Question

In Bohr model of hydrogen atom, the value of potential energy of an electron in nth orbit varies with ‘n’ as

• A. $\frac{1}{n^2}$
• B. $\frac{1}{n}$
• C. $n$
• D. $n^2$

Hint:

In the Bohr model, practically all energy parameters (Kinetic Energy, Potential Energy, Total Energy) for an electron strictly vary proportionally as $1/n^2$.

Answer 1

The Correct Option is A

Concept:
• The Bohr model successfully applies classical mechanics and early quantum quantization rules to deeply describe the hydrogen atom.

• The electron revolves in well-defined circular orbits around the central positive nucleus, with the electrostatic Coulomb force securely providing the requisite centripetal force.

• The absolute radius of the $n$-th allowed Bohr orbit is firmly established to be directly proportional to the square of the principal quantum number $n$ ($r_n \propto n^2$).

• The electrostatic potential energy of the electron-nucleus system depends strictly on their mutual separation distance.

Step 1: State the formula for electrostatic potential energy
For a hydrogen atom, the central nucleus has a charge of $+e$ (one proton) and the revolving electron has a charge of $-e$.
The electrostatic potential energy $U_n$ of this dynamic system at a separation distance $r_n$ is mathematically formulated as:
\[ U_n = \frac{1}{4\pi\epsilon_0} \frac{(+e)(-e)}{r_n} = -\frac{1}{4\pi\epsilon_0} \frac{e^2}{r_n} \]
Here, it is undeniably evident that the potential energy $U_n$ is strictly inversely proportional to the orbital radius $r_n$.

Step 2: Relate the orbital radius to the principal quantum number
From Bohr's rigid quantization condition for angular momentum ($mvr = n\frac{h}{2\pi}$), the explicitly derived formula for the radius of the $n$-th orbit is:
\[ r_n = \frac{\epsilon_0 h^2 n^2}{\pi m e^2} \]
From this fundamental expression, we can clearly isolate the crucial proportionality:
\[ r_n \propto n^2 \]

Step 3: Determine the variation of potential energy with 'n'
We now systematically substitute this proportional relationship for $r_n$ directly back into the potential energy equation.
Since $U_n \propto \frac{1}{r_n}$ and $r_n \propto n^2$, combining these two mathematically yields:
\[ U_n \propto \frac{1}{n^2} \]
This elegantly demonstrates that the absolute magnitude of the potential energy diminishes according to the inverse square of the principal quantum number.

Step 4: Conclusion
The potential energy firmly varies as $1/n^2$. This precisely aligns with option (A). It is also worth noting that the total energy and kinetic energy also scale exactly with $1/n^2$ in the Bohr model.