Experimentally it is found that $12.8 ~\mathrm{eV}$ energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is $\frac{9}{x} \times 10^{-10} \mathrm{~m}$. The value of the $x$ is __________.

$$\left(1 \mathrm{eV}=1.6 \times 10^{-19} \mathrm{~J}, \frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right.$$ and electronic charge $\left.=1.6 \times 10^{-19} \mathrm{C}\right)$

<p>The binding energy of an electron in a hydrogen atom is given by the formula:</p> <p>$E = \frac{k e^2}{2 r}$</p> <p>where:</p> <ul> <li>$E$ is the energy of the electron,</li> <li>$k$ is Coulomb&#39;s constant ($9 \times 10^9 \, \text{Nm}^2/\text{C}^2$),</li> <li>$e$ is the charge of the electron ($1.6 \times 10^{-19} \, \text{C}$), and</li> <li>$r$ is the radius of the orbit.</li> </ul> <p>In this scenario, the energy $E$ required to separate a hydrogen atom into a proton and an electron is given as $12.8 \, \text{eV}$, which needs to be converted into joules using the conversion factor $1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J}$. So, </p> <p>$12.8 \, \text{eV} = 12.8 \times 1.6 \times 10^{-19} \, \text{J}$</p> <p>We can then substitute the given values into the energy equation and solve for $r$:</p> <p>$$ 12.8 \times 1.6 \times 10^{-19} \, \text{J} = \frac{9 \times 10^9 \times (1.6 \times 10^{-19})^2}{2r} $$</p> <p>Solving for $r$, we get:</p> <p>$$ r = \frac{9 \times 10^9 \times (1.6 \times 10^{-19})^2}{2 \times 12.8 \times 1.6 \times 10^{-19}} $$</p> <p>This simplifies to:</p> <p>$r = \frac{9 \times 10^{-10}}{16}$</p> <p>Comparing this with the given form of the radius, which is $\frac{9}{x} \times 10^{-10}$, we find that the value of $x$ is 16.</p>