Question

If one of the two electrons of a H2 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 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.

Answer 1

two protons are sperated by roughly 1.5 A° and electron is roughly sperated 1A° from each proton. hence, it seems like we have three charges two protons and one electrons, so it is the system of three points charges as shown in figure.

now, potential energy stored is this system of charges is given by

$U=\frac{1}{4\pi\epsilon_0}\left[\begin{array}{c}\frac{q_1q_2}{r_{12}}+\frac{q_2q_3}{r_{23}}+\frac{q_3q_1}{r_{31}}\end{array}\right]$

so, $U=9\times10^9\left[\begin{array}{c}\frac{e\times -e}{1}+\frac{e\times e}{1.5} + \frac{e\times -e}{1}\end{array}\right]\times\frac{1}{10\times10^{-10}}$

[ we know, 1A° = 10 × 10^-10 m]

or, U = 9 × 10^9 × (1.6 × 10^-19) × 10^10[ -1 + 1/1.5 - 1] Joule

[ we know, 1e = 1.6 × 10^-19 C]

or, U = 9 × 10^9 × (1.6 × 10^-19)²/1.6 × 10^-19 × (-4/3) eV

[ we know, 1 eV = 1/1.6 × 10^-19 Joule ]

or, U = -19.2 eV

hence, potential energy is -19.2 eV

potential energy is inversely proportional to speration between charge. so, potential energy will be zero, if charges are sperated infinte distance from each other.

hence, with zero potential energy at infinity.