Question

What is the most probable value of r, in the ground state of hydrogen? (The answer is not zero!) Hint: First you must figure out the probability that the electron would be found between r and r + dr.

Short answer

The most probable value of r is in the ground state of hydrogen.

Step-by-step solution

Expression of probability

The expression for the probability is given as follows,

$$\begin{gathered} \mathbf{P}\mathbf{=}{\left|\psi \right|}^{\mathbf{2}}\mathbf{}\mathbf{4}{\mathbf{\pi r}}^{\mathbf{2}}\mathbf{dr} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\frac{\mathbf{4}}{{\mathbf{a}}^{\mathbf{3}}}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{r}\mathbf{/}\mathbf{a}}{\mathbf{r}}^{\mathbf{2}}\mathbf{dr} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{p}\mathbf{\left(}\mathbf{r}\mathbf{\right)}\mathbf{dr} \end{gathered}$$

Here, the value of is , $\frac{4}{{\mathrm{a}}^3}{\mathrm{r}}^2{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}$and the wave function in the ground state of hydrogen,$\mathrm{\psi }=\frac{1}{\sqrt{{\mathrm{\pi a}}^3}}{\mathrm{e}}^{-\mathrm{r}/\mathrm{a}}$.

Determination of the most probable value of r

Take the derivative of $\mathrm{p}\left(\mathrm{r}\right)=\frac{4}{{\mathrm{a}}^3}{\mathrm{r}}^2{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}$with respect to r .

role="math" localid="1657772519231" $$\begin{gathered} \frac{\mathrm{dp}\left(\mathrm{r}\right)}{\mathrm{dr}}=\frac{\mathrm{d}}{\mathrm{dr}}\left(\frac{4}{{\mathrm{a}}^3}{\mathrm{r}}^2{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}\right) \\ =\frac{4}{{\mathrm{a}}^3}\left[2{\mathrm{re}}^{-2\mathrm{r}/\mathrm{a}}+{\mathrm{r}}^2\left(\frac{2}{\mathrm{a}}{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}\right)\right] \\ =\frac{8\mathrm{r}}{{\mathrm{a}}^3}{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}\left(1-\frac{\mathrm{r}}{\mathrm{a}}\right) \end{gathered}$$

Equate the obtained value to zero.

$$\begin{gathered} \frac{\mathrm{dp}\left(\mathrm{r}\right)}{\mathrm{dr}}=0 \\ \frac{8\mathrm{r}}{{\mathrm{a}}^3}{\mathrm{e}}^{-2\mathrm{r}/\mathrm{a}}\left(1-\frac{\mathrm{r}}{\mathrm{a}}\right)=0 \end{gathered}$$

For the value of r equate role="math" localid="1657772026855" $\left(1-\frac{\mathrm{r}}{\mathrm{a}}\right)$to zero.

$$\begin{gathered} 1-\frac{\mathrm{r}}{\mathrm{a}}=0 \\ \mathrm{r}=\mathrm{a} \end{gathered}$$