Question

Considering the Bohr model of hydrogen like atoms, the ratio of the radius $5^{\text {th }}$ orbit of the electron in $\mathrm{Li}^{2+}$ and $\mathrm{He}^{+}$is

• A. $\frac{3}{2}$
• B. $\frac{4}{9}$
• C. $\frac{9}{4}$
• D. $\frac{2}{3}$

Hint:

The radius of an orbit in the Bohr model depends on the principal quantum number and the atomic number.

Answer 1

The Correct Option is D

1. Radius of the $5^{\text{th}}$ orbit for $\mathrm{Li}^{2+}$: \[ r_{5} = \frac{5^2}{3} a_0 \]
2. Radius of the $5^{\text{th}}$ orbit for $\mathrm{He}^{+}$: \[ r_{5} = \frac{5^2}{2} a_0 \]
3. Ratio of the radii: \[ \frac{r_{\mathrm{Li}^{2+}}}{r_{\mathrm{He}^{+}}} = \frac{2}{3} \] Therefore, the correct answer is (4) $\frac{2}{3}$.

Answer 2

We are asked to find the ratio of the radius of the 5^th orbit of an electron in \( \mathrm{Li}^{2+} \) and \( \mathrm{He}^{+} \) ions according to the Bohr model.

Concept Used:

In the Bohr model, the radius of the \( n^{\text{th}} \) orbit for a hydrogen-like atom is given by:

\[ r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2 Z} = a_0 \frac{n^2}{Z} \]

where:

• \( a_0 = 0.529 \times 10^{-10} \, \text{m} \) (Bohr radius)
• \( n \) = principal quantum number
• \( Z \) = atomic number of the element

Step-by-Step Solution:

Step 1: Write the expression for the radius of the 5^th orbit for both ions.

\[ r_{5,\mathrm{Li}^{2+}} = a_0 \frac{5^2}{Z_{\mathrm{Li}}} \] \[ r_{5,\mathrm{He}^{+}} = a_0 \frac{5^2}{Z_{\mathrm{He}}} \]

Step 2: Substitute the atomic numbers:

• For \( \mathrm{Li}^{2+} \), \( Z = 3 \)
• For \( \mathrm{He}^{+} \), \( Z = 2 \)

Step 3: Substitute into the formula and simplify.

\[ r_{5,\mathrm{Li}^{2+}} = a_0 \frac{25}{3} \] \[ r_{5,\mathrm{He}^{+}} = a_0 \frac{25}{2} \]

Step 4: Take the ratio.

\[ \frac{r_{5,\mathrm{Li}^{2+}}}{r_{5,\mathrm{He}^{+}}} = \frac{\frac{25}{3}}{\frac{25}{2}} = \frac{2}{3} \]

Final Computation & Result:

The ratio of the radii of the 5^th orbit is:

\[ \boxed{\frac{r_{5,\mathrm{Li}^{2+}}}{r_{5,\mathrm{He}^{+}}} = \frac{2}{3}} \]

Final Answer: \( \dfrac{2}{3} \)