Question

According to Bohr's theory the radius of the second stationary state of hydrogen atom is

• A. 105.8 pm
• B. 423.2 pm
• C. 5.29 pm
• D. 21.16 pm
• E. 211.6 pm

Hint:

The radius increases with the square of the shell number (\(r \propto n^2\)). If you remember the first shell is ~53 pm, the second is \(4 \times 53 \approx 212\) pm, and the third is \(9 \times 53 \approx 477\) pm.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Bohr's model provides a formula to calculate the radius of stationary orbits in hydrogen and hydrogen-like species. The radius depends on the principal quantum number \(n\) and the atomic number \(Z\).

Step 2: Key Formula or Approach:
The radius of the \(n^{th}\) orbit is given by:
\[ r_n = a_0 \times \frac{n^2}{Z} \]
Where \(a_0 = 52.9 \text{ pm}\) (Bohr radius) and \(Z\) is the atomic number.

Step 3: Detailed Explanation:
1. For Hydrogen atom, \(Z = 1\).
2. For the second stationary state, \(n = 2\).
3. Substitute the values into the formula:
\[ r_2 = 52.9 \times \frac{2^2}{1} \]
\[ r_2 = 52.9 \times 4 \]
\[ r_2 = 211.6 \text{ pm} \]

Step 4: Final Answer:
The radius of the second stationary state is 211.6 pm.