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.

Answer 1

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.