We then obtain:
dS
dU
¼
1
ν
Á f
U
ν
ð4:210Þ
And integrated:
S ¼ f
U
ν
ð4:211Þ
That is, the entropy of a resonator vibrating in an arbitrary diathermic medium
depends only on the variable U/ν, containing besides this only universal constants.
This is the simplest form of Wien’s displacement law known to me.
§10. If we apply Wien’s displacement law in the latter form to Eq. (4.199) for the
entropy S, we then find that the energy element E must be proportional to the
frequency ν, thus:
ε ¼ hν
ð4:212Þ
And consequently:
S ¼ k 1 þ
U
hν
log 1 þ
U
hν
À
U
hν
log
U
hν
n
o
ð4:213Þ
Here h [Planck constant] and k [Boltzmann constant] are universal constants. By
substitution into Eq. (4.208) one obtains:
1
θ
¼
k
hν
log 1 þ
hν
U
ð4:214Þ
U ¼
hν
e
hν
kθ À 1
ð4:215Þ
Moreover, from Eq. (4.206) there then follows the energy distribution law sought
for:
u ¼
8πhν
3
c 3 Á
1
e
hν
kθ À 1
ð4:216Þ
On the other hand, by introducing the substitutions given in (4.203), in terms of
wavelength λ instead of the frequency:
E ¼
8πch
λ
5
1
e ch=kλθ À 1
½
Š
ð4:217Þ
188
4 Unified Mechanics Theory