Question

How do you calculate the ionization energy $(\mathrm{in}\mathrm{kJ}{\mathrm{mol}}^{-1})$ of the ${\mathrm{He}}^{+}$ ion?

Answer 1

Step 1: Formula for calculating Ionization energy:-

The amount of energy required to extract an electron from an isolated gaseous atom to form a positive ion is called Ionization energy.

For a hydrogen-like ion or an ion that contains only one electron, the energy of the electron is given by;

$\mathrm{E}=-(2.18\times {10}^{-18})\left({\mathrm{Z}}^2\right)\left(\frac{1}{{\mathrm{n}}^2}\right)$

(Where n= the principle of quantum number, Z = the atom number of the element)

Here, the negative sign of the energy of the electron in a hydrogen atom in an orbit refers to the energy of the electron in the atom is lower than the energy of a free electron at rest, which is taken as zero.

Step 2: Calculation of ionization energy for ${\mathrm{He}}^{+}$ ion:-

For the higher energy levels, we can see that the value of $\frac{1}{{\mathrm{n}}^2}$ gets progressively smaller, hence the gap in energy between successive energy levels gets less and finally they converge and eventually coalesce.

At this point, as $\mathrm{n}$tends to infinity so $\frac{1}{{\mathrm{n}}^2}\to 0$, so an electron excited to this point can be regarded as ionized i.e it has left the atom.

Hence the difference in energy between these levels basically represents the ionization energy, which can be given as:

$$\begin{gathered} \mathrm{E}=-(2.18\times {10}^{-18})\left({\mathrm{Z}}^2\right)\times 0-\left[-(2.18\times {10}^{-18})\left({\mathrm{Z}}^2\right)\left(\frac{1}{{\mathrm{n}}^2}\right)\right] \\ =(2.18\times {10}^{-18})\left({\mathrm{Z}}^2\right)\left(\frac{1}{{\mathrm{n}}^2}\right) \end{gathered}$$

For ${\mathrm{He}}^{+}$ ion$(\mathrm{Z}=2)$the ionization energy can be calculated as follows;

$$\begin{gathered} \therefore \mathrm{I}.\mathrm{E}=\frac{2.18\times {10}^{-18}\times 2^2}{1}=8.72\times {10}^{-18}\mathrm{J} \\ =8.72\times {10}^{-21}\mathrm{kJ} \\ =8.72\times {10}^{-21}\times 6.02\times {10}^{23} \\ =5250\mathrm{kJ}{\mathrm{mol}}^{-1} \end{gathered}$$

Hence, the ionization energy of the ${\mathrm{He}}^{+}$ ion can be calculated as $5250\mathrm{kJ}{\mathrm{mol}}^{-1}$.