Question

The Bohr orbit radius for the hydrogen atom (n = 1) is approximately 0.530 \AA. The radius for the first excited state (n = 2) orbit is (in \AA)

• A. 0.13
• B. 1.06
• C. 4.77
• D. 2.12

Hint:

Always remember the sequence of squared integers for Bohr radii: 1, 4, 9, 16... Just multiply the base ground state radius by the square of the principal quantum number. \

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the radius of the second orbit (n=2, first excited state) given the radius of the ground state (n=1).

Step 2: Key Formula or Approach:
According to Bohr's model, the radius of the \( n \)-th orbit of a hydrogen-like atom is: \[ r_n = r_0 \frac{n^2}{Z} \] For Hydrogen, \( Z = 1 \), so \( r_n \propto n^2 \).

Step 3: Detailed Explanation:
Given ground state radius \( r_1 = 0.530 \text{ \AA} \).
For the first excited state, \( n = 2 \).
\[ r_2 = r_1 \times (2)^2 \] \[ r_2 = 0.530 \times 4 \] \[ r_2 = 2.12 \text{ \AA} \]

Step 4: Final Answer:
The radius for the first excited state is \(2.12 \text{ \AA}\).