How Planck's quantum theory of energy solved the ultraviolet catastrophe? [duplicate]

Question

Hello I am a high school student and I am stucked at a concept in chemistry . I have studied black body radiation and Planck's quantum theory which is written in very short description but I have to understand it clearly so that I can understand further concepts in detail.

I know Planck's theory of quantisation which was given to account for blackbody radiation that energy from radiation is produced in discrete quantities in the form of packets .But I can't figure out how it was able to solve the ultraviolet catastrophe problem.This is not given in detail in my textbook and I am stuck at that.I have read other books but it is too much difficult for me to understand at this level . So please help me by giving an easy explanation.

Answer 1

Before Planck, the blackbody distribution was derived entirely classically. The result was the Rayleigh-Jeans equation, given by $$ I(\lambda, T) = \frac{2ck_B T}{\lambda^4} $$ where $T$ is the temperature and $\lambda$ is the wavelength of radiation. This expression gives the radiance per wavelength, so that $I(\lambda, T) d\lambda$ would give the radiance resulting from the electromagnetic waves with wavelengths between $\lambda$ and $\lambda + d\lambda$.

However, for small $\lambda$, this expression diverges to infinity! This is the ultraviolet catastrophe.

Planck resolved this, since by postulating that light came in individual packets of energy equal to $hc/\lambda$, the principles of statistical mechanics allowed him to predict a different distribution function:

$$ I(\lambda, T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^{hc / (\lambda k_B T)} - 1 } $$

This expression has no problem as $\lambda \to 0$ (it goes to zero, as we would expect, giving a well-behaved distribution) so it resolved the ultraviolet catastrophe. Moreover, it has been very well experimentally tested.