The ratio of radius of second orbit of hydrogen to the radius of its first orbit is :

Options

A. `2:1`
B. `1 : 2`
C. `4 : 1`
D. `6 : 1`

Answer

To find the ratio of the radius of the second orbit of hydrogen to the radius of its first orbit, we can follow these steps:

### Step 1: Understand the Formula for the Radius of Orbits
The radius of the nth orbit of a hydrogen atom is given by the formula:
\[ R_n = \frac{0.53 \, n^2}{Z} \]
where:
- \( R_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 2: Calculate the Radius of the Second Orbit (\( R_2 \))
For the second orbit, \( n = 2 \):
\[ R_2 = \frac{0.53 \times (2^2)}{1} = 0.53 \times 4 = 2.12 \, \text{Å} \]

### Step 3: Calculate the Radius of the First Orbit (\( R_1 \))
For the first orbit, \( n = 1 \):
\[ R_1 = \frac{0.53 \times (1^2)}{1} = 0.53 \, \text{Å} \]

### Step 4: Find the Ratio of \( R_2 \) to \( R_1 \)
Now, we need to find the ratio \( \frac{R_2}{R_1} \):
\[ \frac{R_2}{R_1} = \frac{2.12 \, \text{Å}}{0.53 \, \text{Å}} \]

### Step 5: Simplify the Ratio
Since both values are in angstroms, they cancel out:
\[ \frac{R_2}{R_1} = \frac{2.12}{0.53} = 4 \]

### Conclusion
Thus, the ratio of the radius of the second orbit to the radius of the first orbit is:
\[ \text{Ratio} = 4:1 \]

### Final Answer
The ratio of the radius of the second orbit of hydrogen to the radius of its first orbit is \( 4:1 \).

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