Question

Identify correct statement(s) among the following statements.

• Energy of the electron in first excited state of $He^+$ ion is equal to the energy of the electron in ground state of hydrogen atom

• In the Lyman series as the energy liberated during the transition increases then the distance between the spectral lines goes on decreasing

• If the radius of the $3^{rd}$ orbit of $Li^{2+}$ is x, the expected radius of $3^{rd}$ orbit of $Be^{3+}$ is $\frac{3}{4}x$

• In $He^+$ ion, 3p and 3d orbitals are degenerate orbitals

• A. Energy of the electron in first excited state of $He^+$ ion is equal to the energy of the electron in ground state of hydrogen atom
• B. In the Lyman series as the energy liberated during the transition increases then the distance between the spectral lines goes on decreasing
• C. If the radius of the $3^{rd}$ orbit of $Li^{2+}$ is x, the expected radius of $3^{rd}$ orbit of $Be^{3+}$ is $\frac{3}{4}x$
• D. In $He^+$ ion, 3p and 3d orbitals are degenerate orbitals

Answer 1

Answer: (A), (B), (C), (D)

All the statements (1, 2, 3, and 4) are correct.

Answer 2

Solution:

1. Statement 1:
   For a hydrogen-like ion, the energy is given by

   $$E_n = -\frac{Z^2 \times 13.6}{n^2}\, \text{eV}.$$

   For He⁺ (Z = 2) in the first excited state (n = 2),

   $$E_2 = -\frac{(2)^2 \times 13.6}{2^2} = -13.6\,\text{eV}.$$

   This is equal to the ground state energy of H (n = 1, Z = 1), which is –13.6 eV.
   → Statement 1 is correct.

2. Statement 2:
   In the Lyman series, transitions occur to n = 1. The energy of the photon emitted is

   $$\Delta E = 13.6\left(1 - \frac{1}{n^2}\right)\, \text{eV}.$$

   As n increases, although the energy ΔE increases (approaching a limiting value), the difference between consecutive transitions (spectral lines) decreases because the levels converge near the series limit.
   → Statement 2 is correct.

3. Statement 3:
   The Bohr radius for a hydrogen-like ion is given by

   $$r_n = \frac{n^2 a_0}{Z}.$$

   For Li²⁺ (Z = 3) and orbit n = 3,

   $$r_3(\text{Li}^{2+}) = \frac{9a_0}{3} = 3a_0.$$

   For Be³⁺ (Z = 4) and n = 3,

   $$r_3(\text{Be}^{3+}) = \frac{9a_0}{4} = \frac{9}{4}a_0.$$

   Their ratio is

   $$\frac{r_3(\text{Be}^{3+})}{r_3(\text{Li}^{2+})} = \frac{9/4}{3} = \frac{9}{12} = \frac{3}{4}.$$

   → Statement 3 is correct.

4. Statement 4:
   In a hydrogen-like ion such as He⁺, the energy depends only on the principal quantum number n. Hence, all orbitals with the same n (e.g. 3p and 3d for n = 3) are degenerate (ignoring fine-structure effects).
   → Statement 4 is correct.