The radius of Bohr 's first orbit in hydrogen atom is 0.53 Å the radius of second orbit in He+ will be

Options

A. 0.0265 nm
B. 0.0530 nm
C. 0.1060 nm
D. 0.2120 nm

Answer

To solve the problem of finding the radius of the second orbit in He+ (helium ion), we will use the formula for the radius of Bohr's orbits. Here’s a step-by-step solution:

### Step 1: Understand the Formula
The formula for the radius of the nth orbit in a hydrogen-like atom is given by:

\[ R_n = R_0 \cdot \frac{n^2}{Z} \]

Where:
- \( R_n \) is the radius of the nth orbit.
- \( R_0 \) is the radius of the first orbit in hydrogen (given as 0.53 Å).
- \( n \) is the principal quantum number (orbit number).
- \( Z \) is the atomic number of the element.

### Step 2: Identify Given Values
From the problem:
- \( R_0 = 0.53 \) Å
- For He+, the atomic number \( Z = 2 \) (since helium has 2 protons).
- We are looking for the radius of the second orbit, so \( n = 2 \).

### Step 3: Substitute Values into the Formula
Now, we can substitute the known values into the formula to find \( R_2 \):

\[ R_2 = R_0 \cdot \frac{n^2}{Z} = 0.53 \cdot \frac{2^2}{2} \]

### Step 4: Calculate \( R_2 \)
Calculating the value:

\[ R_2 = 0.53 \cdot \frac{4}{2} = 0.53 \cdot 2 = 1.06 \, \text{Å} \]

### Step 5: Convert to Nanometers
Since the answer options are in nanometers, we need to convert Ångstroms to nanometers.

1 Å = 0.1 nm, therefore:

\[ 1.06 \, \text{Å} = 0.106 \, \text{nm} \]

### Final Answer
The radius of the second orbit in He+ is:

\[ \text{Answer: } 0.106 \, \text{nm} \]