Deriving the Stefan-Boltzmann law (j* = σT⁴) from Planck's Law is a classic calculation in thermodynamic

Deriving the Stefan-Boltzmann law (j* = σT⁴) from Planck's Law is a classic calculation in thermodynamics and statistical mechanics. Here's the process:

1. Start with Planck's Law:

Planck's law gives the spectral radiance B_ν(ν, T) of a blackbody. This is the power emitted per unit area, per unit solid angle, per unit frequency, at frequency ν and temperature T.

B_ν(ν, T) = (2hν³) / (c² * (exp(hν / kT) - 1))

• h: Planck constant

• c: Speed of light

• k: Boltzmann constant

• ν: Frequency

• T: Absolute temperature

2. Integrate over all Frequencies:

To get the total radiance (power per unit area per unit solid angle) across all frequencies, we need to integrate B_ν from ν = 0 to ν = ∞:

B(T) = ∫[0 to ∞] B_ν(ν, T) dν = ∫[0 to ∞] (2hν³ / (c² * (exp(hν / kT) - 1))) dν

This integral is easier to solve with a substitution. Let:
x = hν / kT
Then:
ν = (kT/h)x
dν = (kT/h)dx

Substitute these into the integral:

B(T) = ∫[0 to ∞] (2h * [(kT/h)x]³ / (c² * (exp(x) - 1))) * (kT/h) dx

Now, pull the constants (relative to x) out of the integral:

B(T) = (2h / c²) * (kT/h)³ * (kT/h) ∫[0 to ∞] (x³ / (exp(x) - 1)) dx
B(T) = (2h / c²) * (k⁴T⁴ / h⁴) ∫[0 to ∞] (x³ / (exp(x) - 1)) dx
B(T) = (2k⁴T⁴ / (c²h³)) ∫[0 to ∞] (x³ / (exp(x) - 1)) dx

The definite integral ∫[0 to ∞] (x³ / (exp(x) - 1)) dx is a standard integral related to the Riemann zeta function, specifically Γ(4)ζ(4) = 3! * (π⁴ / 90) = 6 * π⁴ / 90 = π⁴ / 15.

Substituting this value back:

B(T) = (2k⁴T⁴ / (c²h³)) * (π⁴ / 15)
B(T) = (2π⁴k⁴ / (15c²h³)) * T⁴

This B(T) is the total power per unit area per unit solid angle.

3. Integrate over Solid Angle (Hemisphere):

The Stefan-Boltzmann law gives the total power emitted per unit area (j*, the radiant exitance) integrated over all outward directions (a hemisphere, 2π steradians). However, the emission follows Lambert's cosine law (intensity is proportional to cos(θ) where θ is the angle to the surface normal).

The relationship between radiance B(T) and exitance j* for a Lambertian surface is:

j* = π * B(T)

(The factor is π, not 2π, because of the integration of .

Substituting the expression for B(T):

j* = π * [(2π⁴k⁴ / (15c²h³)) * T⁴]
j* = (2π⁵k⁴ / (15c²h³)) * T⁴

4. Identify the Stefan-Boltzmann Constant (σ):

The Stefan-Boltzmann law is j* = σT⁴. By comparing this to the derived expression for j*, we can identify the constant σ:

σ = 2π⁵k⁴ / (15c²h³)

This derivation shows how the constant σ emerges directly from integrating Planck's distribution of radiated energy over all frequencies and emission angles, fundamentally linking it to h, c, and k.