Chemistry

Problem

Answer the following for a single-electron atom: (i) Derive an expression for the orbital speed

v_n

of the electron in the

n

th stationary orbit as a function of

n

and

Z

. (ii) Hence show that, for hydrogen

(Z=1)

v_n = \dfrac{2.19\times10^{6}\,\text{m\,s}^{-1}}{n}

. (iii) Calculate the speed of the electron in the ground state

(n=1)

of doubly-ionised lithium

(\text{Li}^{2+},\, Z=3)

. Use

e=1.60\times10^{-19}\,\text{C},\; m=9.11\times10^{-31}\,\text{kg},\; h=6.63\times10^{-34}\,\text{J\,s},\; \varepsilon_0=8.85\times10^{-12}\,\text{C}^2\text{N}^{-1}\text{m}^{-2}

. b. (ii) Show that, for hydrogen

(Z=1)

v_n = \dfrac{2.19\times10^{6}\,\text{m\,s}^{-1}}{n}

. c. (iii) Calculate the speed of the electron in the ground state of

\text{Li}^{2+}

using the given constants.