Question

Energy of hydrogen atom in ground state is −13·6 eV. What will be the kinetic energy and the potential energy of the electron in the state?

Hint:

For hydrogen use \(K = -E\) and \(U = 2E\); the kinetic energy is positive and the potential energy is twice the total energy.

Answer 1

Step 1: Use the virial theorem for a Coulomb field.
For an inverse-square attractive force, the virial theorem gives \(2K = -U\), i.e. the kinetic energy is half the magnitude of the potential energy and opposite in usage.

Step 2: Combine with total energy.
Since \(E = K + U\) and \(U = -2K\), we get \(E = K - 2K = -K\). Hence \(K = -E\).

Step 3: Insert the ground-state total energy.
Given \(E = -13.6\ \text{eV}\), the kinetic energy is \(K = -(-13.6) = 13.6\ \text{eV}\).

Step 4: Get the potential energy.
From \(U = -2K\), \(U = -2(13.6) = -27.2\ \text{eV}\). The negative sign shows the electron is bound to the nucleus.

Step 5: Verify the total.
\(K + U = 13.6 - 27.2 = -13.6\ \text{eV}\), matching the given total energy.

\[\boxed{K = 13.6\ \text{eV},\quad U = -27.2\ \text{eV}}\]