Question

Using the functions given in Table 7.4, verify that for the more circular electron orbit in hydrogen $\mathbf{(}\mathit{i}\mathbf{.}\mathit{e}\mathbf{.}\mathbf{,}\mathit{l}\mathbf{=}\mathit{n}\mathbf{-}\mathbf{1}\mathbf{)}$, the radial probability is of the form

$\mathit{P}\mathbf{\left(}\mathit{r}\mathbf{\right)}\mathbf{\propto }{\mathit{r}}^{\mathbf{2}}\mathit{n}{\mathit{e}}^{\mathbf{-}\mathbf{2}\mathbf{r}\mathbf{/}\mathbf{n}{\mathbf{a}}_{\mathbf{o}}}$

Show that the most probable radius is given by

${\mathit{r}}_{\mathbf{m}\mathbf{o}\mathbf{s}\mathbf{t}\mathbf{p}\mathbf{r}\mathbf{o}\mathbf{b}\mathbf{a}\mathbf{b}\mathbf{l}\mathbf{e}}\mathbf{=}{\mathit{n}}^{\mathbf{2}}{\mathit{a}}_{\mathbf{o}}$

Short answer

The most probable radius is$r=n^2{a}_0$.

Step-by-step solution

 Given data

The radial function for $l=n-1is,$

role="math" localid="1659182273339" $R_{n,n-1}\propto r^{n-1}{e}^{-r/\left(n{a}_o\right)}$

 Concept

The most probable radius is the radius of the orbit in which the probability of finding the electron is maximum.

The probability density is

$P\left(r\right)=r^2{R^2}_{n,n-1}$

 Solution

$$\begin{gathered} P\left(r\right)=r^2{R^2}_{n,n-1} \\ =r^2\left(r^{n-1}{e}^{-r/\left(n{a}_o\right)}\right) \\ =r^2{r}^{2n-2}{e}^{-r/\left(n{a}_o\right)} \\ =r^{2n}{e}^{-r/\left(n{a}_o\right)} \end{gathered}$$

To find the most probable location, take the derivative of the probability density,

$$\begin{gathered} \frac{dP\left(r\right)}{dr}=\frac{d}{dr}{r}^{2n}{e}^{-2rl\left(n{a}_{°}\right)} \\ =2n{r}^{2n-1}{e}^{-2rl\left(n{a}_{°}\right)}+r^{2n}\left(\frac{-2}{n{a}_{°}}\right)e^{-2rl\left(n{a}_{°}\right)} \\ =e^{-2rl\left(n{a}_{°}\right)}{r}^{2n-1}\left(2n+\frac{-2r}{n{a}_{°}}\right) \end{gathered}$$

For the most probable location, the derivative should be equal to zero.

role="math" localid="1659182748704" $=e^{-2rl\left(n{a}_{°}\right)}{r}^{2n-1}\left(2n+\frac{-2r}{n{a}_{°}}\right)$

Hence,

$$\begin{gathered} 2n+\frac{-2r}{n{a}_{°}}=0 \\ r=n^2{a}_{°} \end{gathered}$$

The most probable radius is $r=n^2{a}_{°}$.