Rutherford's Model of Atom – Electron Orbits | Limitations | Expression For Radius And Total Energy

Rutherford's Atomic Model

According to Rutherford's nuclear model, an atom consists of a tiny, dense, positively charged nucleus at its centre, around which electrons revolve in circular orbits.

Salient Features

1. An atom has a small, dense, positively charged nucleus.
2. Almost the entire mass of the atom is concentrated in the nucleus.
3. Electrons revolve around the nucleus in circular orbits.
4. The electrostatic force of attraction between the nucleus and electrons provides the necessary centripetal force.
5. Most of the space inside an atom is empty.

Electron Orbits

Let,

• e = charge on electron
• Ze = charge on nucleus
• m = mass of electron
• v = velocity of electron
• r = radius of orbit
• ε₀ = permittivity of free space

1. Electrostatic Force of Attraction

The electrostatic force between the nucleus and electron is given by:

F_e = Ze² / (4πε₀r²)

2. Centripetal Force

The centripetal force required for circular motion is:

F_c = mv² / r

3. Condition for Stable Orbit

For a stable orbit,

F_e = F_c

Therefore,

Ze² / (4πε₀r²) = mv² / r

Multiplying both sides by r,

mv² = Ze² / (4πε₀r)

4. Radius of Orbit

From the above equation,

r = Ze² / (4πε₀mv²)

For hydrogen atom (Z = 1),

r = e² / (4πε₀mv²)

Total Energy of Electron

The total energy of the electron is the sum of kinetic energy and potential energy.

E = K + U

(A) Kinetic Energy of Electron

Kinetic energy is given by:

K = ½mv²

Using,

mv² = Ze² / (4πε₀r)

Therefore,

K = ½ × Ze² / (4πε₀r)

K = Ze² / (8πε₀r)

For hydrogen atom,

K = e² / (8πε₀r)

(B) Potential Energy of Electron

Potential energy is given by:

U = (1 / 4πε₀) × (q₁q₂ / r)

Here,

q₁ = +Ze

q₂ = -e

Therefore,

U = -Ze² / (4πε₀r)

For hydrogen atom,

U = -e² / (4πε₀r)

(C) Total Energy of Electron

Total energy is:

E = K + U

Substituting the values of K and U,

E = Ze² / (8πε₀r) − Ze² / (4πε₀r)

E = (Ze² − 2Ze²) / (8πε₀r)

E = −Ze² / (8πε₀r)

For hydrogen atom,

E = −e² / (8πε₀r)

Significance of Negative Energy

• Kinetic energy is positive.
• Potential energy is negative.
• Total energy is negative.
• Negative total energy indicates that the electron is bound to the nucleus.
• If the total energy becomes positive, the electron escapes from the atom.

Limitations of Rutherford's Model

1. It failed to explain the stability of atoms.
2. It could not explain the discrete line spectra of atoms.
3. It did not provide the concept of quantized energy levels.

Frequently Asked Questions (FAQ)

Q1. Why do electrons revolve around the nucleus?

The electrostatic force of attraction between the nucleus and electron provides the necessary centripetal force.

Q2. What provides the centripetal force in Rutherford's model?

The electrostatic force between the nucleus and electron acts as the centripetal force.

Q3. Why is the potential energy negative?

Because the force between the nucleus and electron is attractive, energy must be supplied to separate them to infinity.

Q4. Why is the total energy negative?

Negative total energy indicates that the electron is bound to the nucleus.

Q5. What happens if the total energy becomes positive?

The electron escapes from the atom and no stable orbit exists.

Q6. Why did Rutherford's model fail?

According to Maxwell's theory, accelerating electrons should continuously radiate energy and collapse into the nucleus, making atoms unstable.

Q7. Which experiment led to Rutherford's model?

The α-particle scattering experiment led to Rutherford's nuclear model of the atom.

Important Formulae

Quantity	Formula
Electrostatic Force	F = Ze² / (4πε₀r²)
Centripetal Force	F = mv² / r
Stable Orbit Condition	Ze² / (4πε₀r²) = mv² / r
Kinetic Energy	K = Ze² / (8πε₀r)
Potential Energy	U = −Ze² / (4πε₀r)
Total Energy	E = −Ze² / (8πε₀r)
Relation between K and U	K = −U/2
Relation between E and K	E = −K
Relation between E and U	E = U/2