Question

If radius of second Bohr orbit of the He⁺ ion is 105.8pm, What is the radius of third Bohr Orbit of Li²⁺ ion?

• A. 158.7 pm
• B. 15.87 pm
• C. 1.587 pm
• D. 158.7 Å

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
According to Bohr's model, the radius of the \(n^{th}\) orbit of a hydrogen-like species is proportional to \(\frac{n^{2}}{Z}\), where \(n\) is the principal quantum number and \(Z\) is the atomic number.
Key Formula or Approach:
\[ r_{n} = a_{0} \frac{n^{2}}{Z} \]
Where \(a_{0}\) is the Bohr radius (radius of the 1st orbit of Hydrogen).
Step 2: Detailed Explanation:
For \(He^{+}\) ion (\(Z = 2\)), \(n = 2\):
\(r_{2}(He^{+}) = a_{0} \frac{2^{2}}{2} = 2a_{0} = 105.8\) pm.
From this, \(a_{0} = \frac{105.8}{2} = 52.9\) pm.
Now, for \(Li^{2+}\) ion (\(Z = 3\)), \(n = 3\):
\(r_{3}(Li^{2+}) = a_{0} \frac{3^{2}}{3} = 3a_{0}\).
Substitute the value of \(a_{0}\):
\(r_{3}(Li^{2+}) = 3 \times 52.9 = 158.7\) pm.
Step 3: Final Answer:
The radius of the third Bohr orbit of \(Li^{2+}\) is 158.7 pm.