Question

In Bohr's model of the atom, the radius of a hydrogen atom in the ground state is $r_1$ and the radius of a He+ ion in the ground state is $r_2$. Which of the following is correct?

• A. r1/r2 =4
• B. r1/r2 = 1/2
• C. r2/r1 =1/4
• D. r2/r1 = 1/2

Answer 1

The Correct Option is D

In Bohr’s model, the radius of an orbit is given by:

$r_n = \dfrac{n^2 h^2}{4\pi^2 m e^2 Z} \propto \dfrac{n^2}{Z}$ 

Where:
- $n$ is the principal quantum number
- $Z$ is the atomic number
- For ground state, $n = 1$ 

Let’s compare the radii: 
Hydrogen (H): $Z = 1$, so $r_1 \propto \dfrac{1^2}{1} = 1$
Helium ion (He⁺): $Z = 2$, so $r_2 \propto \dfrac{1^2}{2} = \dfrac{1}{2}$ 

So, 
$\dfrac{r_2}{r_1} = \dfrac{1/2}{1} = \mathbf{\dfrac{1}{2}}$ 

Answer: $\dfrac{r_2}{r_1} = \dfrac{1}{2}$