Question

Radius of the first excited state of Helium ion is given as:
\(a_0\) = radius of first stationary state of hydrogen atom.

• A. \( r = \frac{a_0}{2} \)
• B. \( r = \frac{a_0}{4} \)
• C. \( r = 4a_0 \)
• D. \( r = 2a_0 \)

Hint:

To find the radius of the excited state of an atom or ion, use the formula \( r = \frac{a_0 n^2}{Z} \), where \( a_0 \) is the Bohr radius.

Answer 1

The Correct Option is D

To find the radius of the first excited state of the Helium ion (\(He^+\)), we must consider the Bohr model of the atom. According to the Bohr model, the radius of an electron's orbit in a hydrogen-like ion is given by the formula:

\[ r_n = a_0 \frac{n^2}{Z} \]

where \( r_n \) is the radius of the orbit, \( a_0 \) is the Bohr radius, \( n \) is the principal quantum number, and \( Z \) is the atomic number of the ion.

For the first excited state, \( n = 2 \), and for the helium ion (\(He^+\)), \( Z = 2 \). Inserting these values into the formula:

\[ r_2 = a_0 \frac{2^2}{2} = a_0 \frac{4}{2} = 2a_0 \]

Thus, the radius of the first excited state of the helium ion is \( 2a_0 \).

The correct answer is:

\( r = 2a_0 \)

Answer 2

Step 1 — Formula for radius of nth orbit: 
For a hydrogen-like ion, $$r_n = \dfrac{n^2 a_0}{Z}$$ where:
\( r_n \) = radius of nth orbit,
\( a_0 \) = Bohr radius of hydrogen atom,
\( Z \) = atomic number of the ion. 

Step 2 — For Helium ion (He⁺):
Helium ion has \( Z = 2 \). 

Step 3 — For first excited state:
First excited state means \( n = 2 \). 

Step 4 — Substituting values:
$$r = \dfrac{n^2 a_0}{Z} = \dfrac{(2)^2 a_0}{2} = \dfrac{4a_0}{2} = 2a_0$$ 
Therefore, the radius of the first excited state of He⁺ ion is:
$$\boxed{r = 2a_0}$$ 
Correct Option: (4)