Question

Derive an expression for the radius of the n^th Bohr orbit for the hydrogen atom.

Answer 1

Let, m_e = mass of electron,

−e = charge on electron,

r_n = radius of n^th Bohr’s orbit,

+e = charge on nucleus,

v_n = linear velocity of electron in n^th orbit,

Z = number of electrons in an atom,

n = principal quantum number.

∴ The angular momentum = m_ev_nr_n

According to second postulate.

m_ev_nr_n = `"nh"/("2"pi)`    ...(i)

From Bohr’s first postulate,

Centripetal force = Electrostatic force

∴ `("m"_"e""v"_"n"^2)/"r"_"n" = "Ze"^2/(4piepsilon_0"r"_"n"^2)`

∴ `"v"_"n"^2 = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")`  ....(ii)

Squaring equation (i) we get

`"m"_"e"^2 "v"_"n"^2 "r"_"n"^2 = ("n"^2"h"^2)/(4pi^2)`

∴ `"v"_"n"^2 = ("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2)` ….(iii)

Equating equations (ii) and (iii), we get,

`("n"^2"h"^2)/(4pi^2"m"_"e"^2"r"_"n"^2) = "Ze"^2/(4piepsilon_0"r"_"n""m"_"e")`

∴ r_n = `("n"^2"h"^2epsilon_0)/(pi"m"_"e""Z"_"e"^2)`

This is the required expression for radius of n^th Bohr orbit of the electron.