The radius of $2^{\text {nd }}$ orbit of $\mathrm{He}^{+}$ of Bohr's model is $r_{1}$ and that of fourth orbit of $\mathrm{Be}^{3+}$ is represented as $r_{2}$. Now the ratio $\frac{r_{2}}{r_{1}}$ is $x: 1$. The value of $x$ is ___________.

<p>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:</p> $r_n = \frac{n^2 \cdot h^2 \cdot \epsilon_0}{\pi \cdot Z \cdot e^2 \cdot m_e}$ <p>Where:</p> <ul> <li>$r_n$ is the radius of the nth orbit</li> <li>$n$ is the principal quantum number (orbit number)</li> <li>$h$ is the Planck's constant</li> <li>$\epsilon_0$ is the vacuum permittivity</li> <li>$Z$ is the atomic number (number of protons in the nucleus)</li> <li>$e$ is the elementary charge</li> <li>$m_e$ is the mass of the electron</li> <li>$\pi$ is the mathematical constant pi</li> </ul> <p>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:</p> <p>For the 2nd orbit of $\mathrm{He}^{+}$ ($n_1 = 2$ and $Z_1 = 2$):</p> $$r_{1} = \frac{n_1^2 \cdot h^2 \cdot \epsilon_0}{\pi \cdot Z_1 \cdot e^2 \cdot m_e}$$ <p>For the 4th orbit of $\mathrm{Be}^{3+}$ ($n_2 = 4$ and $Z_2 = 4$):</p> $$r_{2} = \frac{n_2^2 \cdot h^2 \cdot \epsilon_0}{\pi \cdot Z_2 \cdot e^2 \cdot m_e}$$ <p>We are asked to find the ratio $\frac{r_{2}}{r_{1}}$, which is equal to $x: 1$:</p> $$\frac{r_{2}}{r_{1}} = \frac{\frac{n_2^2 \cdot h^2 \cdot \epsilon_0}{\pi \cdot Z_2 \cdot e^2 \cdot m_e}}{\frac{n_1^2 \cdot h^2 \cdot \epsilon_0}{\pi \cdot Z_1 \cdot e^2 \cdot m_e}}$$ <p>By simplifying the expression, we get:</p> $$\frac{r_{2}}{r_{1}} = \frac{n_2^2 \cdot Z_1}{n_1^2 \cdot Z_2} = \frac{4^2 \cdot 2}{2^2 \cdot 4}$$ <p>Now we can calculate the value of $x$:</p> $x = \frac{r_{2}}{r_{1}} = \frac{16 \cdot 2}{4 \cdot 4} = \frac{32}{16} = 2$ <p>Therefore, the value of $x$ in the ratio $\frac{r_{2}}{r_{1}} = x: 1$ is $\boxed{2}$.