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Modern Physics

Bohr's Theory of Hydrogen Atom

Bohr's model of the hydrogen atom applies quantum conditions to the classical planetary model, postulating that electrons orbit the nucleus only in discrete stable orbits without radiating energy. It successfully accounts for the spectral lines of atomic hydrogen by associating frequency with the energy difference between stationary states.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Postulates of Bohr's model, expressions for radius, velocity, and energy of electron orbits in hydrogen, explanation of hydrogen spectral series (Lyman, Balmer, Paschen, etc.), and calculation of ionization energy.

PGT extension

Reduced mass correction for finite nuclear mass, Rydberg constant derivation, angular momentum quantization extensions, and limitations of the Sommerfeld-Wilson quantization and vector model connections.

Prerequisites

Coulomb's law of electrostatic force
Centripetal force and classical mechanics
Planck's quantum hypothesis and photon energy

Postulates of Bohr's Theory

The electron in a hydrogen atom revolves around the nucleus in circular orbits under the influence of electrostatic attraction, obeying classical mechanics.
Only those orbits are permitted in which the orbital angular momentum (L) of the electron is an integral multiple of h / (2π), i.e., $L = nh / (2\pi)$.
An electron in an allowed stationary state does not radiate electromagnetic energy; radiation is emitted or absorbed only when an electron jumps from one stationary state to another according to h$\nu = E_{initial} - E_{final}$.

Quantization of Radius and Velocity

The radius of the n-th Bohr orbit is directly proportional to the square of the principal quantum number (r_n ∝ n²) and inversely proportional to atomic number Z.
The orbital velocity of the electron in the n-th orbit varies inversely with the principal quantum number (v_n ∝ Z / n), being approximately 1/137th of the speed of light for the ground state of hydrogen.
The total energy of the electron is negative, indicating a bound system, and varies inversely as the square of the principal quantum number (E_n ∝ Z² / n²).

Hydrogen Spectrum and Spectral Series

Transitions terminating at the ground state (n = 1) yield the Lyman series in the ultraviolet region.
Transitions terminating at n = 2 yield the Balmer series in the visible region, while Paschen (n = 3), Brackett (n = 4), and Pfund (n = 5) series lie in the infrared region.
The wave number of emitted radiation is governed by the generalized Rydberg formula containing the Rydberg constant R.

Finite Nuclear Mass Correction

Assuming the nucleus is infinitely massive is an approximation; accounting for finite nuclear mass replaces the electron mass m with the reduced mass $\mu = (m * M) / (m + M)$.
The Rydberg constant depends explicitly on the reduced mass, accounting for isotope shifts observed between ordinary hydrogen and deuterium.
The corrected energy states scale proportionally with the reduced mass μ.

Derivations

Derivation of Radius of n-th Bohr Orbit

Conditions: Circular orbit of an electron of mass m and charge e around a nucleus of charge Ze, assuming infinite nuclear mass.

Equate electrostatic force to centripetal force: \frac{1}{4piepsilon_0} \frac{Ze^2}{$r^2 = \frac{mv^2}{r}$
Apply Bohr's quantization condition for angular momentum: mvr = \frac{nh}{2pi} \implies v = \frac{nh}{2pi mr}
Substitute the expression for velocity v into the force balance equation and solve for radius r.

Result: $r_n = \frac{\epsilon_0 n^2 h^2}{\pi m e^2 Z}$

Derivation of Total Energy of Electron in n-th Orbit

Conditions: Circular orbit with electrostatic potential energy and kinetic energy.

Write kinetic energy $K = \frac{1}{2}mv^2 = \frac{1}{4piepsilon_0} \frac{Ze^2}{2r}$
Write potential energy $V = -\frac{1}{4piepsilon_0} \frac{Ze^2}{r}$
Sum kinetic and potential energies: $E = K + V = -\frac{1}{4piepsilon_0} \frac{Ze^2}{2r}$
Substitute the expression for r_n into the total energy equation.

Result: $E_n = -\frac{m e^4 Z^2}{8 \epsilon_0^2 h^2 n^2}$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Quantized Angular Momentum	$$L = mvr = \frac{nh}{2\pi}$$	m = mass, v = velocity, r = radius, n = principal quantum number, h = Planck constant J s	Stationary orbit condition
Radius of n-th Orbit	$$r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA}$$	n = principal quantum number, Z = atomic number Angstroms	Hydrogen-like atoms
Velocity in n-th Orbit	$$v_n = \frac{c}{137} \frac{Z}{n}$$	c = speed of light, Z = atomic number, n = principal quantum number m s^{-1}	Non-relativistic orbit
Total Energy of n-th Orbit	$$E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$$	Z = atomic number, n = principal quantum number eV	Bound electron state
Rydberg Formula for Wave Number	$$\bar{\nu} = \frac{1}{\lambda} = R Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$$	R = Rydberg constant, n_1 and $n_2 = principal quantum numbers (n_2 > n_1)$ m^{-1}	Emission or absorption transition
Rydberg Constant Expression	$$R = \frac{m e^4}{8 \epsilon_0^2 c h^3}$$	m = mass of electron, e = electronic charge, \epsilo$n_0 = permittivity$, c = speed of light, h = Planck constant m^{-1}	Infinite nuclear mass limit
Reduced Mass Correction	$$\mu = \frac{m M}{m + M}$$	m = electron mass, M = nuclear mass kg	Finite nuclear mass consideration
Ionization Energy	$$E_{ion} = E_{\infty} - E_1 = 13.6 Z^2 \text{ eV}$$	Z = atomic number eV	Transition from n = 1 to n = \infty

Graphs and experiments

Energy Level Diagram of Hydrogen Atom

Draw: Horizontal lines representing energy states n = 1, 2, 3, 4, &infty; with spacing decreasing as n increases, showing downward transition arrows for Lyman, Balmer, and Paschen series.

Interpret: Visualizes quantized energy states, convergence of levels at E = 0, and energy differences corresponding to spectral photon emissions.

Exam use: Frequently tested in identifying maximum/minimum wavelength transitions and calculating series limit wavelengths.

Franck-Hertz Experiment

Draw: A circuit diagram with a heated cathode, grid, and collector plate, alongside a graph of collector current versus accelerating potential showing periodic sharp drops.

Interpret: Proves the existence of discrete atomic energy levels through inelastic electron-atom collisions.

Exam use: Directly tested as experimental verification of Bohr's discrete energy state postulate.

Hydrogen Spectrum Line Series

Draw: A logarithmic or linear wavelength scale marking series boundaries for Lyman (UV), Balmer (Visible), and Paschen (IR).

Interpret: Shows how spectral lines crowd together near the series limit as n₂ approaches infinity.

Exam use: Numerical problems involving shortest and longest wavelengths in a given series.

Solved numericals

TGT core

Calculate the wavelength of the first line of the Balmer series in hydrogen atom. (Given $R = 1.097 x 10^7 m^-1)$

Identify the transitions for the first line of the Balmer series: $n_1 = 2$ and $n_2 = 3$.
Use the Rydberg formula for wave number: 1/$\lambda = R (1/n_1^2 - 1/n_2^2)$.
Substitute values: 1/$\lambda = 1.097 \times 10^7 (1/2^2 - 1/3^2) = 1.097 \times 10^7 (1/4 - 1/9) = 1.097 \times 10^7 (5/36)$.
Calculate wavelength: $\lambda = 36 / (5 \times 1.097 \times 10^7) = 6.563 \times 10^{-7} \text{ m}$.

Answer: 656.3 nm (or 6563 Angstroms)

PGT extension

Determine the ionization potential of a singly ionized helium atom (He+) in volts.

Identify atomic number for helium: Z = 2.
Use the ground state energy formula: $E_1 = -13.6 \times (Z^2 / 1^2) \text{ eV}$.
Substitute $Z = 2: E_1 = -13.6 \times (4) = -54.4 \text{ eV}$.
Ionization energy is the energy required to take the electron from n = 1 to n = \infty, which is $E_{ion} = 0 - (-54.4) = 54.4 \text{ eV}$.
Convert energy to potential: Ionization potential = 54.4 V.

Answer: 54.4 V

PGT extension

Find the ratio of the frequency of revolution of an electron in the first Bohr orbit of a hydrogen atom to that in the second Bohr orbit.

Frequency of revolution $f = v / (2\pi r)$.
Since v \propto 1/n and r \propto n^2, the frequency scales as f \propto 1 / n^3.
Write the ratio for n = 1 and $n = 2: f_1 / f_2 = (2^3) / (1^3)$.
Evaluate the cube: 8 / 1.

Answer: 8 : 1

Assumptions and limitations

Assumes circular orbits, whereas Sommerfeld later introduced elliptical orbits to account for fine structure.
Fails to explain multi-electron atoms (like helium and beyond) because it neglects electron-electron repulsion and exchange forces.
Cannot explain the relative intensities of spectral lines or the Zeeman and Stark effects without relativistic and quantum mechanical extensions.
Violates the Heisenberg Uncertainty Principle by simultaneously assigning exact position (radius) and momentum (velocity) to the electron.

Common misconceptions

Common error: Bohr's model treats electrons as stationary particles in fixed shells.

Correct: Electrons revolve around the nucleus with high speeds in quantized orbits.

Stationary states refer to constant energy over time, not lack of motion.

Common error: Bohr's theory successfully explains the fine structure of spectral lines in hydrogen.

Correct: Bohr's model predicts single lines for each transition; fine structure requires relativistic corrections and electron spin.

Fine structure arises from spin-orbit coupling and relativistic mass variation, which are absent in pure Bohr theory.

Common error: The total energy of an orbiting electron in a Bohr orbit is positive.

Correct: The total energy is negative.

A negative total energy signifies a bound state where external work is required to separate the electron from the nucleus.

Quick revision

Radius r_n scales as n^2/Z while total energy E_n scales as -Z^2/n^2.
Quantized Angular Momentum: L = mvr = \frac{nh}{2\pi}
Radius of n-th Orbit: $r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA}$

Must remember

The orbital angular momentum of an electron in the n-th Bohr orbit is quantized as $L = nh / (2\pi)$.
The radius of the n-th Bohr orbit varies as $r_n \propto n^2 / Z$ with the ground state value of $0.529 \text{ \AA}$ for hydrogen.
The orbital velocity of an electron in the n-th orbit varies inversely with the principal quantum number $n$ ($v_n \propto Z / n$).
The total energy of an electron in a Bohr orbit is negative, indicating a bound state, and is given by $E_n = -13.6 Z^2 / n^2 \text{ eV}$.
Transitions terminating at $n = 1$ give the Lyman series (ultraviolet), while $n = 2$ gives the Balmer series (visible).
Finite nuclear mass correction replaces the electron mass $m$ with the reduced mass $\mu = (m \cdot M) / (m + M)$.

Glossary

Stationary StatesPermitted orbits in which an electron does not radiate electromagnetic energy despite continuous acceleration.

Reduced Mass ($\mu$)Effective mass used to correct for finite nuclear mass when the nucleus is not infinitely heavy compared to the orbiting electron.

Rydberg Constant ($R$)A fundamental physical constant governing the wave numbers of emitted or absorbed spectral lines in hydrogen-like atoms.

Ionization Energy ($E_{\text{ion}}$)The energy required to completely remove an electron from the ground state ($n = 1$) to infinity ($n = \infty$).

Hydrogen Spectral Series Comparison

Series	Lower State ($n_1$)	Spectral Region
Lyman	1	Ultraviolet
Balmer	2	Visible
Paschen	3	Infrared
Brackett	4	Infrared

Practice Quiz

Choose an option to reveal the explanation.

1. What is the orbital angular momentum of an electron in the third Bohr orbit ($n = 3$) of a hydrogen atom?

Explanation: According to Bohr's postulate, angular momentum is quantized as $L = \frac{nh}{2\pi}$. For $n = 3$, $L = \frac{3h}{2\pi}$.

2. How does the radius ($r_n$) of the n-th Bohr orbit depend on the principal quantum number $n$?

Explanation: The radius of the n-th Bohr orbit is directly proportional to the square of the principal quantum number: $r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA}$.

3. What is the total energy of an electron in the ground state of a hydrogen atom?

Explanation: The total energy is given by $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$. For hydrogen ($Z = 1$) in the ground state ($n = 1$), $E_1 = -13.6 \text{ eV}$.

4. Which spectral series of hydrogen falls entirely within the visible region of the electromagnetic spectrum?

Explanation: Transitions terminating at $n = 2$ yield the Balmer series, which lies in the visible region.

5. Why is the total energy of an electron in a Bohr orbit negative?

Explanation: A negative total energy signifies a bound state where external work or energy is required to separate the electron from the nucleus to infinity.

6. How does the orbital velocity ($v_n$) of an electron vary with the principal quantum number $n$ in a hydrogen-like atom?

Explanation: The orbital velocity is given by $v_n = \frac{c}{137} \frac{Z}{n}$, meaning $v_n$ varies inversely with $n$.

7. Which transition in a hydrogen atom emits a photon of the longest wavelength within the Balmer series?

Explanation: Longest wavelength corresponds to minimum energy difference, which occurs for the transition between adjacent levels $n = 3$ to $n = 2$.

8. When accounting for finite nuclear mass, the electron mass $m$ is replaced by the reduced mass $\mu$ given by:

Explanation: For finite nuclear mass, the reduced mass is defined as $\mu = \frac{m M}{m + M}$.

9. What is the ionization energy of a hydrogen atom in its ground state?

Explanation: Ionization energy is the energy required to take the electron from $n = 1$ to $n = \infty$, which equals $E_{\infty} - E_1 = 0 - (-13.6 \text{ eV}) = 13.6 \text{ eV}$.

10. Which region of the electromagnetic spectrum contains the Lyman series of hydrogen?

Explanation: Transitions terminating at the ground state ($n = 1$) yield the Lyman series, which lies in the ultraviolet region.

11. What is the physical meaning of 'stationary states' in Bohr's theory?

Explanation: Stationary states refer to allowed orbits where an electron does not radiate energy, contrary to classical electrodynamics.

12. What is the ratio of the radii of the first three Bohr orbits of hydrogen ($r_1 : r_2 : r_3$)?

Explanation: Since $r_n \propto n^2$, the ratio is $1^2 : 2^2 : 3^2 = 1 : 4 : 9$.

13. How does the Rydberg constant change when finite nuclear mass is considered compared to infinite nuclear mass?

Explanation: Since the reduced mass $\mu = \frac{mM}{m+M}$ is slightly less than the electron mass $m$, the corrected Rydberg constant is slightly smaller.

14. What is the potential energy of an electron in the n-th Bohr orbit of a hydrogen-like atom?

Explanation: Total energy is $E = K + U$. Since $K = -E$ and $U = 2E$, the potential energy is $2 \times (-13.6 \frac{Z^2}{n^2}) = -27.2 \frac{Z^2}{n^2} \text{ eV}$ (or commonly written as $-27.2$ scaling).

15. Which of the following limitations is associated with Bohr's theory of the hydrogen atom?

Explanation: Bohr's model is restricted to single-electron systems, assumes circular orbits, and cannot predict fine structure without further relativistic and spin modifications.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The wavelength of the first line of the Balmer series is $\lambda$. What is the wavelength of the first line of the Lyman series?

Explanation: For Balmer first line ($n_1 = 2, n_2 = 3$): $\frac{1}{\lambda_B} = R(\frac{1}{4} - \frac{1}{9}) = \frac{5}{36}R$. For Lyman first line ($n_1 = 1, n_2 = 2$): $\frac{1}{\lambda_L} = R(1 - \frac{1}{4}) = \frac{3}{4}R = \frac{27}{36}R$. Thus $\frac{\lambda_L}{\lambda_B} = \frac{5}{27} \implies \lambda_L = \frac{5}{27}\lambda$.

2. If the velocity of an electron in the first Bohr orbit of hydrogen is $v$, what is its velocity in the second Bohr orbit?

Explanation: Orbital velocity is inversely proportional to $n$ ($v_n \propto 1/n$). Therefore, for $n = 2$, the velocity is $v / 2$.

3. The total energy of an electron in the ground state of hydrogen is $-13.6 \text{ eV}$. What is its kinetic energy in the second excited state?

Explanation: The second excited state corresponds to $n = 3$. Total energy $E_3 = -13.6 / 3^2 = -1.51 \text{ eV}$. Kinetic energy is equal to $-E_3$, which is $+1.51 \text{ eV}$.

4. What is the maximum number of emission lines possible when an excited hydrogen atom returns from the $n = 4$ state to the ground state?

Explanation: The number of spectral lines emitted when an electron drops from level $n$ to ground state is given by $\frac{n(n-1)}{2}$. For $n = 4$, $\frac{4(3)}{2} = 6$.

5. In a hydrogen-like atom, the radius of the $n$-th orbit is given by $r_n = r_0 n^2$. If the atom is singly ionized helium ($He^+$), what is the radius of its ground state orbit in terms of $r_0$?

Explanation: For helium, $Z = 2$. The radius formula is $r_n = 0.529 \frac{n^2}{Z} \text{ \AA}$. For $n = 1$ and $Z = 2$, $r_1 = \frac{r_0}{2} = 0.5 r_0$.

6. The Rydberg constant $R$ is expressed in terms of fundamental constants as $R = \frac{m e^4}{8 \epsilon_0^2 c h^3}$. What is the dimensional formula for $R$?

Explanation: Since wave number $\bar{\nu} = \frac{1}{\lambda}$, its SI unit is $\text{m}^{-1}$. Therefore, the dimensional formula of the Rydberg constant is $[L^{-1}]$ or $[M^0 L^{-1} T^0]$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. Calculate the frequency of the radiation emitted when an electron jumps from the $n = 4$ state to the $n = 2$ state in a hydrogen atom ($h = 6.63 \times 10^{-34} \text{ J s}$).

Explanation: Energy difference $\Delta E = E_4 - E_2 = -0.85 \text{ eV} - (-3.4 \text{ eV}) = 2.55 \text{ eV} = 2.55 \times 1.6 \times 10^{-19} \text{ J}$. Frequency $\nu = \frac{\Delta E}{h} \approx 6.17 \times 10^{14} \text{ Hz}$.

2. If the mass of the nucleus is finite, the corrected Rydberg constant $R'$ is related to the infinite-mass Rydberg constant $R_{\infty}$ by:

Explanation: Since $\mu = \frac{m}{1 + \frac{m}{M}}$, and Rydberg constant is proportional to $\mu$, we get $R' = \frac{R_{\infty}}{1 + m/M}$.

3. What is the ratio of the frequency of revolution of an electron in the first Bohr orbit to that in the second Bohr orbit of hydrogen?

Explanation: Frequency $f = \frac{v}{2\pi r}$. Since $v \propto \frac{1}{n}$ and $r \propto n^2$, frequency $f \propto \frac{1/n}{n^2} = \frac{1}{n^3}$. Thus, $\frac{f_1}{f_2} = \frac{2^3}{1^3} = \frac{8}{1}$.

4. Which of the following transitions in a hydrogen atom will give rise to the most energetic photon?

Explanation: The transition from $n = \infty$ to $n = 1$ spans the entire potential well of the ground state, releasing maximum energy ($13.6 \text{ eV}$).

5. According to Bohr's model, the magnetic dipole moment $\mu_m$ of an electron revolving in the n-th orbit is proportional to:

Explanation: Magnetic dipole moment $\mu_m = I A = \left(\frac{e}{T}\right) (\pi r^2)$. Since $T \propto \frac{n^3}{Z^2}$ and $r \propto \frac{n^2}{Z}$, $\mu_m \propto n$.

6. What is the shortest wavelength present in the Paschen series of hydrogen?

Explanation: For Paschen series ($n_1 = 3$), the shortest wavelength corresponds to $n_2 = \infty$: $\frac{1}{\lambda} = R (\frac{1}{9} - 0) = \frac{R}{9} \implies \lambda = \frac{9}{R} \approx 820 \text{ nm}$.

7. In terms of Bohr radius $a_0$, what is the radius of the second orbit of doubly ionized lithium ($Li^{2+}$)?

Explanation: For $Li^{2+}$, $Z = 3$. The radius is $r_n = a_0 \frac{n^2}{Z}$. For $n = 2$, $r_2 = a_0 \frac{2^2}{3} = \frac{4}{3} a_0$.

8. The concept of quantization of angular momentum in Bohr's theory was later explained wave-mechanically by considering:

Explanation: de Broglie explained that stationary orbits are those that contain an integral number of electron de Broglie wavelengths along their circumference ($2\pi r = n\lambda$).

9. What is the potential energy to kinetic energy ratio for an electron in any Bohr orbit?

Explanation: Kinetic energy $K = -E$ and potential energy $U = 2E$, so $\frac{U}{K} = \frac{2E}{-E} = -2$ (or $-2:1$).

10. What happens to the speed of an electron in a hydrogen atom if it jumps to a higher orbit?

Explanation: Since orbital velocity $v_n \propto \frac{1}{n}$, as $n$ increases (higher orbit), the orbital velocity decreases.