Question

The ratio of radii of second orbit of hydrogen atom to fourth orbit of \(He^+\) ion is:

• A. \(1:4\)
• B. \(2:1\)
• C. \(1:2\)
• D. \(2:3\)

Hint:

For hydrogen-like species, \[ r_n=\frac{n^2a_0}{Z} \] Thus, orbital radius is directly proportional to \(n^2\) and inversely proportional to the atomic number \(Z\).

Answer 1

The Correct Option is C

Step 1: Use Bohr's radius formula for hydrogen-like atoms.
The radius of the \(n^{\text{th}}\) orbit of a hydrogen-like atom is given by \[ r_n=\frac{n^2a_0}{Z} \] where \[ n=\text{principal quantum number} \] \[ a_0=\text{Bohr radius} \] \[ Z=\text{atomic number} \]

Step 2: Calculate the radius of the second orbit of hydrogen.
For hydrogen atom, \[ Z=1 \] and \[ n=2 \] Therefore, \[ r_H=\frac{(2)^2a_0}{1} \] \[ r_H=4a_0 \]

Step 3: Calculate the radius of the fourth orbit of \(He^+\).
For \(He^+\), \[ Z=2 \] and \[ n=4 \] Therefore, \[ r_{He^+}=\frac{(4)^2a_0}{2} \] \[ r_{He^+}=\frac{16a_0}{2} \] \[ r_{He^+}=8a_0 \]

Step 4: Find the required ratio.
\[ \frac{r_H}{r_{He^+}} = \frac{4a_0}{8a_0} \] \[ \frac{r_H}{r_{He^+}} = \frac{1}{2} \] Hence, \[ r_H:r_{He^+}=1:2 \]

Step 5: Final conclusion.
Therefore, the required ratio is \[ \boxed{1:2} \] and the correct option is \[ \boxed{(3)} \]