If radius of first orbit of hydrogen atom is `5.29 ** 10^(-11) m`, the radius of fourth orbit will be

Options

A. 8.46 A
B. 10.23 A
C. 9.22 A
D. 9.48 A

Answer

To find the radius of the fourth orbit of a hydrogen atom, we can use the formula for the radius of the nth orbit:

\[ R_n = a_0 \cdot n^2 \]

where:
- \( R_n \) is the radius of the nth orbit,
- \( a_0 \) is the radius of the first orbit (also known as the Bohr radius),
- \( n \) is the principal quantum number (the orbit number).

Given:
- The radius of the first orbit \( a_0 = 5.29 \times 10^{-11} \, \text{m} \)
- We want to find the radius of the fourth orbit, so \( n = 4 \).

### Step 1: Substitute the values into the formula
We substitute \( a_0 \) and \( n \) into the formula:

\[ R_4 = a_0 \cdot 4^2 \]

### Step 2: Calculate \( 4^2 \)
Calculate \( 4^2 \):

\[ 4^2 = 16 \]

### Step 3: Multiply \( a_0 \) by \( 16 \)
Now, we multiply \( a_0 \) by \( 16 \):

\[ R_4 = 5.29 \times 10^{-11} \, \text{m} \cdot 16 \]

### Step 4: Perform the multiplication
Calculating the multiplication:

\[ R_4 = 5.29 \times 16 \times 10^{-11} \, \text{m} \]

Calculating \( 5.29 \times 16 \):

\[ 5.29 \times 16 = 84.64 \]

So,

\[ R_4 = 84.64 \times 10^{-11} \, \text{m} \]

### Step 5: Convert to a more standard form
To express this in a more standard scientific notation:

\[ R_4 = 8.464 \times 10^{-10} \, \text{m} \]

### Step 6: Convert to angstroms
Since \( 1 \, \text{Å} = 10^{-10} \, \text{m} \):

\[ R_4 = 8.464 \, \text{Å} \]

Thus, the radius of the fourth orbit of the hydrogen atom is approximately \( 8.464 \, \text{Å} \).

### Final Answer
The radius of the fourth orbit is \( 8.464 \, \text{Å} \).

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