If radius of second stationary orbit (in Bohr's atom) is R. Then radius of third orbit will be :-

Options

A. `R//3`
B. `9R`
C. `R//9`
D. `2.25 R`

Explanation

To find the radius of the third stationary orbit in a Bohr atom, we can use the formula for the radius of the nth orbit:

\[ r_n = \frac{n^2 \cdot R}{Z} \]

where:
- \( r_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (1 for the first orbit, 2 for the second, etc.),
- \( R \) is a constant (the radius of the first orbit),
- \( Z \) is the atomic number of the element.

Given that the radius of the second orbit (when \( n = 2 \)) is \( R \), we can express this as:

\[ r_2 = \frac{2^2 \cdot R}{Z} = \frac{4R}{Z} \]

Now, we need to find the radius of the third orbit (when \( n = 3 \)):

\[ r_3 = \frac{3^2 \cdot R}{Z} = \frac{9R}{Z} \]

To express \( r_3 \) in terms of \( r_2 \), we can relate the two:

Since \( r_2 = \frac{4R}{Z} \), we can find \( R \) in terms of \( r_2 \):

\[ R = \frac{r_2 \cdot Z}{4} \]

Substituting this back into the equation for \( r_3 \):

\[ r_3 = \frac{9 \cdot \left(\frac{r_2 \cdot Z}{4}\right)}{Z} = \frac{9r_2}{4} \]

Thus, the radius of the third orbit is:

\[ r_3 = \frac{9}{4} r_2 \]

**Final Answer:**
If the radius of the second stationary orbit is \( R \), then the radius of the third orbit will be \( \frac{9}{4} R \).

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