Question

Calculate the energy required for the process $H{{e}^{+}}(g)\to H{{e}^{2+}}(g)+{{e}^{-}}$
The ionization energy for the H atom in the ground state is $2.18\times {{10}^{-18}}J/atom$

Answer 1

Hint: Remove an electron from an isolated atom or molecule, the required amount of energy is called ionization potential or ionization energy. The ionization energy is expressed in joules or electron volts and is measured in an electric discharge when a fast-moving electron colloids gaseous elements resulting in the ejection of one electron.

Complete step by step solution:
The general features of the hydrogen atom and its spectrum are quantitatively explained by Bohr’s model for a hydrogen atom.
This model explained based on the following postulates:
(i) The circular path of fixed radius around the nucleus by the electron in the hydrogen atom will move along the paths known as orbits, or stationary states, or allowed energy states. These circular paths are around the nucleus based on their energy.
 (ii) Irrespective with time, there is no change in the energy of an electron in the orbit. When the required amount of energy is absorbed by electron or energy emitted by electron, then electron moves from stationary states to a higher energy state.
When an electron is removed from a hydrogen atom, the required amount of energy is the ionization energy for the H atom in the ground state is $2.18\times {{10}^{-18}}J/atom$ .
The formula of the ionization energy of H atom =
$\dfrac{2{{\pi }^{2}}m{{e}^{4}}}{{{h}^{2}}}=2.18\times {{10}^{-18}}J/atom$
Given process, $H{{e}^{+}}(g)\to H{{e}^{2+}}(g)+{{e}^{-}}$
For $H{{e}^{+}}$ ion, ${{Z}^{2}}={{2}^{2}}=4$
Then the energy required for the given process =
$\dfrac{2Z{{\pi}^{2}}m{{e}^{4}}}{{{h}^{2}}}=Z\times\dfrac{2{{\pi}^{2}}m{{e}^{4}}}{{{h}^{2}}}=4\times \dfrac{2Z{{\pi }^{2}}m{{e}^{4}}}{{{h}^{2}}}$
= $4\times 2.18\times {{10}^{-18}}J/atom$
= $8.72\times {{10}^{-18}}J/ion$

Hence, the energy required for the process $H{{e}^{+}}(g)\to H{{e}^{2+}}(g)+{{e}^{-}}$ is $8.72\times {{10}^{-18}}J/ion$

Note: The characteristic of the ionization energy of an element depends on the combined effects of the electric charge of the nucleus, its electronic configuration, and, the size of the atom. The highest ionization potential for noble gases and lowest ionization potential for the alkali metals. It is a measure of the capability of an element in chemical reactions as ion formation or donation of electrons.