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`

Answer

To find the ratio of the radii of the first orbits of hydrogen (H), helium ion (He⁺), and lithium ion (Li²⁺), we can use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ r_n = \frac{n^2}{Z} \times 0.529 \, \text{Å} \]

Where:
- \( r_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (1 for the first orbit),
- \( Z \) is the atomic number of the element.

### Step 1: Calculate the radius for Hydrogen (H)
- For hydrogen, \( Z = 1 \) and \( n = 1 \):
\[ r_H = \frac{1^2}{1} \times 0.529 \, \text{Å} = 0.529 \, \text{Å} \]

### Step 2: Calculate the radius for Helium ion (He⁺)
- For helium ion, \( Z = 2 \) and \( n = 1 \):
\[ r_{He^+} = \frac{1^2}{2} \times 0.529 \, \text{Å} = \frac{0.529}{2} \, \text{Å} = 0.2645 \, \text{Å} \]

### Step 3: Calculate the radius for Lithium ion (Li²⁺)
- For lithium ion, \( Z = 3 \) and \( n = 1 \):
\[ r_{Li^{2+}} = \frac{1^2}{3} \times 0.529 \, \text{Å} = \frac{0.529}{3} \, \text{Å} = 0.1763 \, \text{Å} \]

### Step 4: Write the ratios of the radii
Now we have the radii:
- \( r_H = 0.529 \, \text{Å} \)
- \( r_{He^+} = 0.2645 \, \text{Å} \)
- \( r_{Li^{2+}} = 0.1763 \, \text{Å} \)

The ratio of the radii is:
\[ r_H : r_{He^+} : r_{Li^{2+}} = 0.529 : 0.2645 : 0.1763 \]

### Step 5: Simplify the ratio
To simplify the ratio, we can multiply each term by a common factor to eliminate the decimals. Let's multiply by 1000:
\[ 529 : 264.5 : 176.3 \]

Now, to make it simpler, we can multiply each term by 2 to avoid decimals:
\[ 1058 : 529 : 352.6 \]

Finally, we can round these to the nearest whole numbers:
\[ 6 : 3 : 2 \]

### Conclusion
The ratio of the radii of the first orbits of H, He⁺, and Li²⁺ is:
\[ 6 : 3 : 2 \]