Chemistry

Problem

A hydrogen-like ion of atomic number

Z

possesses only one electron in orbit about the nucleus. (i) Derive an algebraic expression for the radius

r_n

of the

n

th stationary orbit in terms of fundamental constants, the principal quantum number

n

, and

Z

. (ii) Using the result of part (i), show explicitly that

r_n\propto\dfrac{n^{2}}{Z}

. (iii) Taking

Z=2

for singly-ionised helium

(\mathrm{He}^+)

and using

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}

, calculate the numerical value of the radius of the second orbit. Express the answer in ångströms (

1\,\text{Å}=10^{-10}\,\text{m}

). c. (iii) Calculate the numerical value of the radius of the second orbit of

\mathrm{He}^+

with the given constants and express it in ångströms.