Chemistry

Problem

Hydrogen-like ions with nuclear charge

+Ze

have allowed energy levels that vary as

E_{n}\propto -\dfrac{Z^{2}}{n^{2}}

. Which of the following statements best explains the

Z^{2}

dependence? a) Because the potential energy term contains the nuclear charge only once, the energy should scale as

-Z/n^{2}

rather than

-Z^{2}/n^{2}

. b) The electrostatic attraction increases linearly with

Z

, while the quantized orbit radius decreases in direct proportion to

1/Z

; together these two factors make the total energy proportional to

-Z^{2}/n^{2}

. c) Energy levels depend only on the electron’s mass, so any apparent

Z

dependence must cancel, giving

E_{n}

independent of

Z

. d) If the orbit radius were to scale as

1/Z^{2}

, combining this with the single

Z

factor from the Coulomb potential would make the total energy vary as

-Z^{3}/n^{2}