The ratio of radii of first orbits of H, `He^(+)` and `Li^(2+)` is:

Options

A. `1:2:3`
B. `6:3:2`
C. `1:4:9`
D. `9:4:1`

Explanation

To find the ratio of the radii of the first orbits of H, He\(^+\), and Li\(^{2+}\), we can use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ r_n = \frac{n^2}{Z} \cdot r_0 \]

where:
- \(r_n\) is the radius of the nth orbit,
- \(Z\) is the atomic number,
- \(r_0\) is the radius of the first orbit of hydrogen, which is approximately \(0.529 \, \text{Å}\),
- \(n\) is the principal quantum number (for the first orbit, \(n = 1\)).

### Step 1: Calculate the radius for Hydrogen (H)
For hydrogen, \(Z = 1\):
\[ r_H = \frac{1^2}{1} \cdot r_0 = 1 \cdot r_0 = r_0 \]

### Step 2: Calculate the radius for Helium ion (He\(^+\))
For helium ion, \(Z = 2\):
\[ r_{He^+} = \frac{1^2}{2} \cdot r_0 = \frac{1}{2} r_0 \]

### Step 3: Calculate the radius for Lithium ion (Li\(^{2+}\))
For lithium ion, \(Z = 3\):
\[ r_{Li^{2+}} = \frac{1^2}{3} \cdot r_0 = \frac{1}{3} r_0 \]

### Step 4: Form the ratio of the radii
Now, we can express the ratio of the radii of H, He\(^+\), and Li\(^{2+}\):
\[ \text{Ratio} = r_H : r_{He^+} : r_{Li^{2+}} = r_0 : \frac{1}{2} r_0 : \frac{1}{3} r_0 \]

### Step 5: Simplify the ratio
To simplify the ratio, we can multiply each term by \(6\) (the least common multiple of the denominators 1, 2, and 3):
\[ \text{Ratio} = 6 : 3 : 2 \]

### Final Answer
Thus, the ratio of the radii of the first orbits of H, He\(^+\), and Li\(^{2+}\) is:
\[ \boxed{6 : 3 : 2} \]