Question 377: Atoms & Nuclei

Question

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 $$\frac{9}{x} 	imes 10^{-10} \mathrm{~m}$$. The value of the $$x$$ is __________.
$$\left(1 \mathrm{eV}=1.6 	imes 10^{-19} \mathrm{~J}, \frac{1}{4 \pi \epsilon_{0}}=9 	imes 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}ight.$$ and electronic charge $$\left.=1.6 	imes 10^{-19} \mathrm{C}ight)$$

Accepted Answer

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 ($9 	imes 10^9 \, 	ext{Nm}^2/	ext{C}^2$),
$e$ is the charge of the electron ($1.6 	imes 10^{-19} \, 	ext{C}$), 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 $12.8 \, 	ext{eV}$, which needs to be converted into joules using the conversion factor $1 \, 	ext{eV} = 1.6 	imes 10^{-19} \, 	ext{J}$. So,
$$ 12.8 \, 	ext{eV} = 12.8 	imes 1.6 	imes 10^{-19} \, 	ext{J} $$
We can then substitute the given values into the energy equation and solve for $r$:
$$ 12.8 	imes 1.6 	imes 10^{-19} \, 	ext{J} = \frac{9 	imes 10^9 	imes (1.6 	imes 10^{-19})^2}{2r} $$
Solving for $r$, we get:
$$ r = \frac{9 	imes 10^9 	imes (1.6 	imes 10^{-19})^2}{2 	imes 12.8 	imes 1.6 	imes 10^{-19}} $$
This simplifies to:
$$ r = \frac{9 	imes 10^{-10}}{16} $$
Comparing this with the given form of the radius, which is $\frac{9}{x} 	imes 10^{-10}$, we find that the value of $x$ is 16.

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