# Binding energy of hydrogen ion

• Antonija
In summary, the question asks for the binding energy of the second electron in a two-electron system with a lower charge nucleus, similar to a helium atom. The approach involves using hydrogen wavefunctions and solving for the potential, which includes the Coulomb potential between the electron and nucleus, as well as the potential between the two electrons. However, the integral with the wavefunction for the first electron is difficult to solve due to the presence of r12 in the denominator.
Antonija

## Homework Statement

The negative H− ion is a two-electron system like the He atom. How large is the binding energy of the second electron?

## Homework Equations

For hydrogen-like atoms we can use hydrogen wavefunctions so I did it here. First there is potential to be found and then, multipled with charge, it gives energy. Symbols: r2 is distance from second electron to nucleus, r12 is distance between first and second electron.

## The Attempt at a Solution

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Potential is sum of Coulomb potential between one electron and nucleus, and potential between 2 electrons themselves:

Φ(r2)= -e/(4πε0) * 1/r2 + e/(4πε0) ∫ (Ψ1s(r1))2 /r12 d3r1
I know 1s wavefunction for hydrogen so when I write it, it is proportional to exp(-r1/a0)I can put it into my potential but I don't know how to solve integral with wavefunction, as I have r12 in denominator, which is equal to Ir1-r2II can use cosine law but still can't solve integral. I don't know how to behave with this in integral. Sorry if my equation writing is bad, I'm posting for the first time so still learning...

Antonija said:
For hydrogen-like atoms we can use hydrogen wavefunctions so I did it here.
Well, you do not have a hydrogen-like atom.
Your system has some similarity with a helium atom, but with a lower charge of the nucleus. You might be able to transfer some things from the helium system to the hydrogen atom, although I'm not sure how good that approximation will be. It does not look like the problem asks for a full solution of the two-electron wave function.

## 1. What is the binding energy of a hydrogen ion?

The binding energy of a hydrogen ion, also known as the ionization energy or ionization potential, is the minimum amount of energy needed to completely remove an electron from a hydrogen atom, resulting in a positively charged hydrogen ion (H+).

## 2. How is the binding energy of a hydrogen ion calculated?

The binding energy of a hydrogen ion can be calculated using the Rydberg formula, which takes into account the fundamental constants of nature and the charge of the electron and proton.

## 3. What is the significance of the binding energy of a hydrogen ion?

The binding energy of a hydrogen ion is an important concept in understanding the behavior of atoms and molecules. It helps explain the stability of atoms and the formation of chemical bonds between atoms to form molecules.

## 4. Does the binding energy of a hydrogen ion change with different energy levels?

Yes, the binding energy of a hydrogen ion varies depending on the energy level of the electron being removed. The higher the energy level, the higher the binding energy will be.

## 5. How does the binding energy of a hydrogen ion compare to other atoms?

The binding energy of a hydrogen ion is relatively low compared to other atoms, as it only has one electron in its outermost energy level. Atoms with more electrons have higher binding energies due to stronger interactions between the nucleus and electrons.

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