GATE 2024 CY – Question 33

Physical Chemistry · Hydrogen and Hydrogen-like Atoms: Atomic orbitals; radial and angular distribution functions and their probabilities; Atomic units, Multi-electron atoms: Orbital approximation; electron spin; Pauli exclusion principle; Slater determinants. Variational method and secular determinants; first order non-degenerate perturbation techniques. · 1 mark · Numerical answer

The value of $\frac{e^2}{2\pi\epsilon_0a_0}$ in atomic unit of energy is _____. ($e$: charge of electron; $a_0$: Bohr radius; $\epsilon_0$: permittivity of vacuum) (rounded off to the nearest integer)

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Correct answer: 2

Explanation

The atomic unit of energy is the **hartree**:
$$E_h=\frac{e^2}{4\pi\epsilon_0a_0}\approx27.2\text{ eV}.$$

The given expression has $2\pi\epsilon_0$ in the denominator, which is half of $4\pi\epsilon_0$, so it is twice as large:
$$\frac{e^2}{2\pi\epsilon_0a_0}=2\cdot\frac{e^2}{4\pi\epsilon_0a_0}=2E_h .$$

In atomic units the value is **2**.