Question

determine the speed of the electron in its orbit around the carbon atom

Answer 1

We take the following assumptions:

r_n : radius of nth orbit

v_n : velocity of electron in nth state (orbit)

E_n : energy of nth state

m: mass of an electron (9.1 x 10^–31 Kg)

Z: atomic number (No. of Protons)

K = 1/(4πε₀) = constant = 9 x 10⁹ N m² C^–2

h: Planck's constant (6.67 x 10^–34 J-s)

c: velocity of light (3 x 10⁸ m/s)

R: Rydberg constant (1.097 x 10⁷ m^–1)

e: Charge on an electron (1.6 x 10^–19 C)

: frequency of the radiation emitted or absorbed

: wave number of the spectral line in the atomic spectra

From Bohr’s first postulate, the centripetal force is equal to the electrostatic force so we get,

From Bohr’s second postulate, angular momentum is an integral multiple of

Solving for and , we have:

Radius,

Velocity,

Here for Carbon atom the electrons exist in two different orbits so we can calculate the velocity of electrons in each orbit by substituting

For all electrons in First orbit,  , we have

For all electrons in Second orbit,  , we have