# Rayleigh–Jeans Law

In McQuarrie's Quantum Chemistry, the Rayleigh–Jeans law is given as: $$\operatorname{d}\!\rho(\nu,T)=\rho_v(T)\operatorname{d}\!\nu=\frac{8\pi k_bT\nu^2\operatorname{d}\!\nu}{c^3}$$

where $\rho_v(T)\operatorname{d}\!\nu$ is the radiant energy density between $\nu$ and $\nu+\operatorname{d}\!\nu$.

1. Why does the author consider the range of the energy density, and why does he take the differential of the function $\rho(\nu,T)$?

2. Also, what is the difference between the two functions $\rho_v(T)$ and $\rho(\nu,T)$?

## 1 Answer

Why would one be interested in knowing radiation between certain frequencies?

In this context, radiation is understand as a continuous function of wavelength or, equivalently, frequency. At an exact frequency $\nu$, according to Rayleigh-Jeans law, radiation is

$$\frac{8\pi k_bT\nu^2}{c^3}\text{.}$$

In practice, it is hard to measure a single frequency. We are thus interested in frequency intervals. An exact frequency is the limit of a sequence of smaller and smaller intervals, so there is no problem here.

If we make the assumption that, for a sufficiently small interval, $\rho(\nu, T)$ doesn't vary, we get your definition for the differential $\operatorname{d}\!\rho(\nu,T)$:

$$\operatorname{d}\!\rho(\nu,T) = \frac{8\pi k_bT\nu^2}{c^3}\operatorname{d}\!\nu\text{.}$$

The assumption is fair due to the continuity of $\rho(\nu, T)$. This is the approximation of an integral on a very small interval $\operatorname{d}\!\nu$ by the height of a point inside this interval ($\frac{8\pi k_bT\nu^2}{c^3}$) times its length ($\operatorname{d}\!\nu$). No magic here.

So, if we sum an infinite amount of small intervals like the one above we get an integral. The total radiation between $\nu_1$ and $\nu_2$ will be:

$$\int_{\nu_1}^{\nu_2}\operatorname{d}\!\rho(\nu,T) = \int_{\nu_1}^{\nu_2}\rho(\nu, T)\operatorname{d}\!\nu = \int_{\nu_1}^{\nu_2}\frac{8\pi k_bT\nu^2}{c^3}\operatorname{d}\!\nu = 8 \pi k_b T \frac{v_2^3 - v_1^3}{3 c^3}\text{.}$$

Observe that $\rho(\nu, T)$ is quadratic in $\nu$.

Also what is the difference between the two functions $$\rho_v(T)\text{ and }\rho(\nu,T)\text{?}$$

None. Compare the definitions. I changed the nomenclature because, in my opinion, there is no reason not to treat $\nu$ and $T$ on equal footing.