Question

In the Bohr model of the hydrogen atom, let R, V and E represent the radius of the orbit, speed of the electron and total energy of the electron respectively. Which of the following quantity is proportional to the quantum number n ?

• (A) $E/V$
• (B) $R/E$
• (C) $VR$
• (D) $RE$

Answer 1

The Correct Option is C

In the Bohr model of the hydrogen atom, the relationship between quantum numbers and physical quantities can be derived from the following formulas:

• The radius of the orbit \( R_n \) is given by: R_n = n^2 \frac{h^2}{4\pi^2 m e^2 k}, where \( h \) is Planck's constant, \( m \) is the electron mass, \( e \) is the electron charge, and \( k \) is the Coulomb's constant.
• The speed of the electron \( V_n \) is given by: V_n = \frac{e^2 k}{nh}.
• The total energy \( E_n \) is given by: E_n = -\frac{m e^4 k^2}{2 n^2 h^2}.

Now, let's evaluate each of the given options to determine which is proportional to the quantum number \( n \):

1. Option (A): E/V
   E_n/V_n = \left(-\frac{m e^4 k^2}{2 n^2 h^2}\right) \bigg/ \left(\frac{e^2 k}{nh}\right) = -\frac{m e^2 k}{2 n h}.

   This is inversely proportional to \( n \), thus not the correct option.

2. Option (B): R/E
   R_n/E_n = \left(n^2 \frac{h^2}{4\pi^2 m e^2 k}\right) \bigg/ \left(-\frac{m e^4 k^2}{2 n^2 h^2}\right).

   After simplification, it is found to be proportional to \( n^4 \), thus incorrect.

3. Option (C): V R
   V_n R_n = \left(\frac{e^2 k}{nh}\right) \times \left(n^2 \frac{h^2}{4\pi^2 m e^2 k}\right) = \frac{nh}{4\pi m}.

   This is directly proportional to \( n \), making it the correct option.

4. Option (D): R E
   R_n E_n = \left(n^2 \frac{h^2}{4\pi^2 m e^2 k}\right) \times \left(-\frac{m e^4 k^2}{2 n^2 h^2}\right).

   After simplification, this becomes constant and is not proportional to \( n \), hence incorrect.

Therefore, the correct answer is Option (C): V R.