In 1884, 40-year-old Austrian physicist Ludwig Boltzmann (1844–1906) published a paper in Annalen der Physik (Vol. 22) titled “Ableitung des Stefan’schen Gesetzes, betreffend die Abhängigkeit der Wärmestrahlung von der Temperatur aus der electrodynamischen Lichttheorie”.

Boltzmann derived Josef Stefan’s 1879 empirical $T^4$ radiation law theoretically (Stefan-Boltzmann Law), combining James Clerk Maxwell’s electromagnetic radiation pressure with the Second Law of Thermodynamics.

Mathematical Thermodynamic Derivation

Boltzmann considered an ideal photon gas enclosed inside a cylinder with perfectly reflecting walls and a movable piston:

1. Radiation Pressure Relation

From Maxwell’s electrodynamics, the radiation pressure $P$ of isotropic photon gas equals one-third of its total volumetric energy density $u(T)$:

$$P = \frac{1}{3} u(T)$$

2. First and Second Laws of Thermodynamics

The fundamental thermodynamic relation $dU = T dS - P dV$ for total internal energy $U = u(T) V$ yields:

$$\frac{\partial u}{\partial T} = \frac{4 u}{T}$$

3. Integration & The $T^4$ Law

Integrating this linear differential equation:

$$\int \frac{du}{u} = 4 \int \frac{dT}{T} \implies \ln u = 4 \ln T + \text{constant} \implies u(T) = a \cdot T^4$$

Where $a = 4\sigma / c$, proving that total blackbody radiant power per unit area is $P = \sigma T^4$.

Key Takeaways

  • Year: 1884
  • Key Figure: Ludwig Boltzmann (Austrian Physicist)
  • Core Discovery: Derived the Stefan-Boltzmann Law ($u = a T^4$) theoretically using Maxwellian radiation pressure and thermodynamics.
  • Photon Gas Foundation: Treated blackbody radiation as a thermodynamic working fluid.
  • Quantum Relevance: Exact integral of Max Planck’s 1900 quantum blackbody spectrum ($\int_0^\infty u(\nu) d\nu = a T^4$).