Results · §4 · dilute radiation and the volume law

Page 139

How the entropy of monochromatic radiation depends on its volume

Taking as its basis Wien's law, which experiment confirms for large $\nu/T$ , the paper finds that the entropy of radiation of energy $E$ between $\nu$ and $\nu + d\nu$ depends on the volume it occupies as $S - S_0 = \frac{E}{\beta\nu}\lg\left(\frac{v}{v_0}\right)$ , a result valid only within certain limits.

As printed

Dasselbe wurde aber für große Werte von $\nu/T$ sehr vollkommen durch das Experiment bestätigt. Wir legen diese Formel unseren Rechnungen zugrunde, behalten aber im Sinne, daß unsere Resultate nur innerhalb gewisser Grenzen gelten.
Beschränken wir uns darauf, die Abhängigkeit der Entropie von dem von der Strahlung eingenommenen Volumen zu untersuchen, und bezeichnen wir die Entropie der Strahlung mit $S_0$ , falls dieselbe das Volumen $v_0$ besitzt, so erhalten wir: $S - S_0 = \frac{E}{\beta \nu} \lg \left( \frac{v}{v_0} \right).$ Radiation entropyRadiation entropyRadiation energy in the bandSpectral constant BFrequency of the lightVolumeVolumeExplain this equation

Text on page 139 of the German source, § 4, paragraph 1 · Text on page 139 of the German source, § 4, paragraph 2

In modern notation

u_\nu = A\,\nu^{3}\,\exp\left(-\frac{B\,\nu}{T}\right)

s_\nu = -\frac{u_\nu}{B\,\nu}\,\left(\ln\left(\frac{u_\nu}{A\,\nu^{3}}\right) - 1\right) + C\left(\nu\right)

S\left(V\right) - S\left(V_{0}\right) = \frac{E}{B\,\nu}\,\ln\left(\frac{V}{V_{0}}\right)

Premise: The calculation rests on Wien's law, which the paper says is not exactly valid; its results hold only within certain limits, where Wien's law is confirmed, for large $\nu/T$ .
Premise: The energy and the frequency band are held fixed; only the volume changes.

Decoder

$E$: The energy of radiation whose frequency lies between $\nu$ and $\nu + d\nu$ .
$v$ , $v_0$ , $S_0$: The volume the radiation occupies, a reference volume, and the radiation's entropy when it occupies $v_0$ .
$\beta$: The constant of Wien's law, $\varrho = \alpha\nu^3 e^{-\beta\nu/T}$ .
$\lg$: The natural logarithm, as the paper prints it.

Try it

LQ-04: Halve the volume the radiation occupies at fixed energy: how much does its entropy change? Choose the preset “Halve the volume (the golden state)” there.

LQ-04: Try a dense state, outside the range of Wien's law: why does the laboratory refuse the comparison? Choose the preset “Dense state (outside the Wien regime)” there.

Where this stops: A moving mirror generally changes frequency and energy, so it is not this held-fixed comparison. The result is not valid merely because one of the two endpoint states is dilute. Read the argument

Where this stops: A positive inferred temperature alone is not a guarantee that the Wien approximation is accurate. The spectral constants are not initially interpreted as particle energies. Read the argument

Argument: A derivation from Wien's law
Model: Within the range of validity of Wien's law
Evidence: A derivation, not evidence
History: Introduced in this paper

Reading preferences

§4 · dilute radiation and the volume law

The paper's results