First Bohr radius of an atom with Z=82 is r. radius its third orbit is

Options

A. 9r
B. 6r
C. 3r
D. r

Explanation

To find the radius of the third orbit of an atom with atomic number \( Z = 82 \), we can use the formula for the radius of the nth orbit in the Bohr model of the atom. The formula for the radius of the nth orbit is given by:

\[ r_n = \frac{n^2 h^2}{4 \pi^2 k e^2 m Z} \]

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.),
- \( h \) is Planck’s constant,
- \( k \) is Coulomb's constant,
- \( e \) is the charge of the electron,
- \( m \) is the mass of the electron,
- \( Z \) is the atomic number.

### Step 1: Identify the first orbit radius
Given that the radius of the first orbit (\( r_1 \)) is \( r \) for \( Z = 82 \), we have:

\[ r_1 = r \]

### Step 2: Write the formula for the third orbit radius
Now, we need to find the radius of the third orbit (\( r_3 \)). According to the formula, we have:

\[ r_3 = \frac{3^2 h^2}{4 \pi^2 k e^2 m Z} \]

### Step 3: Relate the first and third orbit radii
Since the constants \( h \), \( k \), \( e \), and \( m \) remain the same for both orbits, we can express the ratio of the radii of the first and third orbits as follows:

\[ \frac{r_1}{r_3} = \frac{n_1^2}{n_3^2} \]

Where \( n_1 = 1 \) and \( n_3 = 3 \):

\[ \frac{r}{r_3} = \frac{1^2}{3^2} = \frac{1}{9} \]

### Step 4: Solve for the third orbit radius
Now, rearranging the equation gives:

\[ r_3 = 9r \]

### Conclusion
Thus, the radius of the third orbit for the atom with \( Z = 82 \) is:

\[ \boxed{9r} \]