If the radius of first, second, third and fourth orbit of hydrogen atom are `r_(1),r_(2),r_(3) and r_(4)` respectively .Then their correct increasing order will be:

Options

A. (1) `r_(4)` `lt`` r_(3)` `lt` `r_(2)` `lt``r_(1)`
B. (2) `r_(1)` `lt` `r_(2)` `lt` `r_(3)` `lt``r_(4)`
C. (3) `r_(4)` `lt` `r_(3)` `lt` `r_(2)` `gt``r_(1)`
D. (4) Equal in all

Explanation

To solve the problem of determining the correct increasing order of the radii of the first, second, third, and fourth orbits of a hydrogen atom, we will use the formula for the radius of an orbit in a hydrogen atom.

### Step-by-Step Solution:

1. **Understanding the Formula**:
The radius of the nth orbit of a hydrogen atom is given by the formula:
\[ r_n = 0.529 \times \frac{n^2}{Z} \text{ angstroms} \]
where \( Z \) is the atomic number (for hydrogen, \( Z = 1 \)) and \( n \) is the principal quantum number.

2. **Calculating the Radius for Each Orbit**:
- **First Orbit (n=1)**:
\[ r_1 = 0.529 \times \frac{1^2}{1} = 0.529 \text{ angstroms} \]
- **Second Orbit (n=2)**:
\[ r_2 = 0.529 \times \frac{2^2}{1} = 0.529 \times 4 = 2.116 \text{ angstroms} \]
- **Third Orbit (n=3)**:
\[ r_3 = 0.529 \times \frac{3^2}{1} = 0.529 \times 9 = 4.761 \text{ angstroms} \]
- **Fourth Orbit (n=4)**:
\[ r_4 = 0.529 \times \frac{4^2}{1} = 0.529 \times 16 = 8.464 \text{ angstroms} \]

3. **Comparing the Values**:
Now we have the radii:
- \( r_1 = 0.529 \) angstroms
- \( r_2 = 2.116 \) angstroms
- \( r_3 = 4.761 \) angstroms
- \( r_4 = 8.464 \) angstroms

4. **Determining the Increasing Order**:
The increasing order of the radii is:
\[ r_1 < r_2 < r_3 < r_4 \]

5. **Final Answer**:
Therefore, the correct increasing order of the radii of the first, second, third, and fourth orbits of a hydrogen atom is:
\[ r_1 < r_2 < r_3 < r_4 \]