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 $\rho(\nu,T)$?





> 
> 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](https://en.wikipedia.org/wiki/Rayleigh%E2%80%93Jeans_law)


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


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.




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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.




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So, if we *sum* an infinite amount of small intervals like the one above [we get an *integral*](https://en.wikipedia.org/wiki/Riemann_integral#Riemann_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 $\nu$.




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> 
> 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.


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