Question

According to Bohr's model of the hydrogen atom, (A) the radius of the orbit of an electron is directly proportional to \(n\). (B) the speed of the orbiting electron is directly proportional to \(\dfrac{1}{n}\). (C) the total energy of the electron is directly proportional to \(\dfrac{1}{n^2}\). (D) the radius of the orbit of an electron is directly proportional to \(n^2\). Choose the correct answer from the options given below:

• A. (A), (B) and (C) only
• B. (B), (C) and (D) only
• C. (A), (C) and (D) only
• D. (C) and (D) only

Hint:

For Bohr's model: \[ r_n\propto n^2,\qquad v_n\propto \frac1n,\qquad E_n\propto -\frac1{n^2} \] These three relations are frequently asked in MCQs.

Answer 1

The Correct Option is B

Concept: According to Bohr's model of hydrogen atom, Radius of the \(n^{\text{th}}\) orbit: \[ r_n=n^2a_0 \] Speed of electron: \[ v_n\propto \frac{1}{n} \] Total energy: \[ E_n=-\frac{13.6}{n^2}\;\text{eV} \]

Step 1: Check Statement (A). \[ r_n=n^2a_0 \] Radius is proportional to \(n^2\), not \(n\). Therefore, \[ \boxed{\text{Statement (A) is false.}} \]

Step 2: Check Statement (B). \[ v_n\propto \frac{1}{n} \] Hence, \[ \boxed{\text{Statement (B) is true.}} \]

Step 3: Check Statement (C). \[ E_n=-\frac{13.6}{n^2} \] Thus, the magnitude of energy varies as \[ \frac{1}{n^2} \] Hence, \[ \boxed{\text{Statement (C) is true.}} \]

Step 4: Check Statement (D). \[ r_n=n^2a_0 \] Therefore, \[ \boxed{\text{Statement (D) is true.}} \]

Step 5: Choose the correct option. True statements are: \[ (B),\ (C),\ (D) \] \[ \boxed{ \begin{array}{c} (B),\ (C)\text{ and }(D)\text{ only} \end{array} } \]