The ratio of areas within the elctron orbits for the first excited state to the ground sate for hydrogen atom is

Options

A. `16:1`
B. `4:1`
C. `8:1`
D. `1:8`

Answer

To solve the problem of finding the ratio of areas within the electron orbits for the first excited state to the ground state for a hydrogen atom, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Quantum Numbers**:
- For the ground state of hydrogen, the principal quantum number \( n_1 = 1 \).
- For the first excited state, the principal quantum number \( n_2 = 2 \).

2. **Understand the Relationship of Radius and Quantum Number**:
- The radius of the electron orbit in a hydrogen atom is given by the formula:
\[ r_n \propto n^2 \]
- This means that the radius is directly proportional to the square of the principal quantum number.

3. **Calculate the Radii for Both States**:
- For the ground state (\( n_1 = 1 \)):
\[ r_1 \propto 1^2 = 1 \]
- For the first excited state (\( n_2 = 2 \)):
\[ r_2 \propto 2^2 = 4 \]

4. **Determine the Areas of the Orbits**:
- The area \( A \) of the orbit is given by the formula:
\[ A = \pi r^2 \]
- Therefore, the areas for both states can be expressed as:
- For the ground state:
\[ A_1 = \pi r_1^2 \propto \pi (1^2) = \pi \]
- For the first excited state:
\[ A_2 = \pi r_2^2 \propto \pi (4^2) = 16\pi \]

5. **Calculate the Ratio of Areas**:
- Now, we can find the ratio of the areas:
\[ \text{Ratio} = \frac{A_2}{A_1} = \frac{16\pi}{\pi} = 16 \]

6. **Final Result**:
- The ratio of the areas within the electron orbits for the first excited state to the ground state is:
\[ \text{Ratio} = 16:1 \]

### Conclusion:
The correct answer is that the ratio of areas within the electron orbits for the first excited state to the ground state for a hydrogen atom is \( 16:1 \).