Question

If one of the two electrons of a ${H_2}$ molecule is removed, we get a hydrogen molecular ion$H_2^ + $. In the ground state of an$H_2^ + $ , the two protons are separated by roughly$1.5{A^0}$ , and the electron is roughly $1{A^0}$ from each proton. Determine the potential energy of the system. Specify your choice of the zero of the potential energy.

Answer 1

Hint- The energy that is stored is known as the potential energy. The potential energy has the ability to change its form around which the force acts. The existence of the potential energy is based on the virtue of the relative positions of the given system.

Formula used: To solve this type of problems we use the following formulas.
$V = \dfrac{1}{{4\pi {\varepsilon _0}}}\left( {\dfrac{{{Q_1}{Q_2}}}{{{r_{12}}}} + \dfrac{{{Q_2}{Q_3}}}{{{r_{23}}}} + \dfrac{{{Q_4}{Q_5}}}{{{r_{34}}}}} \right)$ ; This is the formula of potential energy of system of charges. ${r_{12}},{r_{23}},{r_{34}}$ Represents the separation between the respective charges.

Complete step by step answer:
Let us write the information given in the question: ${Q_1} = 1.6 \times {10^{ - 19}}C = {Q_2}$ and ${Q_3} = - 1.6 \times {10^{ - 19}}C$
And ${r_{12}} = 1.5\mathop A\limits^0 = 1.5 \times {10^{ - 10}}m$, ${r_{23}} = {r_{31}} = 1\mathop A\limits^0 = {10^{ - 10}}m$
Let zero of potential energy is infinity.
Now, let us calculate the potential energy using the formula $V = \dfrac{1}{{4\pi {\varepsilon _0}}}\left( {\dfrac{{{Q_1}{Q_2}}}{{{r_{12}}}} + \dfrac{{{Q_2}{Q_3}}}{{{r_{23}}}} + \dfrac{{{Q_4}{Q_5}}}{{{r_{34}}}}} \right)$
Let us substitute the values in the above formula.
$V = 9 \times {10^9}\left( {\dfrac{{{{(1.6)}^2} \times {{10}^{ - 38}}}}{{1.5 \times {{10}^{ - 10}}}} + \dfrac{{ - {{(1.6)}^2} \times {{10}^{ - 38}}}}{{{{10}^{ - 10}}}} + \dfrac{{ - {{(1.6)}^2} \times {{10}^{ - 39}}}}{{{{10}^{ - 10}}}}} \right)$
Let us further simplify the equation.
$V = 9 \times {10^9} \times {(1.6)^2} \times {10^{ - 38}}\left( {\dfrac{{1 - 2 \times 1.5}}{{1.5 \times {{10}^{ - 10}}}}} \right)$
Let us solve the above expression and find the value of potential.
$V = - 30.72 \times {10^{ - 19}}J = - 19.2eV$
Hence, required potential energy is -19.2eV.

Additional information:
Electric potential energy is the potential energy that results from the conservative Coulomb force and depends on the particular set of the configuration of charges.
When a system is formed, the potential energy of the first charge, brought from infinity to some point, is zero. As there is no other charge present around it. So, work done is almost zero.
When the second charge is brought, it will feel force either attractive or repulsive that means some work needs to be done. That work is stored as the potential energy of the second particle.

Note: At infinity, the potential energy is considered to be zero. Potential energy is the work done to bring the charge from infinity to some system of charges.