For hydrogen like species, which of the following graphs provides the most appropriate representation of E vs Z plot for a constant n ? [E: Energy of the stationary state, Z : atomic number, n = principal quantum number]

For a one-electron (hydrogen-like) species the energy of the stationary state with principal quantum number $$n$$ is given by

$$E=-\frac{13.6\;{\rm eV}\;Z^{2}}{n^{2}} \qquad -(1)$$

where $$Z$$ is the atomic number of the nucleus.

For the present question $$n$$ is kept constant. Therefore from $$-(1)$$ we get the functional dependence

$$E \propto -Z^{2} \qquad -(2)$$

Key features of the relation $$-(2)$$:

1. The energy is always negative (bound state), so every point of the curve lies below the $$E=0$$ axis.
2. As $$Z$$ increases, $$|E|$$ increases quadratically; hence the curve falls more and more steeply—in other words, it is a downward-opening parabola in the fourth quadrant (positive $$Z$$, negative $$E$$).
3. If one extrapolates mathematically to $$Z=0$$ the energy tends to $$0$$, so the parabola’s vertex touches the origin, but for real atoms the physically relevant domain is $$Z\ge 1$$ where the curve starts at $$E=-13.6/n^{2}$$ eV and descends.

Among the given sketches, the only graph that shows a parabolic fall completely in the negative $$E$$ region, becoming more negative as $$Z$$ increases, is Option A.

Hence the correct choice is:
Option A which is: the downward-opening parabolic $$E$$ vs $$Z$$ plot.