Question

In the hydrogen atom, what is the physical significance of the state for which \(n=\infty\) and \(E=0 ?\)

Short answer

The physical significance of the hydrogen atom state when the principal quantum number \(n\) approaches infinity and the energy \(E\) is zero is that the electron is on the verge of being detached from the hydrogen atom. This state corresponds to the transition of the hydrogen atom from an atomic existence to its ionized state.

Step-by-step solution

Recall the hydrogen atom energy levels formula

The energy levels of a hydrogen atom can be calculated using the following formula: \[ E = -\frac{13.6 eV}{n^2} \] where \(E\) is the energy of the electron in the hydrogen atom, \(n\) is the principal quantum number, and 13.6 eV is the ionization energy of the hydrogen atom.

Calculate the energy for n approaching infinity

As the principal quantum number, \(n\), approaches infinity, we calculate the energy as follows: \[ E_{n \to \infty} = -\frac{13.6 eV}{n^2} \] Since \(n\) approaches infinity, the expression \(n^2\) will also approach infinity. Therefore: \[ E_{n \to \infty} = -\frac{13.6 eV}{\infty} = 0 \]

Consider the electron's behavior for E=0

When the energy of an electron in a hydrogen atom approaches zero, it means that the electron is moving further away from the nucleus of the hydrogen atom. At a very high value of the principal quantum number, the atom's electron moves away from the nucleus and the distance between the electron and the nucleus increases, eventually reaching a point where the electron is almost detached from the hydrogen atom.

Physical Significance of n approaching infinity and E=0

The physical significance of the hydrogen atom state when the principal quantum number approaches infinity and the energy is zero indicates that the electron is on the verge of being detached from the hydrogen atom. The hydrogen atom becomes ionized as the electron reaches the ionization energy, which is equal to 13.6 eV. Essentially, this state corresponds to the transition of the hydrogen atom from anatomic existence to its ionized state.