For hydrogen-like species, a graph of the square root of the magnitude of total energy ( |E_n| ) versus atomic number ( Z ) is plotted for two different principal quantum numbers, n=2 and n=3 . Both plots yield straight lines passing through the origin. What is the ratio of the slope of the line for n=2 to the slope of the line for n=3 ?

Step-by-step solution

The energy of the n^ th stationary state for a hydrogen-like species is: E_n = -13.6 Z^2 n^2 eV The magnitude of the energy is: |E_n| = 13.6 Z^2 n^2 Taking the square root on both sides: |E_n| = 13.6 n Z This represents a straight line equation y = mZ passing through the origin, where the slope m = 13.6 n . Thus, the slope is inversely proportional to n ( m 1 n ). The ratio of the slopes for n=2 and n=3 is: m_ n=2 m_ n=3 = 1 2 1 3 = 3 2 Therefore, the ratio is 3 : 2 . Answer: 3 : 2