When hydrogen atom is in first excited level, its radius is….its ground state radius

Options

A. four times, its gound state radius
B. twice, its gound state radius
C. same as its ground sate radius
D. half of its ground state radius

Answer

To determine the radius of a hydrogen atom in its first excited state in relation to its ground state radius, we can use Bohr's model of the hydrogen atom. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Understanding the Formula**:
According to Bohr's model, the radius of the nth orbit (Rn) of a hydrogen atom is given by the formula:
\[ R_n = \frac{n^2}{Z} R_0 \]
where \( R_0 \) is the ground state radius (the radius of the first orbit), \( n \) is the principal quantum number, and \( Z \) is the atomic number.

2. **Ground State Radius**:
For the ground state of hydrogen:
- The principal quantum number \( n = 1 \)
- The atomic number \( Z = 1 \) (since hydrogen has one proton)

Using the formula:
\[ R_1 = \frac{1^2}{1} R_0 = R_0 \]
Thus, the radius of the ground state \( R_1 \) is equal to \( R_0 \).

3. **First Excited State Radius**:
For the first excited state:
- The principal quantum number \( n = 2 \)
- The atomic number \( Z = 1 \)

Using the formula:
\[ R_2 = \frac{2^2}{1} R_0 = \frac{4}{1} R_0 = 4 R_0 \]
Thus, the radius of the first excited state \( R_2 \) is equal to \( 4 R_0 \).

4. **Comparison of Radii**:
Now, we can compare the radius of the first excited state to the ground state radius:
\[ R_2 = 4 R_1 \]
Therefore, the radius of the hydrogen atom in its first excited state is 4 times the radius of its ground state.

### Final Answer:
The radius of the hydrogen atom in its first excited state is **4 times** its ground state radius.

---