Which of the following statements regarding the energy of the stationary state is true in the following one - electron systems?

Determine which statement about the energy of stationary states is true.

The energy of the $$n$$th orbit in a hydrogen-like atom with atomic number $$Z$$ is given by $$E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV}$$. Using $$1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$$, this can be written as $$E_n = -2.18 \times 10^{-18} \times \frac{Z^2}{n^2} \text{ J}$$.

For the third orbit of $$Li^{2+}$$ (where $$Z=3$$ and $$n=3$$), the energy is $$E_3 = -2.18 \times 10^{-18} \times \frac{3^2}{3^2} = -2.18 \times 10^{-18} \text{ J}$$, which matches the value stated in Option A.

Option B refers to the second orbit of hydrogen ($$Z=1$$, $$n=2$$), giving $$E = -2.18 \times 10^{-18}/4 = -0.545 \times 10^{-18}$$ J, not $$-1.09 \times 10^{-18}$$ J. Option C is incorrect because the energy of a bound state is always negative, never positive. Option D is false since the first orbit of $$He^+$$ has energy $$-2.18 \times 10^{-18} \times 4 = -8.72 \times 10^{-18}$$ J, not a positive value.

The correct answer is Option A: $$-2.18 \times 10^{-18}$$ J for the third orbit of $$Li^{2+}$$.