Question

The ground state energy of hydrogen atom is -13.6 eV. The potential energy of the electron in this state is:

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

Hint:

In the ground state of a hydrogen atom, the potential energy is twice the total energy and opposite in sign. Use this relationship to find the potential energy when the total energy is given.

Answer 1

The Correct Option is B

In the context of the hydrogen atom, the total energy \( E \) of an electron in a certain state is the sum of its kinetic energy \( T \) and potential energy \( V \). For the ground state, the total energy given is \( E = -13.6 \, \text{eV} \).

The potential energy \( V \) of the electron in this state is calculated by using the relationship between total energy and potential energy in a Coulomb potential. The potential energy is twice the magnitude but opposite in sign to the total energy, as per the Virial Theorem, which states:

\( E = T + V \) and \( T = -\frac{1}{2}V \).

Substituting for \( T \) gives:

\( E = -\frac{1}{2}V + V \) which simplifies to \( E = \frac{1}{2}V \).

Thus, \( V = 2E \).

With \( E = -13.6 \, \text{eV} \), we find:

\( V = 2 \times (-13.6 \, \text{eV}) = -27.2 \, \text{eV} \).

Therefore, the potential energy of the electron in the ground state of the hydrogen atom is \(-27.2 \, \text{eV}\).

Answer 2

In the ground state of a hydrogen atom, the total energy \( E \) is given by the sum of the kinetic energy \( K \) and potential energy \( U \), such that: \[ E = K + U \] The total energy in the ground state is given as: \[ E = -13.6 \, {eV} \] For a hydrogen atom, the potential energy \( U \) is twice the negative value of the kinetic energy, i.e., \[ U = 2K \] Additionally, since the total energy is the sum of kinetic and potential energy, we have: \[ E = K + U = K + 2K = 3K \] Thus, the kinetic energy \( K \) is: \[ K = \frac{E}{3} = \frac{-13.6}{3} = -4.53 \, {eV} \] Since the potential energy \( U = 2K \), we can calculate the potential energy: \[ U = 2 \times (-4.53) = -9.06 \, {eV} \] Thus, the potential energy of the electron in this state is \( -27.2 \, {eV} \), which is twice the total energy of the system.