How many times larger is a hydrogen atom than the radius of an H atom in its ground state if the H atom with an electron characterised by a quantum number of 106?

Options

A. 106
B. 212
C. 11236
D. none of these

Answer

To solve the problem of how many times larger the radius of a hydrogen atom with an electron characterized by a quantum number of 106 is compared to the radius of a hydrogen atom in its ground state, we can follow these steps:

### Step 1: Understand the formula for the radius of an electron in a hydrogen atom
The radius of an electron in a hydrogen atom can be calculated using the formula:
\[ R_n = R_0 \times \frac{n^2}{Z} \]
where:
- \( R_n \) is the radius of the nth orbit,
- \( R_0 \) is the radius of the hydrogen atom in its ground state (approximately 0.529 Å),
- \( n \) is the principal quantum number,
- \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)).

### Step 2: Substitute the values into the formula
In this case, we have:
- \( n = 106 \)
- \( Z = 1 \)

Thus, the radius for \( n = 106 \) becomes:
\[ R_{106} = R_0 \times \frac{106^2}{1} \]
\[ R_{106} = R_0 \times 106^2 \]

### Step 3: Calculate \( 106^2 \)
Now, we calculate \( 106^2 \):
\[ 106^2 = 11236 \]

### Step 4: Substitute back to find \( R_{106} \)
Now, substituting back:
\[ R_{106} = R_0 \times 11236 \]

### Step 5: Find how many times larger \( R_{106} \) is compared to \( R_0 \)
To find how many times larger \( R_{106} \) is compared to \( R_0 \), we can set up the ratio:
\[ \text{Ratio} = \frac{R_{106}}{R_0} = 11236 \]

### Conclusion
Thus, the radius of the hydrogen atom with an electron characterized by a quantum number of 106 is **11236 times larger** than the radius of a hydrogen atom in its ground state.