91Ó°ÊÓ

The distance between two protons is $0.11\text{ }\text{nm}$.

The potential energy between the two protons is given by a relation, $\mathbf{U}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{Ï€Î\mu }}_{\mathbf{0}}}\frac{{\mathbf{e}}^{\mathbf{2}}}{\mathbf{a}}$.

Here, ${\mathbf{Î\mu }}_{\mathbf{0}}$ is permittivity of the free space,$\mathbf{e}$ is charge on each proton, and role="math" localid="1658384048724" $\mathit{a}$is the distance between the two protons.

Substitute$8.85×{10}^{−12}{\text{ C}}^{\text{2}}{\text{/Nm}}^{\text{2}}$ for${\mathrm{Î\mu }}_0,1.6×{10}^{−19}\text{ C}$for$e,\text{ }0.11\text{ n³¾}$for$a$ in the equation$\mathrm{U}=\frac{1}{4{\mathrm{Ï€Î\mu }}_0}\frac{{\mathrm{e}}^2}{\mathrm{a}}$and solve for$U$.

$\begin{array}{c}\mathrm{U}=\frac{{(1.6×{10}^{−19}\text{ C})}^2}{\left(4\mathrm{Ï€}\right)(8.85×{10}^{−12}{\text{ C}}^{\text{2}}{\text{/Nm}}^{\text{2}})(0.11\text{ n³¾})}\\ \mathrm{U}=\frac{{(1.6×{10}^{−19}\text{ C})}^2}{\left(4\mathrm{Ï€}\right)(8.85×{10}^{−12}{\text{ C}}^{\text{2}}{\text{/Nm}}^2)(0.11×{10}^{−9}\text{″¾})}\\ \mathrm{U}=2.09×{10}^{−18}\text{J}\end{array}$

Convert joules to electron volte.

$\begin{array}{c}\mathrm{U}=2.09×{10}^{−18}\text{J}\left(\frac{1\text{eV}}{1.6×{10}^{−19}\text{J}}\right)\\ \mathrm{U}=13.1\text{eV}\end{array}$

The energy of the electron is given by the difference of the total energy and the proton's energy, ${\mathrm{E}}_{\mathrm{e}}={\mathrm{E}}_{\mathrm{T}}−\mathrm{U}$.

Here, ${\mathrm{E}}_{\mathrm{e}}$ is the energy of the electron, ${\mathrm{E}}_{\mathrm{T}}$ is the total energy, and $\mathrm{U}$ is the energy of the proton.

Substitute$−16.3\text{eV}$for${\mathrm{E}}_{\mathrm{T}}$and$13.1\text{eV}$for$\mathrm{U}$in the equation${\mathrm{E}}_{\mathrm{e}}={\mathrm{E}}_{\mathrm{T}}−\mathrm{U}$and solve for${\mathrm{E}}_{\mathrm{e}}$.

$\begin{array}{c}{\mathrm{E}}_{\mathrm{e}}=−16.3\text{eV}−13.1\text{eV}\\ {\mathrm{E}}_{\mathrm{e}}=−29.4\text{eV}\end{array}$

Therefore, the energy of the electron in ${\text{H}}_2^{+}$ is role="math" $−29.4\text{eV}$.