Question

According to Bohr's theory of hydrogen atom, the speed of the electron, its energy and radius of its orbit vary with the principal quantum number \( n \), respectively as

• A. \( n, \; n^2, \; \frac{1}{n} \)
• B. \( \frac{1}{n}, \; \frac{1}{n^2}, \; n^2 \)
• C. \( n, \; \frac{1}{n}, \; n^2 \)
• D. \( \frac{1}{n}, \; n, \; \frac{1}{n^2} \)

Hint:

Bohr model shortcut: \( v \propto \frac{1}{n} \), \( E \propto \frac{1}{n^2} \), \( r \propto n^2 \).

Answer 1

The Correct Option is B

Step 1: Recall Bohr's model relations.
Bohr’s theory gives relations for velocity, energy, and radius as functions of principal quantum number \( n \).

Step 2: Expression for velocity.
\[ v_n \propto \frac{1}{n} \]

Step 3: Expression for energy.
\[ E_n \propto -\frac{1}{n^2} \]
Magnitude wise, it varies as \( \frac{1}{n^2} \).

Step 4: Expression for radius.
\[ r_n \propto n^2 \]

Step 5: Writing combined variation.
\[ v \propto \frac{1}{n}, \quad E \propto \frac{1}{n^2}, \quad r \propto n^2 \]

Step 6: Matching with options.
This matches option (B).

Step 7: Final conclusion.
\[ \boxed{\text{Option (B)}} \]