Question

The radius of the third stationary orbit of an electron in Bohr's atom is \( R \). The radius of the fourth stationary orbit will be:

• A. \(\frac{4}{3} R\)
• B. \(\frac{16}{9} R\)
• C. \(\frac{3}{4} R\)
• D. \(\frac{9}{16} R\)

Answer 1

The Correct Option is B

To determine the radius of the fourth stationary orbit relative to the third orbit in Bohr's atom model, we require the relationship between orbit radius and principal quantum number.

Bohr's model states the radius of the \(n\)-th orbit, denoted as \(r_n\), is defined by the formula:

\(r_n = n^2 \cdot r_1\)

Here, \(r_1\) represents the radius of the first stationary orbit, and \(n\) is the principal quantum number.

Given: The radius of the third orbit is \(R\). Thus:

\(r_3 = 3^2 \cdot r_1 = 9 \cdot r_1 = R\)

From this, we can ascertain \(r_1\):

\(r_1 = \frac{R}{9}\)

We aim to find the radius of the fourth orbit, \(r_4\):

\(r_4 = 4^2 \cdot r_1 = 16 \cdot r_1\)

Substituting the value of \(r_1\):

\(r_4 = 16 \cdot \frac{R}{9} = \frac{16}{9} R\)

Consequently, the radius of the fourth stationary orbit is \(\frac{16}{9} R\).

Therefore, the correct answer is: \(\frac{16}{9} R\).

This result aligns with the provided correct answer option, validating our calculation.