

hc

kBT

)∫∞

0

x^3

ex− 1

dx