Question

Find the radius and energy of a He⁺ ion in the states (a) n = 1, (b) n = 4 and (c) n = 10.

Answer 1

For He⁺ ion,

Atomic number, Z = 2

For hydrogen like ions, radius (r) of the nth state is given by

`r = (0.53 n^2)/ZÅ`

Here

Z=  Atomic number of ions

n = Quantum number of the state

Energy (E) of the nth state is given by

`E_n = -(13.6 Z^2)/(n^2)`

(a)

for n = 1

Radius,

`r = (0.53xx(1)^2)/2`

  `Å = 0.265  A^0`

Energy, E_n =  `(-13.6 xx 4)/1`

= - 54.4 ev

(b)

For n = 4,

Radius, `r = (0.53xx16)/2 = 4.24  Å `

Energy, `E = (-13.6xx4)/16= -3.4 eV`

(c)

For n = 10,

`Radius, r = (0.53xx100)/2`

  = 26.5 Å

Energy, E = `(-13.6 xx 4 )/100 = -0.544  eV`