Light Quanta · Section 6

The radiation entropy law matches the gas entropy law.
The exponent identifies the light quantum.

Why does equating the volume dependence of Wien radiation entropy to Boltzmann's independent-particle entropy law suggest that monochromatic radiation behaves as independent energy quanta of magnitude

R\beta\nu / N = h\nu

Read Section 6 of Einstein’s 1905 paper →

Interactive Critical Edition · Instrument LQ-06

Matching the Entropy Laws to Derive the Light Quantum (§6 The Move)

Select the Subexpression for "Number of Things (n)"

Compare S - S₀ = (R/N) ln[(V/V₀)^n_eff] with S - S₀ = (R/N) n ln(V/V₀). Which term plays the role of n?

State Parameters

Radiation Energy (E)9.056 nJ

Frequency (ν)600.0 THz

Volume Ratio (V / V₀)0.50

Comparison Gas Particles (n)10

Blackbody Temperature (T)3000 K

Fork A: Epistemic Interpretation

Side-by-Side Entropy Volume Laws (§6 The Move)

V/V₀ = 0.50

Wien Spectrum Mean Quantum Energy vs Molecule Kinetic Energy (§6)

T = 3000 K

Integrating over a full Wien spectrum, the average energy of a light quantum is exactly twice the average translational kinetic energy of a gas molecule (at 600 THz, monochromatic h·ν is 3.20× this mean quantum energy):

1. Derivation (Algebra)

The functional forms of Wien radiation entropy and Boltzmann gas entropy agree identically if and only if n = N·E / (R·β·ν) = E / (h·ν).

2. Heuristic Inference

Monochromatic radiation of low density in the Wien regime behaves thermodynamically as though it consists of independent energy quanta of magnitude h·ν.

3. Further Hypothesis

Are the laws of production (Stokes rule §7) and transformation (photoelectric §8, ionization §9) also governed by discrete energy exchanges of size h·ν?

Accepted Telemetry Snapshot

Quantity	Symbol	Status	Value
Radiation Energy	E	value	9.0556 nJ
Frequency	ν	value	600.00 THz
Volume Ratio	V/V₀	value	0.5000
Effective Quanta Count (Never Rounded)	n_eff	value	2.277774e+10
Energy per Quantum (SI)	ε = hν	value	3.975642e-19 J
Energy per Quantum (eV)	ε_eV	value	2.481401 eV
Radiation Entropy Volume Coeff	E / (βν)	value	3.144807e-13 J/K
Gas Entropy Volume Coeff	(R/N) n	value	1.380649e-22 J/K
Radiation Entropy Change	ΔS_rad	value	-2.179814e-13 J/K
Gas Entropy Change	ΔS_gas	value	-9.569930e-23 J/K
Wien Mean Quantum Energy	⟨ε⟩ = 3 k_B T	value	0.775560 eV
Gas Molecule Kinetic Energy	⟨E_kin⟩ = 1.5 k_B T	value	0.387780 eV

The Physical Argument

The Entropy Volume Laws Placed Side by Side

In §4, Einstein showed that for monochromatic radiation of energy

E

and frequency

\nu

in the Wien regime, changing the enclosing volume from

V_0

V

changes the entropy by:

S - S_0 = \frac{E}{\beta\nu}\ln\frac{V}{V_0}

In §5, Einstein evaluated Boltzmann's principle

S - S_0 = \frac{R}{N}\ln W

for a system of

n

independent particles in a container, finding that the statistical probability of finding all

n

particles in a subvolume

V

W = (V/V_0)^n

, leading to:

S - S_0 = \frac{R}{N}\,n\ln\frac{V}{V_0} = k_B\,n\ln\frac{V}{V_0}

The Move: Equating the Functional Forms

To make the two equations directly comparable, Einstein rewrites the radiation entropy formula with Boltzmann's constant factor

R/N

outside the logarithm:

S - S_0 = \frac{R}{N}\ln\left[\left(\frac{V}{V_0}\right)^{\frac{N}{R}\frac{E}{\beta\nu}}\right]

Comparing this with the gas probability law reveals that the statistical probability that all the monochromatic radiation energy

E

is found in subvolume

V

is:

W = \left(\frac{V}{V_0}\right)^{\frac{N}{R}\frac{E}{\beta\nu}}

The exponent

n_{\text{eff}} = \frac{N E}{R\beta\nu} = \frac{E}{h\nu}

plays precisely the role of the particle count

n

Energy per Element and Historical Constants

If a total energy

E

is composed of

n_{\text{eff}}

independent quanta, each quantum carries an energy:

\epsilon = \frac{E}{n_{\text{eff}}} = \frac{R\beta\nu}{N} = h\nu

Using the 1905 experimental values for the gas constant

R = 8{,}31\cdot 10^7\text{ erg/K}

, Wien's constant

\beta = 4{,}866\cdot 10^{-11}\text{ K}\cdot\text{s}

, and Avogadro's number

N = 6{,}17\cdot 10^{23}

, Einstein calculated:

\frac{R\beta}{N} = 6{,}5537\cdot 10^{-27}\text{ erg}\cdot\text{s} \approx 6{,}626\cdot 10^{-27}\text{ erg}\cdot\text{s} = h

This confirmed that the packet energy derived purely from thermodynamic entropy matches Planck's quantum of action

h

to within 1%.

Mean Quantum Energy over a Wien Spectrum

Einstein further calculated the average energy of light quanta in thermal radiation at temperature

T

by integrating over the full Wien spectrum:

\langle \epsilon \rangle = \frac{\int_0^\infty \alpha\nu^3 e^{-\beta\nu/T} d\nu}{\int_0^\infty \frac{N}{R\beta\nu}\alpha\nu^3 e^{-\beta\nu/T} d\nu} = 3\frac{R}{N}T = 3 k_B T

This is exactly twice the average translational kinetic energy of a monoatomic gas molecule,

\langle E_{\text{kin}} \rangle = \frac{3}{2} k_B T

The Three Logical Roles

Derivation (Mathematical Identity): The radiation entropy volume law and the ideal gas entropy volume law agree identically for all volume ratios
$V/V_0$
if and only if
$n = N E / (R\beta\nu) = E / (h\nu)$
.
Heuristic Inference (Thermodynamic Analogy): In the Wien regime of low radiation density, monochromatic radiation behaves thermodynamically as though it consisted of
$n_{\text{eff}}$
mutually independent energy quanta
$h\nu$
.
Further Physical Hypothesis (Emission and Absorption): This analogy suggests investigating whether the processes of emission and absorption of light also proceed by discrete quanta of size
$h\nu$
(demonstrated in §7 for Stokes' rule, §8 for photoelectricity, and §9 for gas ionization).

Return to the Light Quanta Paper →Explore the Discovery Journey