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GATE 2025 PH – Question 36

Atomic and Molecular Physics · Spectra of one-and many-electron atoms · 2 marks · Multiple choice

The screened nuclear charge of neutral helium is $1.7e$, where $e$ is the electronic charge magnitude. Assuming the Bohr model with $E_n=-Z^2/(2n^2)$ atomic units ($Z$ is atomic number), the first ionization potential in atomic units is

  1. 0.89
  2. 1.78
  3. 0.94
  4. 3.16

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Correct answer: (A) 0.89

Explanation

**Bohr model with screening.** Each electron in neutral helium feels an effective nuclear charge $Z_{eff}=1.7$.

**Energy of neutral helium** (two electrons, $n=1$), in atomic units, $E=-Z^2/(2n^2)$ per electron:
$$E_{He}=2\times\left(-\frac{1.7^2}{2}\right)=-1.7^2=-2.89 .$$

**Energy of $\text{He}^+$** (one electron, no screening, $Z=2$):
$$E_{He^+}=-\frac{2^2}{2}=-2 .$$

**First ionisation potential** = energy needed to remove one electron:
$$IP=E_{He^+}-E_{He}=-2-(-2.89)=\mathbf{0.89}\text{ a.u.}$$