If `r_(1)` is the radius of first orbit of hydrogen atom, then the radii of second, third and fourth orbits in terms of `r_(1)` are

Options

A. `r_(1)^(2), r_(1)^(3), r_(1)^(4)`
B. `8r_(1), 27r_(1), 64r_(1)`
C. `4r_(1),9r_(1),16r_(1)`
D. `2r_(1),6r_(1), 8r_(1)`

Answer

To find the radii of the second, third, and fourth orbits of a hydrogen atom in terms of the radius of the first orbit \( r_1 \), we can follow these steps:

### Step 1: Understand the relationship between radius and orbit number
The radius of an orbit in a hydrogen atom is given by the formula:
\[ r_n = k \frac{n^2}{z} \]
where:
- \( r_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (orbit number),
- \( z \) is the atomic number (for hydrogen, \( z = 1 \)),
- \( k \) is a constant.

### Step 2: Calculate the radius of the first orbit
For the first orbit (\( n = 1 \)):
\[ r_1 = k \frac{1^2}{z} = k \frac{1}{1} = k \]
Thus, we have:
\[ r_1 = k \]

### Step 3: Calculate the radius of the second orbit
For the second orbit (\( n = 2 \)):
\[ r_2 = k \frac{2^2}{z} = k \frac{4}{1} = 4k \]
Now, substituting \( k = r_1 \):
\[ r_2 = 4r_1 \]

### Step 4: Calculate the radius of the third orbit
For the third orbit (\( n = 3 \)):
\[ r_3 = k \frac{3^2}{z} = k \frac{9}{1} = 9k \]
Substituting \( k = r_1 \):
\[ r_3 = 9r_1 \]

### Step 5: Calculate the radius of the fourth orbit
For the fourth orbit (\( n = 4 \)):
\[ r_4 = k \frac{4^2}{z} = k \frac{16}{1} = 16k \]
Substituting \( k = r_1 \):
\[ r_4 = 16r_1 \]

### Final Result
Thus, the radii of the second, third, and fourth orbits in terms of \( r_1 \) are:
- \( r_2 = 4r_1 \)
- \( r_3 = 9r_1 \)
- \( r_4 = 16r_1 \)

### Summary
The final expressions for the radii are:
- \( r_2 = 4r_1 \)
- \( r_3 = 9r_1 \)
- \( r_4 = 16r_1 \)