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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 zero potential energy.
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Solution
The system of two protons and one electron is represented in the given figure.
Charge on proton 1, q1 = 1.6 × 10−19 C
Charge on proton 2, q2 = 1.6 × 10−19 C
Charge on electron, q3 = −1.6 × 10−19 C
Distance between protons 1 and 2, d1 = 1.5 × 10−10 m
Distance between proton 1 and electron, d2 = 1 × 10−10 m
Distance between proton 2 and electron, d3 = 1 × 10−10 m
The potential energy at infinity is zero.
Potential energy of the system,
`"V" = ("q"_1"q"_2)/(4piin_0"d"_1) + ("q"_2"q"_3)/(4piin_0"d"_3) + ("q"_3"q"_1)/(4piin_0"d"_2)`
Substituting `1/(4piin_0) = 9 xx 10^9 "N m"^2 "C"^-2`, we obtain
`"V" = (9 xx 10^9 xx 10^-19 xx 10^-19)/(10^-10) [-(16)^2 + (1.6)^2/(1.5) + -(1.6)^2]`
= `-30.7 xx 10^-19 "J"`
= −19.2 eV
Therefore, the potential energy of the system is −19.2 eV.
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