Question

In Bohr model of the hydrogen atom, the potential energy of electron in the ground state is:

• A. 13.6 eV
• B. \(-13.6\) eV
• C. 27.2 eV
• D. \(-27.2\) eV

Hint:

The potential energy is minus two times the kinetic energy. In the ground state the kinetic energy is 13.6 eV, so U is -27.2 eV.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In the Bohr model, the electron is bound to the nucleus by the Coulomb force. The total energy of the electron in the nth orbit is \( E_n = -\dfrac{13.6}{n^2} \text{ eV} \).

Step 2: Relation between energies:
For an orbit of radius r, the kinetic energy is \( K = \dfrac{e^2}{8\pi\varepsilon_0 r} \), and the potential energy is \( U = -\dfrac{e^2}{4\pi\varepsilon_0 r} \).
So \( U = -2K \) and \( E = K + U = -K \).

Step 3: Ground state:
For \( n = 1 \), \( E = -13.6 \text{ eV} \), so \( K = +13.6 \text{ eV} \).
\( U = -2K = -2 \times 13.6 = -27.2 \text{ eV} \).

Step 4: Why the other options are wrong:
(A) +13.6 eV is the kinetic energy, not the potential energy.
(B) \( -13.6 \) eV is the total energy of the ground state.
(C) +27.2 eV has the right size but the wrong sign. The potential energy of a bound electron is negative.

Final Answer:
The potential energy of the electron in the ground state is \( -27.2 \) eV, option (D). \[ \boxed{U = -27.2 \text{ eV}} \]