**59.** Radius of the first excited state of Helium ion is given as: \( a_0 \rightarrow \) radius of first stationary state of hydrogen atom. Options: - (1) \( r = \frac{a_0}{2} \) - (2) \( r = \frac{a_0}{4} \) - (3) \( r = 4a_0 \) - (4) \( r = 2a_0 \)

Key Ideas to Know

1. Bohr Radius ($a_0$)
   The radius of the ground state ($n = 1$) of the hydrogen atom.

2. Bohr Model for Hydrogen-Like Species
   For any one-electron ion (H, He⁺, Li²⁺ …) the radius of the $n^{\text{th}}$ orbit is

   $r_n = \frac{n^2 a_0}{Z}$

   where
   • $n$ = principal quantum number (1, 2, 3 …)
   • $Z$ = nuclear charge (1 for H, 2 for He⁺, 3 for Li²⁺ …).

3. First Excited State
   Means $n = 2$ (because the ground state is $n = 1$).

Reasoning Chain a Student Should Follow

1. Identify Z: For He⁺, $Z = 2$.

2. Identify n: First excited state → $n = 2$.

3. Plug into Formula:

   $r = \frac{(2)^2 a_0}{2}$

4. Simplify:

   $r = \frac{4 a_0}{2} = 2a_0$

5. Match Option: Option (4) is the correct one.

Each step is chosen directly from Bohr’s radius formula, tailored to a hydrogen-like ion with higher nuclear charge.

Imagine Planets and Orbits

Think of an atom like a mini-solar system. The electron is the planet, and the nucleus is the Sun. For hydrogen, the very first circular path (orbit) has a special size called Bohr radius and we name it $a_0$.

Now, a helium ion (He⁺) is like hydrogen but the Sun is twice as strong (because its charge is +2 instead of +1). When the planet jumps to the next bigger orbit (first excited state), you simply ask:

1. How big is that orbit for hydrogen?
2. How does a stronger Sun pull the planet closer?

The math answer says the new size is $2a_0$. So the helium planet’s first excited circle is exactly twice the Bohr radius.

Step-by-Step Solution

1. Write Bohr Radius Formula for Hydrogen-Like Ion

$r_n = \frac{n^2 a_0}{Z}$

2. Insert Helium Ion Data
   Nuclear charge: $Z = 2$
   First excited state: $n = 2$

3. Substitute

$r = \frac{(2)^2 a_0}{2}$

4. Calculate

$r = \frac{4a_0}{2} = 2a_0$

5. Select Correct Option
   Option (4) $r = 2a_0$ is correct.