Which of the following is the energy of a possible excited state of hydrogen?

We recall that in the Bohr model of the hydrogen atom the allowed (quantised) total energies of the electron are given by the well-known expression

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

where $$n$$ is the principal quantum number and can take only positive integer values $$n = 1,2,3,\dots$$. The negative sign shows that the electron is bound to the nucleus; its energy is less than that of a free electron, which is taken to be zero.

Now we list the first few energies by substituting successive integer values of $$n$$ into the formula.

For the ground state we put $$n = 1$$. We have

$$E_1 = -\dfrac{13.6\ \text{eV}}{1^2} = -13.6\ \text{eV}.$$

For the first excited state we put $$n = 2$$. So

$$E_2 = -\dfrac{13.6\ \text{eV}}{2^2} = -\dfrac{13.6\ \text{eV}}{4} = -3.4\ \text{eV}.$$

For the second excited state we put $$n = 3$$. Thus

$$E_3 = -\dfrac{13.6\ \text{eV}}{3^2} = -\dfrac{13.6\ \text{eV}}{9} \approx -1.51\ \text{eV}.$$

We observe an important point: every legitimate energy level obtained from the formula is negative. Positive values like $$+6.8\ \text{eV}$$ or $$+13.6\ \text{eV}$$ cannot correspond to any bound state of hydrogen.

Next, let us inspect the proposed options one by one and compare them with the allowed values we have just derived.

A. $$+6.8\ \text{eV}$$ – This is positive, whereas all allowed energies are negative. Hence this cannot be an energy level.

B. $$+13.6\ \text{eV}$$ – Again positive, so not permissible.

C. $$-6.8\ \text{eV}$$ – This is negative, but we test whether it fits the formula. Setting

$$-6.8 = -\dfrac{13.6}{n^2}$$

gives $$n^2 = \dfrac{13.6}{6.8} = 2$$ and therefore $$n = \sqrt{2}$$, which is not an integer. Because $$n$$ must be an integer, $$-6.8\ \text{eV}$$ is not an allowed level.

D. $$-3.4\ \text{eV}$$ – We already found that for $$n = 2$$ the energy is exactly $$-3.4\ \text{eV}$$. Thus this value satisfies the quantisation condition and represents the first excited state.

Among the given choices, only $$-3.4\ \text{eV}$$ corresponds to an allowed excited state of hydrogen.

Hence, the correct answer is Option 4.