Animated Solution for Physics - Atoms and Nuclei: In the Bohr model of the hydrogen atom, the ratio of the kinetic energy to the total energy of the electron in a quantum state $n$ is ........... .

Imagine you are looking deep into the heart of a hydrogen atom. At the center lies a positively charged proton, massive and stationary. Orbiting around it, much like a planet around the sun, is a tiny, negatively charged electron. This is the classic Bohr model, a beautiful stepping stone between classical mechanics and quantum physics.

The electron doesn't just fly off into space; it is held in a delicate dance by the invisible, yet immensely powerful, electrostatic force of attraction. This force acts as the centripetal force, constantly pulling the electron towards the nucleus and keeping it in a stable, circular orbit.

But what does it mean for the electron to be in an orbit? It means it possesses energy. Energy of motion, because it is whizzing around at incredible speeds, and energy of position, because it is trapped in the electrostatic field of the nucleus. Understanding the interplay between these energies is the key to unlocking the secrets of the atom.

Let's break down the total energy of this orbiting electron. As it moves, it possesses kinetic energy, which we denote as $K$. Through the rigorous application of Bohr's postulates and classical electrodynamics, we can derive the expression for the kinetic energy of an electron in the $n^{\text{th}}$ quantum state.

The kinetic energy is given by the elegant formula:

Here, $m$ is the mass of the electron, $e$ is the elementary charge, ${ϵ}_0$ is the permittivity of free space, $h$ is Planck's constant, and $n$ is the principal quantum number that dictates the specific orbit. Notice that the kinetic energy is inherently positive, as it depends on the square of the velocity.

Now, what about the potential energy, $U$? The electron is in a potential well created by the positive nucleus. Because the electrostatic force is attractive, the potential energy is negative. In fact, it is exactly twice the magnitude of the kinetic energy, but with a negative sign: $U=-2K$.

The total energy, $E$, is simply the sum of the kinetic and potential energies:

Substituting our expression for $K$, we find the total energy:

The negative sign here is profoundly significant. It tells us that the electron is in a bound state. It is trapped. To free the electron from the atom—to ionize it—we would need to supply an amount of energy exactly equal to the magnitude of this total energy.

The question poses a simple yet profound query: what is the ratio of the kinetic energy to the total energy?

Let's set up the ratio using the expressions we just discussed:

Look closely at this fraction. It might seem intimidating at first glance, filled with constants and variables. But watch what happens when we simplify it. The mass $m$ cancels out. The charge $e^4$ cancels out. The constants $8{ϵ}_0^2{h}^2$ cancel out. Even the principal quantum number $n^2$ cancels out!

We are left with a beautifully simple result:

This means that the kinetic energy is exactly equal to the negative of the total energy.

Why did all those complex terms cancel out so perfectly? Is it just a mathematical coincidence? Absolutely not. This result is a direct consequence of the Virial Theorem applied to inverse-square force fields.

Whether you are looking at an electron orbiting a nucleus under the electrostatic Coulomb force, or a satellite orbiting the Earth under the gravitational force, the underlying physics is identical. Both forces follow an inverse-square law ($F\propto 1/r^2$).

For any system bound by an inverse-square attractive force, the time-averaged kinetic energy is always equal to half the magnitude of the time-averaged potential energy, and therefore, exactly equal to the negative of the total energy.

So, the ratio of $-1$ is not just a quirk of the hydrogen atom; it is a universal signature of the fundamental forces that shape our universe, from the microscopic realm of atoms to the macroscopic dance of galaxies. It is a testament to the elegant symmetry and interconnectedness of physical laws.