QJEEM23Ph076d: JEE Main 2023 Physics - Class 12 JEEM23Ph076d Solution: The radius of orbit of of Bohr's model is and that of fourth…

To find the value of $x$ , we need to first determine the expressions for the radii of the specified orbits for $\mathrm{He}^{+}$ and $\mathrm{Be}^{3+}$ according to Bohr's model. The radius of an orbit in a hydrogen-like atom (an atom with only one electron) is given by:

Where:

• $r_n$ is the radius of the nth orbit
• $n$ is the principal quantum number (orbit number)
• $h$ is the Planck's constant
• $\epsilon_0$ is the vacuum permittivity
• $Z$ is the atomic number (number of protons in the nucleus)
• $e$ is the elementary charge
• $m_e$ is the mass of the electron
• $\pi$ is the mathematical constant pi

In this problem, we are looking at the 2nd orbit of $\mathrm{He}^{+}$ (which has an atomic number $Z = 2$ ) and the 4th orbit of $\mathrm{Be}^{3+}$ (which has an atomic number $Z = 4$ ). Let's calculate the radii for these orbits:

For the 2nd orbit of $\mathrm{He}^{+}$ ( $n_1 = 2$ and $Z_1 = 2$ ):

For the 4th orbit of $\mathrm{Be}^{3+}$ ( $n_2 = 4$ and $Z_2 = 4$ ):

We are asked to find the ratio $\frac{r_{2}}{r_{1}}$ , which is equal to $x: 1$ :

By simplifying the expression, we get:

Now we can calculate the value of $x$ :

Therefore, the value of $x$ in the ratio $\frac{r_{2}}{r_{1}} = x: 1$ is $\boxed{2}$ .