Question

An electron is revolving in its Bohr orbit having Bohr radius of 0.529 Å, then the radius of third orbit is

• A. 4.761 Å
• B. 4234 nm
• C. 5125 nm
• D. 4496 Å

Answer 1

The Correct Option is A

To determine the radius of the nth orbit in the Bohr model, we use the formula:

\( r_n = r_1 \cdot n^2 \) 

Where:

• \( r_n \) is the radius of the nth orbit
• \( r_1 \) is the Bohr radius = \( 0.529 \, \text{Å} \)
• \( n \) is the principal quantum number (orbit number)

For the third orbit (\( n = 3 \)):

\( r_3 = r_1 \cdot 3^2 = 0.529 \, \text{Å} \cdot 9 \)

Calculating the value:

\( r_3 = 4.761 \, \text{Å} \)

Therefore, the radius of the third orbit is: \( \boxed{4.761 \, \text{Å}} \)

Correct Option: (A) 4.761 Å

Answer 2

To find the radius of the nth orbit in the Bohr atomic model, we make use of the following equation:

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

Explanation of terms: 

• \( r_n \): Radius of the nth energy level
• \( r_1 \): Radius of the first orbit (Bohr radius), which is equal to \( 0.529 \, \text{Å} \)
• \( n \): The principal quantum number or orbit number

Let’s calculate for the third orbit, i.e., when \( n = 3 \):

\( r_3 = r_1 \cdot 3^2 = 0.529 \, \text{Å} \cdot 9 \)

Now, performing the multiplication:

\( r_3 = 4.761 \, \text{Å} \)

Hence, the radius of the third orbit comes out to be: \( \boxed{4.761 \, \text{Å}} \)

Correct Answer: (A) 4.761 Å

Answer 3

According to Bohr's model, the radius of the \( n^{th} \) orbit for a hydrogen-like atom is given by the formula: \[ r_n = n^2 a_0 \] where:

• \( r_n \) is the radius of the \( n^{th} \) orbit. 
• \( n \) is the principal quantum number (orbit number).
• \( a_0 \) is the Bohr radius, which is the radius of the first orbit (\( n=1 \)) for hydrogen.

(Note: The full formula is \( r_n = \frac{n^2 a_0}{Z} \), where \( Z \) is the atomic number. Since the problem refers to the Bohr radius \( a_0 \) directly, it implies either a hydrogen atom \( (Z=1) \) or that \( a_0 \) is defined as the radius of the first orbit for the specific atom in question. Assuming it's hydrogen or the base Bohr radius is given).

 

We are given:

• The Bohr radius (radius of the first orbit, \( n=1 \)), \( \mathbf{a_0 = 0.529 \text{ Å}} \).

We need to find the radius of the third orbit, which corresponds to \( \mathbf{n=3} \).

 

Using the formula for \( n=3 \): \[ r_3 = 3^2 \times a_0 \] \[ r_3 = 9 \times a_0 \]

Substituting the given value of \( a_0 \): \[ r_3 = 9 \times 0.529 \text{ Å} \] \[ \mathbf{r_3 = 4.761 \text{ Å}} \]

Comparing this result with the given options:

• 4.761 Å
• 4234 nm (which is 42340 Å)
• 5125 nm (which is 51250 Å)
• 4496 Å

The calculated radius of the third orbit matches the first option.