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 that relates the radius of the orbits to the principal quantum number \( n \). The radius of the \( n \)-th orbit is given by:

\[ R_n = n^2 \cdot R_1 \]

where:
- \( R_n \) is the radius of the \( n \)-th orbit,
- \( R_1 \) is the radius of the first orbit,
- \( n \) is the principal quantum number.

### Step 1: Identify the given values
- The radius of the first orbit \( R_1 = 5.29 \times 10^{-11} \, \text{m} \)
- The principal quantum number for the fourth orbit \( n = 4 \)

### Step 2: Calculate the radius of the fourth orbit
Using the formula, we can find \( R_4 \):

\[ R_4 = n^2 \cdot R_1 \]

Substituting the values:

\[ R_4 = 4^2 \cdot (5.29 \times 10^{-11} \, \text{m}) \]

Calculating \( 4^2 \):

\[ 4^2 = 16 \]

Now substituting back into the equation:

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

### Step 3: Perform the multiplication
Now we perform the multiplication:

\[ R_4 = 16 \cdot 5.29 \times 10^{-11} \, \text{m} = 84.64 \times 10^{-11} \, \text{m} \]

### Step 4: Convert to scientific notation
To express \( 84.64 \times 10^{-11} \) in proper scientific notation:

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

### Step 5: Convert to angstroms
Since \( 1 \, \text{angstrom} = 10^{-10} \, \text{m} \):

\[ R_4 = 8.464 \, \text{angstroms} \]

### Final Answer
The radius of the fourth orbit of the hydrogen atom is:

\[ R_4 \approx 8.46 \, \text{angstroms} \]

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