A singly ionized helium atom He has only one electron in orb

A singly ionized helium atom (He⁺) has only one electron in orbit about the nuclues. What is the radius of the ion when it is in the excited state defined by n=4?

Solution

for circular motion,

k (2e)(e) / r^2 = m v^2 / r

v^2 = 2 k e^2 / mr

and for angular momentum,

m v r = n h / 2pi

putting v from above equation,

n^2 h^2 / (4 pi^2 r^2 m^2 ) = 2 k e^2 / m r

(4^2) ( 6.626 x 10^-34)^2 / (4 x pi^2 x r x 9.109 x 10^-31 ) = ( 2 x 9 x 10^9 x (1.6 x 10^-19)^2)

r = 4.239 x 10^-10 m