Question

Consider the hydrogen atom to be a proton embedded in a cavity of radius ${a_0}$ (Bohr radius) whose charge is neutralized by the addition of an electron to the cavity in vacuum infinity slowly. Estimate the average total energy of an electron in its ground state in a hydrogen atom as the work done in the above neutralization process. Also, if the magnitude of average $KE$ is half the magnitude of average potential energy, find the average potential energy.

Answer 1

Hint : The total energy is the sum of potential energy and the kinetic energy and to make this easier we can also assume kinetic energy as $x$ and potential energy as $\dfrac{x}{2}$ .

Complete Step By Step Answer:
We know, the total energy is
= kinetic energy $(KE)$ + potential energy $(PE)$
We are given that the magnitude of average kinetic energy is half the magnitude of average potential energy.
From the bohr model
For bringing an electron from infinity to the proton of radius the work done is
= $- \dfrac{{{e^2}}}{{(4\pi {E_0}).{a_0}}}$ .
$= TE = - PE + \dfrac{{PE}}{2}$ .
we can write this as
$= - PE = 2 \times TE$ .
$= - \dfrac{{2{e^2}}}{{4\pi {E_0}{a_0}}}$ .
so, the potential energy is
$- \dfrac{{{e^2}}}{{2\pi {E_0}{a_0}}}$ .

Additional Information:
Let’s know more about kinetic and potential energy
Kinetic energy
Kinetic energy is straightforwardly corresponding to the mass of the substance and to the square of its speed: $KE = \dfrac{1}{2}m{v^2}$ . In the event that the mass has units of kilograms and the speed of meters each second, the kinetic energy has units of kilograms-meters squared each second squared. Kinetic energy is normally estimated in units of Joules, one Joule is equivalent to $1\dfrac{{km{m^2}}}{{{s^2}}}$ .
Potential energy
The formula for potential energy relies upon the power following up on the two objects. For the gravitational power the equation is $PE = mgh$ where m is the mass in kilograms, g is the speed increase because of gravity and h is the height in meters. Notice that gravitational potential energy has similar units as kinetic energy. Indeed, all energy has similar units, and is estimated utilizing the unit Joule.

Note :
The relation between the kinetic and potential energy is that the Energy stored in an object due to its position is Potential Energy. Energy that a moving object has due to its motion is Kinetic Energy.