Annus MirabilisEinstein in 1905
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Read · Light quanta: from entropy to an energy scale · Results

§3 · temperature and entropy

Einstein argues that light, when it is emitted or absorbed, behaves as if its energy came in separate quanta, and calls the idea a heuristic point of view. Follow all nine sections, from the trouble with classical radiation to the photoelectric effect.

The paper's results

Pages 138, 139

The entropy of radiation and its temperature

Comparing two expressions for the entropy gained when black radiation of unit volume is warmed gives ∂φ/∂ϱ=1/T\partial\varphi/\partial\varrho = 1/T, which the paper calls the law of black radiation: from the entropy function φ\varphi one can find the radiation law, and from the radiation law, by integration, the function φ\varphi.

As printed

Durch Vergleich erhält man: ∂φ∂ϱ=1T.\frac{\partial \varphi}{\partial \varrho} = \frac{1}{T}.Spectral entropy densitySpectral energy densityTemperature of the radiationExplain this equation Dies ist das Gesetz der schwarzen Strahlung. Man kann also aus der Funktion φ\varphi das Gesetz der schwarzen Strahlung und umgekehrt aus letzterem die Funktion φ\varphi durch Integration bestimmen mit Rücksicht darauf, daß φ\varphi für ϱ=0\varrho = 0 verschwindet.

Text on pages 138–139 of the German source, § 3, paragraph 4

In modern notation

(∂sν∂uν)ν=1T\left(\frac{\partial s_\nu}{\partial u_\nu}\right)_{\nu} = \frac{1}{T}
  • Premise: The warming is reversible and dEdE is the heat supplied; that is what allows dS=dE/TdS = dE/T.
  • Premise: That ∂φ/∂ϱ\partial\varphi/\partial\varrho does not depend on ν\nu comes from the section's condition that the entropy be a maximum at fixed energy.

Decoder

φ(ϱ,ν)\varphi(\varrho, \nu)
The entropy per unit volume and unit frequency of radiation whose energy density at the frequency ν\nu is ϱ\varrho.
ϱ\varrho
The energy density per unit frequency, printed in §§ 3 and 4 without its subscript ν\nu.
φ\varphi vanishes for ϱ=0\varrho = 0
The condition that fixes the constant of integration when φ\varphi is found from the radiation law.

Try it

LQ-04: Take the derivative of the entropy density with respect to the energy density: does it come out as 1/T? Choose the preset “Derivative check: ds_nu/drho_nu against 1/T” there.

Where this stops: This is a thermodynamic comparison, not a microscopic account of every exchange. Integrating the derivative requires a boundary condition. Read the argument

Argument
A thermodynamic relation, used as a tool
Model
Exact for radiation in equilibrium
Evidence
A derivation, not evidence
History
Follows from thermodynamics available before 1905; the paper states it in this form

§3 · temperature and entropy

A spectrum can determine an entropy derivative

At fixed volume and frequency, the entropy increase per added energy is 1/T. Inverting an admitted spectrum therefore determines an entropy derivative, but not yet its integration constant.

Read the argument