Bohr Atom Model

In a process of overcoming the defects of Rutherford nuclear model of the atom Neil Bohr proposed new atomic model called Bohr atomic model.

Postulate-I

An electron cannot revolve round the nucleus in all possible orbits as suggested by classical theory. The electron can revolve round the nucleus only in those allowed or permissible orbits for which the angular momentum of the electrons is an integral multiple of \( \frac {h}{2\pi }\). Here h is Planck's constant whose value is \(6.64 \times 10^{-34} \) js.
If m is the mass of electron and v is velocity of the electron in an orbit of radius r, then
\begin{align*} \text {Angular momentum, L}\: &= m\:vr =mr^2\omega \\ &= \frac {nh}{2\pi } \dots (i) \\ \end{align*}
where n is an integer and can take values n = 1, 2, 3, 4, …... It is called principal quantum number. This equation (i) is called Bohr's quantization condition.

Postulate-II

When electron revolves in permitted orbits they do not radiate energy. An atom radiates energy only when an electron jumps from a higher energy state to the lower energy state and the energy is absorbed, when it jumps from lower to higher energy orbit.
If En₁ and En₂ are energies associated with first and second orbits respectively, then the frequency f of the radiation emitted is given by
$$hf = En_2 - En_1 $$
This is called Bohr's frequency condition.

Bohr's Theory of Hydrogen Atom

Bohr assumed that a hydrogen atom consists of a nucleus with one unit positive charge +e (i.e. a proton)and a single electron of charge -e, revolving around it in a circular orbit of radius r. The electrostatic force of attraction between the proton and the electron is given by
$$F = \frac {1}{4\pi \epsilon _0}\frac {e^2}{r^2} \dots (i) $$
where \(\epsilon _0\) is the permittivity of free space.
If m and v are mass and velocity of the electron in the orbit, then the centripetal force required by the electron to move in circular orbit of radius r is given by
$$F = \frac {mv^2}{r} \dots (ii) $$
Further, the electrostatic force of attraction between the electron and the nucleus provides the necessary centripetal force. Therefore, for circular motion of electron,
\begin{align*} \frac {mv^2}{r} &= \frac {e^2}{4\pi \epsilon _0r^2} \dots (iii) \\ \text {According to Bohr's first postulate} \\ m\:vr &= \frac {nh}{2\pi } \\ \end{align*}

\begin{align*} \text {or,} \: v &= \frac {nh}{2\pi rm} \dots (iv)\\ \text {or}\: v^2 &= \frac {n^2h^2}{4\pi ^2r^2m^2} \\ \text {substituting this value of}\: v^2 \: \text {in equation} \: (iii),\:\text {we get}\\\frac mr \left (\frac {n^2h^2}{4\pi ^2r^2m^2} \right )&= \frac {1}{4\pi \epsilon _0}\frac{e^2}{r^2} \ \\ \text {or,}\: r &= \frac{\epsilon_o n^2h^2}{\pi me^2}\dots (v)\\\text {Radius of the}\: n^{th}\: \text{permissible orbit for hydrogen is given by} \\r_n &= \frac{\epsilon_o n^2h^2}{\pi me^2}\dots (vi) \end{align*}
As n = 1, 2, 3, .... it follows from equation (vi) that the radii of the stationary orbits are proportional to n2.

Bohr Radius (a₀)
the radius of the inner most orbit in hydrogen atom with n = 1 is called Bohr's radius and is denoted by a­0. For \(n =1, r = a_0 = \epsilon_0 \frac {h^2}{\pi me^2}\).
Substituting the known values of \(\epsilon _0\), h, m and e, we get
\begin{align*}a_0 &= 0.529 Amstrong\: \\ r_n &= 0.529 \times n^2 Amstrong\dots (vii) \\ \end{align*}

(1 Amstrong = 10^-10 meter)
This equation gives the radii of the orbits in the Bohr model of atom.

Velocity of the Electron

The velocity of the electron in the nth orbit, Vn is given by
\begin{align*} v_n &= \frac {nh}{2\pi mr_n} \:\:\: [\because \text {By using equation}\:(iv)] \\ \end{align*}

Substituting the value of r_n from equation (vi) we get \begin{align*} v_n &= \frac {nh}{2\pi m}\left (\frac {\pi me^2}{\epsilon_0n^2h^2}\right) \\ &= \frac {e^2}{2\epsilon _0 nh} \dots (vii) \\ \end{align*}
Here the electrons closer to the nucleus move with higher velocity than lying farther.

Energy of the electron in n^th orbit

As electron is revolving round the nucleus, it has kinetic energy.
But K.E. of the electron in nth orbit \(=\frac 12mv_n^2 \)

\begin{align*} \text {From equation}\: (vii), \: v_n &= \frac {e^2}{2\epsilon_0 nh} \\ \therefore \text {KE of the electron in nth orbit} &=\frac 12 m \left (\frac {e^2}{2\epsilon _0 nh^2}\right )^2\\ &= \frac {me^4}{8\epsilon _0^2 n^2h^2} \\ \text {The potential energy}\: (P.E.) &= \text {potential due to nucleus at distance r}\times \text {charge on electron}\\ \end{align*}

\begin{align*} \text {But from equation}\: (vi),\: r_n &= \frac {\epsilon_0n^2h^2}{\pi \:me^2} \\ \therefore P.E. &= -\frac {e^2}{4\pi \epsilon_0}\frac {\pi me^2}{\epsilon_0n^2h^2} \\ &= -\frac {me^4}{4\epsilon_0^2n^2h^2} \\ \text {Therefore, total energy of the electron in the nth orbit is} \\ E_n &= K.E. + P.E. \\ &= \frac {me^4}{8\epsilon _0^2 n^2h^2} - \frac {me^4}{4\epsilon_0^2n^2h^2} \\ &= -\frac {me^4}{8\epsilon_0^2n^2h^2} \dots (ix)\\ \end{align*}

reference

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.