if a is the radius of first Bohr orbit in hydrogen atom, the radius of `3^rd` orbit is

Options

A. (a)3a
B. (b)9a
C. (c)27a
D. (d)81a

Answer

To find the radius of the third Bohr orbit in a hydrogen atom given that the radius of the first Bohr orbit is \( a \), we can follow these steps:

### Step 1: Understand the relationship between the radius and the principal quantum number.
In the Bohr model of the hydrogen atom, the radius of the nth orbit is given by the formula:
\[ r_n = n^2 \cdot r_1 \]
where \( r_n \) is the radius of the nth orbit, \( r_1 \) is the radius of the first orbit, and \( n \) is the principal quantum number.

### Step 2: Identify the radius of the first orbit.
We are given that the radius of the first Bohr orbit \( r_1 \) is equal to \( a \):
\[ r_1 = a \]

### Step 3: Calculate the radius of the third orbit.
To find the radius of the third orbit \( r_3 \), we substitute \( n = 3 \) into the formula:
\[ r_3 = 3^2 \cdot r_1 = 9 \cdot r_1 \]
Substituting \( r_1 = a \):
\[ r_3 = 9 \cdot a \]

### Step 4: Conclusion.
Thus, the radius of the third Bohr orbit is:
\[ r_3 = 9a \]

### Final Answer:
The radius of the third orbit is \( 9a \).

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