Question

In the first excited state of hydrogen atom, the energy of its electron is 10.2 eV. The radial distance of the electron from the hydrogen nucleus in this case is approximately: ____.

• A. 2.1 $\times$ 10⁻¹¹ m
• B. 2.1 $\times$ 10⁻¹⁰ m
• C. 2.1 $\times$ 10⁻⁹ m
• D. 2.1 $\times$ 10⁻⁸ m

Hint:

Always remember: $n=1$ is Ground State, $n=2$ is 1st Excited State, $n=3$ is 2nd Excited State. Using the wrong $n$ is the most common mistake in these problems.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
According to Bohr's model, the radius of the $n$-th orbit of a hydrogen-like atom is proportional to $n^2$. The "first excited state" corresponds to the second orbit ($n=2$).

Step 2: Key Formula or Approach:
The radius of the $n$-th orbit is given by: \[ r_n = a_0 \cdot n^2 \] Where $a_0$ (Bohr radius) $\approx 0.529\,\text{\AA} = 0.529 \times 10^{-10}\,\text{m}$.

Step 3: Detailed Explanation:
1. For the ground state ($n=1$), $r_1 = 0.529 \times 10^{-10}\,\text{m}$. 2. For the first excited state ($n=2$): \[ r_2 = 0.529 \times 10^{-10} \times (2)^2 \] \[ r_2 = 0.529 \times 10^{-10} \times 4 \] \[ r_2 = 2.116 \times 10^{-10}\,\text{m} \]

Step 4: Final Answer:
The radial distance is approximately 2.1 $\times$ 10⁻¹⁰ m.