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

The radius of the first excited state for the Helium ion (\(He^+\)) is determined using the Bohr model. This model provides the orbital radius for electrons in hydrogen-like ions via the formula:

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

In this equation, \( r_n \) denotes the orbital radius, \( a_0 \) represents the Bohr radius, \( n \) is the principal quantum number, and \( Z \) is the atomic number of the ion.

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

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

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

The definitive result is:

\( r = 2a_0 \)