Question

What is the probability of finding a n = 2, ℓ = 1 electron between a0 and 2a0?

Answer 1

The radial wave function of hydrogen atom is $R_{21}=\frac{1}{\sqrt{24a_0^3}}(\frac{r}{a_0})e^{-\frac{r}{2a_0}}$

The probability of finding a n = 2, ℓ = 1 electron between $a_0$ and $2a_0$ is

$P=\intop(R_{21})^2r^2dr =\intop(\frac{1}{\sqrt{24a_0^3}}(\frac{r}{a_0})e^{-\frac{r}{2a_0}})^2r^2dr$

$=\frac{1}{{24a_0^5}}\int r^4e^{-\frac{r}{a_0}}dr$ between the limits $a_0$ and $2a_0$

On solving the above integration, we get

P =0.04899