The Stefan-Boltzmann Law

In the last half of the Nineteenth Century one of the great projects of science was the physicists' efforts to obtain a full understanding of blackbody radiation. Gustav Robert Kirchhoff (1824 Mar 12 - 1887 Oct 17) established that project by suggesting that a full mathematical description of the spectral density of blackbody radiation gave physicists a worthy goal of their efforts and then took the first step toward that goal by deducing the fact that the formula describing blackbody radiation must be a function of the frequency of the radiation, of the absolute temperature of the blackbody, and of nothing else pertaining to the radiation or the blackbody. Taking the second major step in that project, Joseph Stefan (1835 Mar 24 - 1893 Jan 07) inferred from experiments in 1879 and Ludwig Eduard Boltzmann (1844 Feb 20- 1906 Sep 05) deduced from the laws of thermodynamics in 1884 the law that bears their name. I make that deduction of Boltzmann's the topic of this essay.

Imagine a large cavity of volume V inside a blackbody held at an absolute temperature T. Heat radiating from the cavity's walls fills the cavity with a radiation field possessing energy E=E(V,T). That energy can only change in consequence of changes in the volume of the cavity and/or in the temperature of the blackbody; that is,

(Eq'n 1)

in which the subscripts on the partial derivatives represent statements to the effect that we hold the variables that they represent constant as we change the variable in the differentiation operator; so, for example, in the first term on the right side of the equality sign we hold the volume of the cavity unchanged as we vary the temperature of the blackbody and measure the consequent change in the energy of the radiation field in the cavity.

Now we invoke the master equation of classical thermodynamics,

(Eq'n 2)

If we divide that equation by the absolute temperature and make the appropriate substitution for dE from Equation 1, we get

(Eq'n 3)

But, as with the energy of the radiation field, we can only describe the entropy of the radiation field as a function of the volume of the cavity and the temperature of the radiation (S=S(V,T)), so we have

(Eq'n 4)

Comparing that equation to Equation 3 leads us to infer that

(Eq'n 5)

and that

(Eq'n 6)

Next we note the mathematical fact that in a second-order partial derivative with respect to two independent variables the partial differentiation operators commute with each other; that is, the order in which we carry out the differentiations has no effect upon the outcome. Thus, in this case, we have

(Eq'n 7)

which we rewrite more explicitly as

(Eq'n 8)

Making the appropriate substitutions from Equations 5 and 6 transforms that equation into

(Eq'n 9)

We know that the radiation field in the cavity has a uniform energy density u such that E=uV. The radiation also exerts a pressure p=u/3. When we substitute those facts into Equation 9 we get

(Eq'n 10)

which gives us, in turn,

(Eq'n 11)

That equation leads readily to

(Eq'n 12)

which we integrate directly to get

(Eq'n 13)

in which the argument Θ of the constant of integration absorbs the factors necessary to make the arguments of the other natural logarithms pure numbers. We thus get

(Eq'n 14)

which expresses the Stefan-Boltzmann law.

We usually think of the Stefan-Boltzmann law as expressing the flux of radiant energy from a blackbody, not as describing the energy density of a radiation field within a resonant cavity. But if we imagine cutting a small hole in the wall of the cavity, then we see that the flow of radiation through that hole must reflect the nature of the radiation field at its source. Thus we know that the flux of thermal energy from a blackbody must conform to

(Eq'n 15)

in which the Stefan-Boltzmann constant,
σ=5.67x10^{-8} watt per square meter per degree Kelvin raised to the
fourth power.

Finally, note that, although I invoked the entropy of the radiation field in the above derivation, I did so without describing it explicitly. I noted only that we must represent the entropy of the radiation field with a function of only the volume of the cavity and the temperature of the radiation within it: we did not need to know the actual algebraic form of the function itself. If we want to make a weak chemical analogy, then we might compare the entropy in this derivation with an enzyme or a catalyst, something that mediates a chemical reaction without suffering any change itself.

Through that fact we gain a
further insight into the Map of Physics. We see how Boltzmann took very little
information and parlayed it into a deeper understanding of thermal radiation.
Indeed, the theoretical study of blackbody radiation relied so much on almost
pure mathematical reasoning that, as the Nineteenth Century turned into the
Twentieth, physicists might have seen in the science of thermodynamics a plan
for a full Rationalist theoretical physics. That they did not do so likely
reflects the fact that thermodynamics merely describes the relationship between
matter and energy and says nothing about the nature of matter or of forcefields
or of their relationships with space and time. Now that we have begun to
rationalize those deeper natures in the Map of Physics we may finally find
ourselves standing on the threshold of a new age in physics.

habg