The energy of the electron in any given state for the hydrogen atom is $-13.6/n^2\, eV$. Identify the correct statements among the following.

(A) the negative sign in $-13.6\, eV$ signifies that the electron is bound to the nucleus.
(B) the negative sign in $-13.6\, eV$ signifies that the electron requires $13.6\, eV$ energy to be separated from the nucleus.
(C) $3.4\, eV$ energy is required to take the electron from the ground to the first excited state.
(D) to find the energy of third excited state, the value of $n$ is 4.

Choose the correct answer from the options given below:

(A), (B) and (D) only

Explanation:

The energy of an electron in the hydrogen atom is given by

$ E_n = -\frac{13.6}{n^2}\ \text{eV} $

(A) The negative sign indicates that the electron is bound to the nucleus and energy must be supplied to free it. Hence, this statement is correct.

(B) The negative sign also means that the electron requires 13.6 eV to be removed completely from the nucleus (from $n = 1$ to $n = \infty$). Hence, this statement is also correct.

(C) The energy difference between the ground state ($n = 1$) and the first excited state ($n = 2$) is

$ \Delta E = E_2 - E_1 = \left(-\frac{13.6}{2^2}\right) - \left(-\frac{13.6}{1^2}\right) = (-3.4) - (-13.6) = 10.2\ \text{eV} $

Hence, 10.2 eV, not 3.4 eV, is required. Therefore, this statement is incorrect.

(D) The third excited state corresponds to $n = 4$ (since ground is $n = 1$, first excited $n = 2$, second excited $n = 3$, third excited $n = 4$). Hence, this statement is correct.

Correct statements: (A), (B), and (D)