Question

Radiation coming from transition n = 2 to n = 1 of hydrogen atoms falls on helium ions in n = 1 and n = 2 states. What are the possible transitions of helium ions as they absorbs energy from the radiation?

Answer 1

Energy of radiation (E) from the hydrogen atom is given by
$E=13.6\left(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2}\right)$
Hydrogen atoms go through transition, n = 1 to n = 2.
The energy released is given by
$$\begin{gathered} E=13.6\left(\frac{1}{1}-\frac{1}{4}\right) \\ =13.6\times \frac{3}{4}=10.2\mathrm{eV} \end{gathered}$$
For He,
Atomic no, Z = 2
Let us check the energy required for the transition in helium ions from n = 1 to n = 2.

∴ n₁ = 1 to n₂ = 2

Energy $\left(E_1\right)$ of this transition is given by
$$\begin{gathered} E_1=Z^{2}13.6\left(\frac{1}{{n_1}^2}-\frac{1}{{n_2}^2}\right) \\ =4\times 13.6\left(1-\frac{1}{4}\right) \\ =40.8\mathrm{eV} \end{gathered}$$
E₁ > E,
Hence, this transition of helium ions is not possible.

Let us check the energy required for the transition in helium ion from n = 1 to n = 3.
∴ n₁ = 1 to n₂ = 3

Energy$\left(E_2\right)$ for this transition is given by
$$\begin{gathered} E_2=Z^2\times 13.6\left(\frac{1}{{n_2}^2}-\frac{1}{{n_1}^2}\right) \\ =4\times 13.6\times \left(\frac{1}{1}-\frac{1}{9}\right) \\ =48.3\mathrm{eV} \end{gathered}$$
It is clear that E₂ > E.
Hence, this transition of helium ions is not possible.

Similarly, transition from n₁ = 1 to n₂ = 4 is also not possible.

Let us check the energy required for the transition in helium ion from n = 2 to n = 3.
∴ n₁ = 2 to n₂ = 3
Energy $\left(E_3\right)$ for this transition is given by
$$\begin{gathered} E_3=13.6\times 4\left(\frac{1}{4}-\frac{1}{9}\right) \\ =\frac{20\times 13.6}{36}=7.56\mathrm{eV} \end{gathered}$$

Let us check the energy required for the transition in helium ion from n = 2 to n = 3.
∴ n_​​1 = 2 to n₂ = 4
Energy $\left(E_4\right)$ for this transition is given by
$$\begin{gathered} E_4=13.6\times 4\left(\frac{1}{4}-\frac{1}{16}\right) \\ =13.6\times \frac{3}{4}=10.2\mathrm{eV} \end{gathered}$$

We find that
E₃ < E
E₄ = E
Hence, possible transitions are from n = 2 to n = 3 and n = 2 to n = 4.