Considering Bohr’s atomic model for hydrogen atom:

Question

Accepted Answer

Given: We need to identify the correct statements using Bohr's atomic model for hydrogen-like species.

Find: Which pair among (A), (B), (C), (D) is correct.

For a hydrogen-like atom, the energy of the electron in the $n^{\text{th}}$ orbit is

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

where $Z$ is the atomic number.

Also,

Now evaluate the relevant comparisons shown in the solution.

For H atom in ground state:

E_{\text{H, ground}} = -13.6 \times \frac{1^2}{1^2} = -13.6 \, \text{eV}

For He $^+$ in first excited state:

E_{\text{He}^+, \text{1st excited}} = -13.6 \times \frac{2^2}{2^2} = -13.6 \, \text{eV}

So statement (A) is true.

For Li $^{2+}$ in second excited state:

E_{\text{Li}^{2+}, \text{2nd excited}} = -13.6 \times \frac{3^2}{3^2} = -13.6 \, \text{eV}

Thus this is also equal to the ground-state energy of hydrogen, so statement (B) is true.

For He $^+$ in ground state:

E_{\text{He}^+, \text{ground}} = -13.6 \times \frac{2^2}{1^2} = -54.4 \, \text{eV}

This is not equal to $-13.6 \, \text{eV}$ , so statement (C) is false.

For Li $^{2+}$ in ground state:

E_{\text{Li}^{2+}, \text{ground}} = -13.6 \times \frac{3^2}{1^2} = -122.4 \, \text{eV}

This is not equal to the first excited state energy of He $^+$ , so statement (D) is false.

Therefore, only (A) and (B) are correct. The correct option is B.

Given: The question is based on Bohr energy levels for hydrogen-like atoms.

Find: Which statements among (A), (B), (C), (D) are correct.

The key dependence is

E_n \propto \frac{Z^2}{n^2}

So whenever $\frac{Z^2}{n^2}$ is the same for two cases, their energies are equal.

Hence these three states have the same energy, namely $-13.6 \, \text{eV}$ .

So:

Now check the remaining two:

He $^+$ ground state gives $\frac{2^2}{1^2} = 4$ , so its energy is much lower, not equal to hydrogen ground state. Hence (C) is false.
Li $^{2+}$ ground state gives $\frac{3^2}{1^2} = 9$ , so it is also not equal to He $^+$ first excited state. Hence (D) is false.

Therefore, the correct option is B, that is (A), (B) only.

Answer