INTRODUCTION

By the end of the 19th century, significant evidence supported the idea that matter is composed of atoms. Experiments involving the discharge of electricity through gases, notably by J. J. Thomson in 1897, revealed that atoms contain identical negatively charged particles called **electrons**. Since atoms are electrically neutral overall, they must also contain an equal amount of positive charge. A major question arose: how are these positive charges and electrons arranged within the atom?

The first attempt to describe atomic structure was **J. J. Thomson's model** (1898). He proposed that an atom is a sphere of uniformly distributed positive charge with electrons embedded within it, like seeds in a watermelon (often called the **plum pudding model**). However, later experimental findings contradicted this uniform distribution of charge and mass within the atom.

Observations of light emitted by substances also provided clues about atomic structure. Unlike hot, dense matter, which emits a continuous spectrum of wavelengths, rarefied gases when heated or electrically excited emit light at specific, discrete wavelengths, producing a **line spectrum**. Each element exhibits a unique characteristic line spectrum.

The fact that elements emit distinct wavelengths suggested a close relationship between the internal structure of an atom and the light it emits. In 1885, Johann Jakob Balmer found an empirical formula describing the wavelengths of a series of lines in the visible spectrum of hydrogen, the simplest atom.

To further investigate atomic structure, Ernest Rutherford, building on his studies of alpha particles from radioactive elements, proposed a scattering experiment. Conducted by Hans Geiger and Ernst Marsden around 1911, this experiment (detailed in the next section) profoundly impacted the understanding of the atom, leading to Rutherford's **nuclear model**.

In Rutherford's model (also known as the planetary model), the atom's positive charge and most of its mass are concentrated in a small central region (the nucleus), with electrons orbiting around it. This model was a significant step but faced challenges in explaining the observed discrete atomic spectra and the fundamental stability of the atom based on classical physics.

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ALPHA-PARTICLE SCATTERING AND RUTHERFORD’S NUCLEAR MODEL OF ATOM

Following Rutherford's suggestion, H. Geiger and E. Marsden conducted experiments around 1911 to probe atomic structure using alpha particles. They directed a beam of energetic alpha particles (e.g., 5.5 MeV, which are Helium nuclei with a charge of +2e and mass approximately equal to a Helium atom) from a radioactive source towards a very thin gold foil.

A detector (consisting of a zinc sulphide screen and a microscope) was used to observe the scattered alpha particles at various angles by detecting the light flashes (scintillations) produced when the particles struck the screen.

The observations were striking:

• Most $\alpha$-particles passed straight through the foil without significant deviation.
• A small fraction of $\alpha$-particles scattered through moderate angles (greater than 1°).
• A very tiny fraction of $\alpha$-particles (about 1 in 8000) were deflected by more than 90°, some even being scattered backward ($\approx 180^\circ$).

Rutherford interpreted these results. Most $\alpha$-particles passing straight through suggested that most of the atom is **empty space**. The occasional large-angle scattering and backscattering indicated that the $\alpha$-particle must have encountered a very strong repulsive force. This force, based on Coulomb's law for charged particles, could only arise if the atom's positive charge and nearly all its mass were concentrated in a very small volume at the center, rather than being spread throughout the atom (as in Thomson's model). The heavy gold nucleus (much heavier than the $\alpha$-particle) was assumed to remain stationary during the scattering interaction.

This led to the development of **Rutherford's nuclear model** of the atom: the atom consists of a small, dense, positively charged **nucleus** at the center, surrounded by electrons orbiting in relatively large empty space around it. The size of the nucleus was estimated to be around $10^{-15}$ to $10^{-14}$ m, while the atomic size is about $10^{-10}$ m (about 10,000 to 100,000 times larger than the nucleus).

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Alpha-Particle Trajectory

In Rutherford scattering, the path followed by an alpha particle depends on its **impact parameter ($b$)**. The impact parameter is the perpendicular distance between the initial velocity vector of the alpha particle and the center of the nucleus it is approaching.

• For a large impact parameter ($b$), the alpha particle passes far from the nucleus and experiences a weak repulsive force, resulting in a **small scattering angle** ($\theta \approx 0$).
• For a small impact parameter ($b$), the alpha particle approaches close to the nucleus and experiences a strong repulsive force, resulting in a **large scattering angle**.
• For a head-on collision ($b$ is minimum, ideally zero), the alpha particle comes closest to the nucleus, momentarily stops, and is scattered backward ($\theta \approx \pi$ or 180°).

The observation of a small fraction of particles undergoing large-angle scattering (especially backscattering) strongly supported the idea that the positive charge and mass were concentrated in a tiny nucleus, as only small impact parameters lead to such deflections. Rutherford scattering experiments were therefore instrumental in determining an upper limit to the size of the atomic nucleus.

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Electron Orbits

In Rutherford's nuclear model, electrons orbit the nucleus like planets orbit the sun. For a hydrogen atom (one electron, one proton), the electron's motion is a balance between the attractive electrostatic force from the nucleus and the centripetal force required for circular motion.

Electrostatic force ($F_e$) = Centripetal force ($F_c$): $\frac{1}{4\pi\epsilon_0} \frac{e^2}{r^2} = \frac{mv^2}{r}$ (for an electron of mass $m$ and charge $-e$ orbiting at radius $r$ with speed $v$).

From this, the kinetic energy is $K = \frac{1}{2}mv^2 = \frac{1}{8\pi\epsilon_0} \frac{e^2}{r}$.

The electrostatic potential energy is $U = -\frac{1}{4\pi\epsilon_0} \frac{e^2}{r}$ (taking zero PE at infinite separation).

The total energy $E = K + U = \frac{1}{8\pi\epsilon_0} \frac{e^2}{r} - \frac{1}{4\pi\epsilon_0} \frac{e^2}{r} = -\frac{1}{8\pi\epsilon_0} \frac{e^2}{r}$.

The total energy of the orbiting electron is negative, indicating that the electron is bound to the nucleus. Positive total energy would mean the electron is unbound.

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Atomic Spectra

When atomic gases or vapours are excited at low pressure (e.g., by electric current), they emit radiation only at certain specific wavelengths, forming an **emission line spectrum** (bright lines on a dark background). When white light passes through such a gas, the gas absorbs light at specific wavelengths, producing an **absorption spectrum** (dark lines in the continuous spectrum), which corresponds to the wavelengths present in its emission spectrum.

Each element has a unique characteristic spectrum, acting as a "fingerprint", providing information about its atomic structure.

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Spectral Series

The spectral lines of hydrogen, the simplest atom, show regularity and are grouped into series. The first series observed was the **Balmer series** in the visible region. Johann Jakob Balmer (1885) found an empirical formula for its wavelengths:

$\mathbf{\frac{1}{\lambda} = R \left(\frac{1}{2^2} - \frac{1}{n^2}\right)}$, where $n = 3, 4, 5, ...$

$R$ is the **Rydberg constant** ($1.097 \times 10^7 \text{ m}^{-1}$). Different values of $n$ correspond to different lines in the series (H$_\alpha$ for $n=3$, H$_\beta$ for $n=4$, etc.). The series limit is for $n=\infty$, giving the shortest wavelength.

Other series were discovered later, in different spectral regions (ultraviolet and infrared). These include the **Lyman, Paschen, Brackett, and Pfund series**, each corresponding to transitions ending in different principal quantum number levels ($n_f$).

Series	Final level ($n_f$)	Initial levels ($n_i$)	Formula for wavelength	Spectral Region
Lyman	1	2, 3, 4, ...	$\frac{1}{\lambda} = R \left(\frac{1}{1^2} - \frac{1}{n_i^2}\right)$	Ultraviolet
Balmer	2	3, 4, 5, ...	$\frac{1}{\lambda} = R \left(\frac{1}{2^2} - \frac{1}{n_i^2}\right)$	Visible
Paschen	3	4, 5, 6, ...	$\frac{1}{\lambda} = R \left(\frac{1}{3^2} - \frac{1}{n_i^2}\right)$	Infrared
Brackett	4	5, 6, 7, ...	$\frac{1}{\lambda} = R \left(\frac{1}{4^2} - \frac{1}{n_i^2}\right)$	Infrared
Pfund	5	6, 7, 8, ...	$\frac{1}{\lambda} = R \left(\frac{1}{5^2} - \frac{1}{n_i^2}\right)$	Infrared

These formulas describe the observed wavelengths but don't explain the underlying physics (why only these specific wavelengths appear).

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Bohr Model Of The Hydrogen Atom

Rutherford's classical model faced significant problems: (i) Stability issue - an orbiting electron, being an accelerating charge, should continuously radiate energy and spiral into the nucleus according to classical electromagnetic theory. Atoms would collapse. (ii) Spectrum issue - the electron's orbital frequency and hence the frequency of emitted light should change continuously as it spirals, leading to a continuous spectrum, not the observed discrete line spectrum.

Niels Bohr proposed a model in 1913 that combined classical physics with quantum ideas to resolve these issues, specifically for the hydrogen atom. Bohr's postulates are:

• (i) Bohr's First Postulate (Stability): Electrons in an atom revolve around the nucleus only in certain **stable, discrete orbits** without radiating energy. These specific orbits correspond to **stationary states** of the atom, each having a definite total energy.
• (ii) Bohr's Second Postulate (Quantisation of Angular Momentum): The electron revolves around the nucleus only in those orbits for which its angular momentum $L$ is an integral multiple of $h/(2\pi)$. $h$ is Planck's constant.

  $\mathbf{L_n = m v_n r_n = \frac{nh}{2\pi}}$, where $n = 1, 2, 3, ...$.

  $n$ is the principal quantum number, identifying the allowed orbits.

• (iii) Bohr's Third Postulate (Transitions): An electron can make a transition from a higher energy stationary state ($E_i$) to a lower energy state ($E_f$). When it does, it emits a photon whose energy equals the energy difference between the states.

  $\mathbf{h\nu = E_i - E_f}$, where $\nu$ is the photon frequency.

  Absorption of a photon with this specific energy can cause a transition from a lower to a higher energy state ($E_f + h\nu = E_i$).

Using these postulates and the classical force balance (electrostatic attraction provides centripetal force), Bohr derived expressions for the radii of the allowed orbits ($r_n$) and the corresponding total energies ($E_n$) for the hydrogen atom:

$\mathbf{r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2}}$

For $n=1$, $r_1 = \frac{h^2 \epsilon_0}{\pi m e^2}$ is the **Bohr radius ($a_0$)**, the radius of the innermost orbit ($a_0 \approx 5.29 \times 10^{-11}$ m). Radii of higher orbits increase as $r_n = n^2 a_0$.

$\mathbf{E_n = -\frac{me^4}{8\epsilon_0^2 h^2 n^2}}$

Substituting values of constants: $E_n \approx -\frac{2.18 \times 10^{-18}}{n^2}$ J or $\mathbf{E_n \approx -\frac{13.6}{n^2}}$ eV.

The negative energy indicates the electron is bound. Ionisation energy for hydrogen (from the ground state, $n=1$) is $+13.6$ eV.

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Energy Levels

The electron energy is lowest (most negative) in the orbit closest to the nucleus ($n=1$), called the **ground state** ($E_1 = -13.6$ eV). Higher values of $n$ correspond to **excited states** ($n=2, 3, ...$) with progressively higher energies ($E_2 = -3.40$ eV, $E_3 = -1.51$ eV, etc.). The energy levels get closer together as $n$ increases. The highest energy state ($n=\infty$) corresponds to $E_\infty = 0$ eV, representing an ionised atom (electron is free and at rest far from nucleus).

Atoms in the ground state can absorb energy (e.g., from collisions or photons) to excite electrons to higher energy levels. Electrons in excited states can transition back to lower states, emitting photons.

The energy level diagram for hydrogen (Figure 12.8) visualises these discrete allowed energies, showing the convergence of levels at higher $n$ and the continuum of energies above $E=0$ for a free electron.

The existence of these discrete energy levels was experimentally verified by the **Franck-Hertz experiment** in 1914, where electrons lost energy only in discrete amounts when colliding with mercury atoms, and specific wavelengths were emitted.

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The Line Spectra Of The Hydrogen Atom

Bohr's third postulate explains the emission and absorption line spectra of hydrogen. When an electron transitions from a higher energy level ($n_i$) to a lower energy level ($n_f$), a photon is emitted with energy $h\nu = E_{n_i} - E_{n_f}$.

Substituting the derived energy levels $E_n = -\frac{me^4}{8\epsilon_0^2 h^2 n^2}$: $\mathbf{h\nu = \frac{me^4}{8\epsilon_0^2 h^2} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)}$, where $n_i > n_f$.

The frequency $\nu$ (or wavelength $\lambda = c/\nu$) of the emitted photon is therefore determined by the initial and final quantum numbers of the electron's transition. This formula predicts only specific frequencies, which matches the observed discrete line spectrum.

Dividing by $hc$ gives the wave number $\frac{1}{\lambda}$: $\mathbf{\frac{1}{\lambda} = \frac{me^4}{8\epsilon_0^2 h^3 c} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)}$

This formula has the same form as the empirical Rydberg formula. The theoretical expression for the Rydberg constant is $\mathbf{R = \frac{me^4}{8\epsilon_0^2 h^3 c}}$. Calculating this value using known physical constants yields a value ($1.097 \times 10^7 \text{ m}^{-1}$) that is in excellent agreement with the experimentally determined value from the hydrogen spectrum. This provided strong evidence for the correctness of Bohr's model.

Different series in the hydrogen spectrum correspond to transitions ending in a specific lower level $n_f$:

• Lyman series: $n_f=1$, $n_i = 2, 3, 4, ...$ (UV region)
• Balmer series: $n_f=2$, $n_i = 3, 4, 5, ...$ (Visible region)
• Paschen series: $n_f=3$, $n_i = 4, 5, 6, ...$ (IR region)
• Brackett series: $n_f=4$, $n_i = 5, 6, 7, ...$ (IR region)
• Pfund series: $n_f=5$, $n_i = 6, 7, 8, ...$ (IR region)

The different lines within each series arise from transitions from different higher levels $n_i$ to the same lower level $n_f$. The line spectra can be visualised on the energy level diagram as vertical arrows representing transitions (Figure 12.9).

An atom emits a photon when an electron transitions to a lower state (emission lines). An atom absorbs a photon of the exact energy difference to transition to a higher state (absorption lines).

Bohr's model was a great success in explaining the hydrogen spectrum, but it had limitations, especially for atoms with more than one electron and in explaining the intensity of spectral lines.

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DE BROGLIE’S EXPLANATION OF BOHR’S SECOND POSTULATE OF QUANTISATION

Bohr's second postulate, stating that angular momentum is quantised ($L_n = nh/(2\pi)$), lacked a fundamental theoretical basis within Bohr's model itself. Louis de Broglie provided an explanation in 1923 based on his hypothesis of the wave nature of matter.

De Broglie proposed that an electron orbiting the nucleus should be viewed as a **matter wave**. For a stable orbit, this electron wave must form a **standing wave**, meaning the wave must reinforce itself after each revolution. The condition for forming a standing wave in a circular orbit is that the circumference of the orbit ($2\pi r_n$) must be an integral multiple of the electron's de Broglie wavelength ($\lambda_n$).

$\mathbf{2\pi r_n = n \lambda_n}$, where $n = 1, 2, 3, ...$

Using de Broglie's relation $\lambda_n = h/(mv_n)$, where $m$ is the electron mass and $v_n$ is its speed in the $n$-th orbit, we get:

$2\pi r_n = n \frac{h}{mv_n}$

Rearranging this equation gives:

$\mathbf{mv_n r_n = \frac{nh}{2\pi}}$

This is exactly Bohr's second postulate, the quantisation of angular momentum ($L_n = mv_n r_n$). De Broglie's hypothesis explained that the allowed orbits are those where the electron wave forms a stable standing wave pattern, a consequence of the wave nature of the electron.

Bohr's model, despite its successes in explaining the hydrogen spectrum and predicting energy levels, had limitations. It was a semi-classical model that didn't fully align with quantum mechanics. It could not explain multi-electron atoms or spectral line intensities.

Quantum mechanics, developed later, provides a more complete picture where electron location is described by probability distributions (orbitals) rather than definite orbits.

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SUMMARY

This chapter traces the historical development of atomic models and introduces the wave nature of matter.

• Early models: Thomson's plum pudding model was contradicted by later experiments.
• Rutherford's $\alpha$-scattering experiment revealed the **nuclear model**: tiny, dense, positive nucleus at the center, electrons orbiting in empty space. Scattering angle depends on impact parameter. Estimated nucleus size $\sim 10^{-15}$ m.
• Classical Rutherford model fails to explain atomic stability (orbiting electrons should radiate and collapse) and discrete atomic spectra.
• Atomic spectra: Excited gases emit discrete wavelengths (line spectra). Hydrogen spectrum has series (Lyman, Balmer, Paschen, etc.) with wavelengths described by empirical formulas like the Rydberg formula.
• **Bohr's model** (for hydrogenic atoms): Introduced quantum ideas to resolve classical issues. Postulates: (i) Stable stationary orbits (no radiation). (ii) Angular momentum quantised ($L_n = nh/(2\pi)$). (iii) Photon emission/absorption during transitions ($h\nu = E_i - E_f$).
• Bohr model derived quantised radii ($r_n = n^2 a_0$) and energy levels ($E_n = -13.6/n^2$ eV) for hydrogen. Ground state $n=1$. Negative energy means bound.
• Bohr model correctly predicted hydrogen spectral lines and the Rydberg constant value, confirming the origin of discrete spectra from transitions between discrete energy levels.
• **De Broglie hypothesis:** Matter (particles) also has wave properties. **De Broglie wavelength** $\lambda = h/p$.
• De Broglie explained Bohr's quantisation: Electron wave forms **standing wave** in orbit ($2\pi r_n = n\lambda_n$), leading to $L_n = nh/(2\pi)$.
• Experimental verification of electron wave nature by Davisson and Germer, and G.P. Thomson (electron diffraction).
• Matter waves incorporate the uncertainty principle and provide basis for quantum mechanics.

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POINTS TO PONDER

Further reflections on the concepts:

• Thomson's and Rutherford's classical models were unstable (electrostatically for Thomson, due to radiation for Rutherford).
• Bohr's choice of angular momentum quantisation (related to $h$, a constant with angular momentum units) was insightful.
• Bohr's orbital picture is simplified; quantum mechanics uses orbitals (probability distributions).
• Bohr's model applies mainly to hydrogenic atoms because electron-electron interactions are not included.
• In Bohr model, photon frequency is not electron orbital frequency, but energy difference divided by $h$. They coincide for large $n$.
• Bohr's model is simplified but accounts for key hydrogen spectrum features and uses classical concepts, making it historically important despite quantum mechanical refinements.

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Summary table of physical quantities:

Physical Quantity	Symbol	Dimensions	Unit	Remarks
Planck’s constant	$h$	[ML$^2$T$^{–1}$]	J s	$E = h\nu$
Stopping potential	$V_0$	[ML$^2$T$^{–3}$A$^{–1}$]	V	$eV_0 = K_{max}$
Work function	$\phi_0$	[ML$^2$T$^{–2}$]	J; eV	$K_{max} = h\nu – \phi_0$
Threshold frequency	$\nu_0$	[T$^{–1}$]	Hz	$\nu_0 = \phi_0/h$
de Broglie wavelength	$\lambda$	[L]	m	$\lambda = h/p$

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