Bohr's Atomic Model: Postulates, derivation of radii of orbits, velocity, and total energy of an electron in a hydrogen-like atom

📘 Topic Description

Bohr's atomic model was proposed by Niels Bohr in 1913 to explain the stability of atoms and the line spectrum of hydrogen. The model combines Rutherford's nuclear model with the idea of quantisation of angular momentum.

⭐ For a hydrogen-like atom of atomic number Z, the electron moves in certain permitted stationary orbits around the nucleus.

⚛️ Bohr's Postulates

1. Electrons revolve around the nucleus only in certain permitted circular orbits called stationary orbits.
2. An electron moving in a stationary orbit does not radiate energy.
3. The angular momentum of the electron is quantised:

mvr = nh/2π

where n = 1, 2, 3, ... is the principal quantum number.

4. Radiation is emitted or absorbed only when an electron jumps from one stationary orbit to another.

hν = E₂ − E₁

🔬 Hydrogen-Like Atom

A hydrogen-like atom or ion contains only one electron. Examples include:

H
Z = 1

He⁺
Z = 2

Li²⁺
Z = 3

Be³⁺
Z = 4

The following Bohr formulas apply directly to all one-electron hydrogen-like species.

📐 Derivation of Radius of nth Orbit

For an electron of mass m moving around a nucleus of charge +Ze, the electrostatic attraction provides the required centripetal force.

Coulomb Force = Centripetal Force

1/(4πε₀) × Ze²/r² = mv²/r

Therefore:

mv² = 1/(4πε₀) × Ze²/r

Using Bohr's quantisation condition:

mvr = nh/2π

Therefore:

v = nh/(2πmr)

Substituting this value of velocity gives:

⭐ rₙ = ε₀h²n²/(πmZe²)

Since the Bohr radius is:

a₀ = 0.529 Å

the radius becomes:

⭐ rₙ = a₀ n²/Z

Important: Radius is proportional to n² and inversely proportional to Z.

rₙ ∝ n²/Z

🚀 Velocity of Electron in nth Orbit

From the electrostatic force equation:

mv²/r = 1/(4πε₀) × Ze²/r²

Therefore:

v² = 1/(4πε₀) × Ze²/(mr)

Using the radius of nth orbit:

⭐ vₙ = Ze²/(2ε₀hn)

In terms of the fine-structure constant α:

⭐ vₙ = Zαc/n

For the hydrogen atom in the first orbit, the electron speed is approximately 2.18 × 10⁶ m/s.

vₙ ∝ Z/n

⚡ Total Energy of Electron

The total energy of an electron is:

E = K + U

Kinetic energy:

K = ½mv²

From the electrostatic force equation:

K = 1/(8πε₀) × Ze²/r

Potential energy of electron:

U = −1/(4πε₀) × Ze²/r

Hence:

E = K + U

⭐ Eₙ = −1/(8πε₀) × Ze²/rₙ

Substituting the expression for radius:

⭐ Eₙ = −mZ²e⁴/(8ε₀²h²n²)

For hydrogen-like atoms, the commonly used form is:

⭐ Eₙ = −13.6 Z²/n² eV

The negative sign indicates that the electron is bound to the nucleus.

📊 Energy Levels and Transitions

When an electron jumps from a higher orbit to a lower orbit, a photon is emitted.

hν = Eᵢ − E_f

For absorption, the electron moves from a lower energy state to a higher energy state.

hν = E_f − Eᵢ

🎬 Interactive Bohr Orbit Animation

Atomic Number Z: 1

Orbit n: 1

🧪 Animation Interpretation

• Central glowing region represents the nucleus.
• Circular rings represent permitted Bohr orbits.
• The moving dot represents the electron.
• Increasing n increases the orbital radius.
• Increasing Z decreases the radius for the same n.
• The electron remains in a stationary orbit without radiating energy.

📋 Bohr Model Formula Table

Quantity	Formula	Dependence
Angular Momentum	mvr = nh/2π	∝ n
Radius	rₙ = a₀n²/Z	∝ n²/Z
Velocity	vₙ = Zαc/n	∝ Z/n
Kinetic Energy	K = 13.6Z²/n² eV	∝ Z²/n²
Potential Energy	U = −27.2Z²/n² eV	∝ −Z²/n²
Total Energy	Eₙ = −13.6Z²/n² eV	∝ −Z²/n²

📝 MCQ Practice — 15 Questions

1. Bohr's quantisation condition is:

A. mv = nh

B. mvr = nh/2π

C. mr = nh

D. mv²r = nh

✔ Answer: B

2. The radius of nth orbit of a hydrogen-like atom is proportional to:

A. n/Z

B. Z/n²

C. n²/Z

D. Z²/n

✔ Answer: C

3. The radius of first Bohr orbit of hydrogen is approximately:

A. 0.529 Å

B. 5.29 Å

C. 52.9 Å

D. 0.0529 Å

✔ Answer: A

4. Velocity of electron in nth orbit is proportional to:

A. n/Z

B. Z/n

C. nZ

D. Z²/n²

✔ Answer: B

5. Total energy of electron in nth orbit is:

A. +13.6/n² eV

B. −13.6/n² eV

C. −13.6n² eV

D. +13.6n² eV

✔ Answer: B

6. The negative sign of total energy indicates:

A. Electron is free

B. Electron is bound to nucleus

C. Electron has zero kinetic energy

D. Atom is unstable

✔ Answer: B

7. For He⁺ in n = 1, total energy is:

A. −13.6 eV

B. −27.2 eV

C. −54.4 eV

D. −108.8 eV

✔ Answer: C

8. If n is doubled, the orbital radius becomes:

A. 2 times

B. 4 times

C. 1/2 times

D. 1/4 times

✔ Answer: B

9. If atomic number is doubled for the same orbit, radius becomes:

A. Double

B. Four times

C. Half

D. Same

✔ Answer: C

10. Radiation is emitted when an electron:

A. Moves to a higher energy state

B. Moves to a lower energy state

C. Remains in same orbit

D. Stops revolving

✔ Answer: B

11. The energy of nth orbit varies as:

A. n

B. n²

C. 1/n

D. 1/n²

✔ Answer: D

12. Which species is hydrogen-like?

A. He

B. Li

C. He⁺

D. Be

✔ Answer: C

13. The angular momentum in the nth orbit is:

A. nh

B. nh/2π

C. h/n

D. n²h

✔ Answer: B

14. In a stationary orbit, an electron:

A. Radiates continuously

B. Does not radiate energy

C. Stops moving

D. Loses charge

✔ Answer: B

15. Bohr's model successfully explained the spectrum of:

A. Hydrogen atom

B. All atoms

C. Molecules only

D. Solids only

✔ Answer: A

🟣 Assertion–Reason — 5 Questions

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true but R is not the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

1. Assertion: An electron does not radiate energy in a stationary orbit.

Reason: Stationary orbits are allowed states according to Bohr's postulates.

Answer: A

2. Assertion: Radius of the nth orbit increases as n².

Reason: rₙ = a₀n²/Z.

Answer: A

3. Assertion: The velocity of electron decreases as n increases.

Reason: vₙ ∝ 1/n.

Answer: A

4. Assertion: Total energy of electron in a bound orbit is positive.

Reason: Potential energy of the electron is negative.

Answer: D

5. Assertion: He⁺ has a more negative ground-state energy than H.

Reason: Ground-state energy is proportional to −Z².

Answer: A

🟢 2 Marks — 6 Questions

Q1. State Bohr's quantisation condition.

Q2. What is meant by a stationary orbit?

Q3. Define a hydrogen-like atom.

Q4. Write the expression for Bohr radius.

Q5. Write the expression for velocity of electron in nth orbit.

Q6. Why is the total energy of an electron negative?

🟡 3 Marks — 6 Questions

Q1. State three important postulates of Bohr's atomic model.

Q2. Derive the expression for angular momentum of an electron.

Q3. Explain the meaning of stationary orbit.

Q4. Write the formula for radius, velocity and energy of electron.

Q5. Explain why the energy of a bound electron is negative.

Q6. What is meant by hydrogen-like species? Give two examples.

🟠 4 Marks — 6 Questions

Q1. Derive the expression for radius of nth Bohr orbit.

Q2. Derive the expression for velocity of electron in nth orbit.

Q3. Explain the quantisation of angular momentum.

Q4. Derive the expression for kinetic energy of electron.

Q5. Explain emission and absorption of radiation using Bohr's model.

Q6. Explain the energy levels of a hydrogen-like atom.

🔴 5 Marks — 6 Questions

Q1. State and explain Bohr's postulates of atomic model.

Q2. Derive the expression for radius of the nth orbit of a hydrogen-like atom.

Q3. Derive the expression for velocity of an electron in nth orbit.

Q4. Derive the expression for total energy of electron in a hydrogen-like atom.

Q5. Explain how Bohr's model accounts for emission spectrum of hydrogen.

Q6. Compare radius, velocity and energy of electrons in different Bohr orbits.

🔵 6 Marks — 6 Questions

Q1. State Bohr's postulates and explain the significance of each postulate.

Q2. Derive expressions for radius, velocity and total energy of an electron in the nth orbit of a hydrogen-like atom.

Q3. Explain Bohr's atomic model and derive the energy expression Eₙ = −13.6Z²/n² eV.

Q4. Explain the formation of spectral lines using transitions between Bohr energy levels.

Q5. Derive the radius and velocity of electron in a hydrogen-like atom and discuss their dependence on n and Z.

Q6. Discuss the successes and limitations of Bohr's atomic model.

🧮 Numerical Practice — 6 Questions

Q1. Calculate the radius of the first Bohr orbit of hydrogen.

Q2. Find the radius of the second orbit of hydrogen in terms of Bohr radius.

Q3. Find the radius of the first orbit of He⁺.

Q4. Calculate the energy of the electron in the third orbit of hydrogen.

Q5. Calculate the ground-state energy of Li²⁺.

Q6. Compare the speeds of electrons in the first orbit of H and He⁺.

🧠 High-Level Competitive Practice

Q1. If the radius of a hydrogen-like orbit is 9 times the Bohr radius, find possible values of n and Z.

Q2. Two hydrogen-like ions have the same electron velocity. What relation must exist between Z and n?

Q3. Compare the total energies of H in n = 2 and He⁺ in n = 4.

Q4. If an electron jumps from n = 4 to n = 2, determine whether radiation is emitted or absorbed.

Q5. If n becomes three times, how do radius, velocity and total energy change?

Q6. For two hydrogen-like species having equal radii, establish the relationship between their principal quantum numbers and atomic numbers.

📌 One-Minute Formula Revision

Angular Momentum

mvr = nh/2π

Radius

rₙ = a₀n²/Z

Velocity

vₙ = Zαc/n

Kinetic Energy

K = +13.6Z²/n² eV

Potential Energy

U = −27.2Z²/n² eV

Total Energy

Eₙ = −13.6Z²/n² eV

Photon Energy

hν = |E₂ − E₁|

🚀 One-Minute Revision

🔹 Bohr introduced quantised stationary orbits.
🔹 Electron does not radiate energy in a stationary orbit.
🔹 Angular momentum = nh/2π.
🔹 Radius rₙ = a₀n²/Z.
🔹 Radius increases as n².
🔹 Radius decreases as Z increases.
🔹 Velocity vₙ = Zαc/n.
🔹 Velocity increases with Z.
🔹 Velocity decreases with n.
🔹 Total energy Eₙ = −13.6Z²/n² eV.
🔹 Higher n means energy becomes less negative.
🔹 Ground state corresponds to n = 1.
🔹 Electron falling to a lower energy state emits a photon.
🔹 Electron moving to a higher energy state absorbs a photon.
🔹 Hydrogen-like species contain only one electron.

🔥 Important Exam & Search Keywords

Bohr Atomic Model Bohr Postulates Bohr Radius Radius of Bohr Orbit Velocity of Electron Energy of Electron Hydrogen Like Atom Bohr Energy Formula Atomic Model Hydrogen Spectrum Class 12 Physics CBSE Physics JEE Physics NEET Physics Physics Numericals

📚 Class 12 Physics • CBSE • JEE • NEET
⚛️ Bohr's Atomic Model • Bohr Radius • Electron Velocity • Total Energy