Question

2. The Bohr model holds for any one-electron atom. Calculate the lowest energy level of the helium ion, He". (Hint: Examine the parts of the Bohr equation carefully). 3. A fun exercise with units... Given the following information, show that the units for energy derived from the Bohr model can indeed work out to be the Joule unit. . 1 J-1 kg.mº/s? • The charge of an electron is measured in units of Coulombs, C • The permittivity, to, is in units of C/Jºm

Answer 1

Ans 2:

According to Bohr model, energy of hydrogen/hydrogen like systems (single electron systems) is given by,

E = - R_H .(Z²/n²)

Where, E = energy of the electron in ‘n^th’ orbit

               R_H = Rydberg’s constant = (e⁴.m)/(8.h².ꞓ_o²) = 2.18 x 10^-18 J

               e = charge on the electron = 1.6 x 10^-19 C

               m = mass of the electron =9.10938356 × 10^-31 kg

               h = plank’s constant = 6.62607004 × 10^-34 m² kg / s

               .ꞓ_o = relative permittivity (C²/J.m

In the Helium ion (He⁺), number of protons (Z) present = 2, No of electrons = 1

For the ground state of ‘He⁺’ ion, n = 1

E = - R_H .(Z²/n²) = -2.18 x 10^-18 J (4/1) = -8.72 x 10^-18 J

Lowest energy of ‘He⁺’ ion = -8.72 x 10^-18 J

{Here, the negative sign indicates that electron is bound to helium nucleus}

Ans 3:

E = - R_H .(Z²/n²) = - (e⁴.m)/(8.h².ꞓ_o²) (Z²/n²)

Units of right hand side equation:

(e⁴.m)/(8.h².ꞓ_o²) (Z²/n²) = (C⁴ . Kg)/(m⁴ Kg² s^-2 x C⁴ J^-2.m^-2) = Kg^-1. m^-2 . s² . J²

We know that 1J = 1 Kg. m².s^-2

So, Kg^-1. m^-2 . s² . J² = J^-1.J² = J (Unit of energy)