If the radius of first Bohar orbit is x pm, then the radius of the third orbit would be

Options

A. `(3xxx)` pm
B. `(6xxx)` pm
C. `(1/2xxx)` pm
D. `(9xxx)` pm

Answer

To solve the problem of finding the radius of the third Bohr orbit given that the radius of the first Bohr orbit is \( x \) pm, we can follow these steps:

### Step 1: Understand the Formula for the Radius of Bohr Orbits
The radius of the nth Bohr orbit is given by the formula:
\[ r_n = \frac{a_0 \cdot n^2}{Z} \]
where:
- \( r_n \) is the radius of the nth orbit,
- \( a_0 \) is the Bohr radius (approximately 52.9 pm),
- \( n \) is the principal quantum number (1 for the first orbit, 2 for the second, etc.),
- \( Z \) is the atomic number.

### Step 2: Determine the Radius of the First Orbit
From the problem, we know:
\[ r_1 = x \text{ pm} \]
Using the formula for the first orbit (\( n = 1 \)):
\[ r_1 = \frac{a_0 \cdot 1^2}{Z} = \frac{a_0}{Z} \]
Setting this equal to \( x \):
\[ \frac{a_0}{Z} = x \]

### Step 3: Solve for Z
From the equation above, we can solve for \( Z \):
\[ Z = \frac{a_0}{x} \]

### Step 4: Determine the Radius of the Third Orbit
Now, we need to find \( r_3 \) using the formula for the third orbit (\( n = 3 \)):
\[ r_3 = \frac{a_0 \cdot 3^2}{Z} = \frac{9a_0}{Z} \]

### Step 5: Substitute the Value of Z
Substituting \( Z \) from Step 3 into the equation for \( r_3 \):
\[ r_3 = \frac{9a_0}{\frac{a_0}{x}} = 9x \]

### Step 6: Final Result
Thus, the radius of the third Bohr orbit is:
\[ r_3 = 9x \text{ pm} \]

### Summary
If the radius of the first Bohr orbit is \( x \) pm, then the radius of the third orbit would be \( 9x \) pm.

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