The binding energy of an electron in a hydrogen atom is given by the formula:

$$ E = \frac{k e^2}{2 r} $$

where:

• $E$ is the energy of the electron,
• $k$ is Coulomb's constant ($9 \times 10^9 \, \text{Nm}^2/\text{C}^2$),
• $e$ is the charge of the electron ($1.6 \times 10^{-19} \, \text{C}$), and
• $r$ is the radius of the orbit.

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,

$$ 12.8 \, \text{eV} = 12.8 \times 1.6 \times 10^{-19} \, \text{J} $$

We can then substitute the given values into the energy equation and solve for $r$:

$$ 12.8 \times 1.6 \times 10^{-19} \, \text{J} = \frac{9 \times 10^9 \times (1.6 \times 10^{-19})^2}{2r} $$

Solving for $r$, we get:

$$ r = \frac{9 \times 10^9 \times (1.6 \times 10^{-19})^2}{2 \times 12.8 \times 1.6 \times 10^{-19}} $$

This simplifies to:

$$ r = \frac{9 \times 10^{-10}}{16} $$

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.