Question

If one of the two electrons of a H2 molecule is removed, we get ahydrogen molecular ion H+2. In the ground state of an H+2, the twoprotons are separated by roughly 1.5 Å, and the electron is roughly1 Å from each proton. Determine the potential energy of the system.Specify your choice of the zero of potential energy.

Answer 1

Given: The charge on proton 1 is 1.6× 10 −19 , the charge on proton 2 is 1.6× 10 −19 the charge on electron is −1.6× 10 −19 , the distance between two protons is 1.5× 10 −10  m, the distance between electron and proton 1 is 1× 10 −10  m and the distance between electron and proton 2 is 1× 10 −10  m.

The potential energy at infinity is zero.

The potential energy of the system is given as,

V= q 1 q 2 4π ε 0 d 1 + q 3 q 2 4π ε 0 d 2 + q 3 q 3 4π ε 0 d 3 (1)

Where, the charge on proton 1 is q 1 , the charge on proton 2 is q 2 , the charge on electron is q 3 , the distance between proton 1 and 2 is d 1 , the distance between proton 1 and electron is d 2 and the distance between proton 2 and electron is d 3 .

By substituting the given values in the above expression, we get

V= 9× 10 9 × 10 −19 × 10 −19 10 −10 [ − ( 1.6 ) 2 + ( 1.6 ) 2 1.5 − ( 1.6 ) 2 ] =9× 10 −19 ( −3.413 ) =−30.7× 10 −19  J =−19.2 eV

Thus, the potential energy of the system is −19.2 eV.