Chapter 12: Atoms

Dear Class 12 Student! The journey into the heart of matter begins here. Atoms were once thought to be indivisible, solid spheres (Thomson's plum pudding model). But through the elegant scattering experiments of Rutherford and the profound quantum insights of Niels Bohr, we mapped the true structure of the nucleus and orbiting electrons. This chapter is a treasure trove for Boards (derivations & spectral series) and forms the mathematical backbone for JEE Modern Physics. Let's unlock the atom!

1. Alpha-Particle Scattering and Rutherford’s Nuclear Model

In 1911, H. Geiger and E. Marsden, under the direct supervision of Ernest Rutherford, performed the landmark Gold Foil Experiment. They aimed a highly energetic beam of alpha particles ($\text{He}^{2+}$ ions) at an incredibly thin gold foil (thickness $\approx 2.1 \times 10^{-7} \text{ m}$).

Observations and Revolutionary Conclusions

They observed flashes of light on the Zinc Sulfide (ZnS) screen to track where the alpha particles went. The results shattered classical physics:

• Observation 1: Most of the alpha particles ($>99\%$) passed straight through the gold foil without any measurable deflection.
  Conclusion: Most of the space inside an atom is entirely empty.
• Observation 2: A small fraction ($\approx 0.14\%$) of alpha particles were deflected by angles greater than 1°.
  Conclusion: There must be a massive, positively charged center repelling the positively charged alpha particles.
• Observation 3: A highly minute fraction (about 1 in 8,000) deflected by more than 90°, or even rebounded completely back ($\theta = 180^\circ$). Rutherford famously said it was like firing a 15-inch naval shell at tissue paper and having it bounce back!
  Conclusion: The entire positive charge and almost the entire mass of the atom are intensely concentrated in an extremely tiny central volume, which Rutherford named the Nucleus.

Distance of Closest Approach ($r_0$)

Imagine an alpha particle (charge $q_1 = +2e$, mass $m$) fired directly head-on at a gold nucleus (charge $q_2 = +Ze$). As it approaches, the massive electrostatic repulsion slows it down. At the "Distance of Closest Approach" ($r_0$), its velocity becomes zero. Here, its initial Kinetic Energy ($K$) is completely converted into Electrostatic Potential Energy ($U$).

Drawbacks of Rutherford's Model

While correct about the nucleus, Rutherford's model had two fatal flaws regarding electrons:

1. Atom Instability: According to classical electromagnetic theory (Maxwell), any charged particle undergoing acceleration must continuously radiate electromagnetic energy. An electron orbiting a nucleus is constantly changing direction, hence it is constantly accelerating ($v^2/r$). It should radiate energy, spiral inward, and crash into the nucleus within $10^{-8}$ seconds. Matter shouldn't exist!
2. Continuous Spectra: If an electron spirals inward, its orbital frequency changes continuously. It should emit a continuous rainbow spectrum of light. In reality, gases like Hydrogen emit highly specific, discrete line spectra.

2. Bohr Model of the Hydrogen Atom

In 1913, Niels Bohr boldly mixed classical mechanics with Max Planck's new quantum theory to save the atom. He proposed three revolutionary postulates.

A. Derivation of Radius of $n^{th}$ Orbit ($r_n$)

For an electron (mass $m$, charge $e$, velocity $v$) orbiting a nucleus of charge $+Ze$, the required centripetal force is provided exactly by the electrostatic force of attraction:

$$F_c = F_e$$

$$\frac{mv^2}{r} = \frac{1}{4\pi\epsilon_0} \frac{(Ze)(e)}{r^2}$$

$$mv^2 = \frac{Ze^2}{4\pi\epsilon_0 r} \quad \text{--- (Equation 1)}$$

From Bohr's Second Postulate, we extract velocity $v$:
$mvr = \frac{nh}{2\pi} \implies v = \frac{nh}{2\pi mr}$.

Substitute this $v$ into Equation 1:
$$m \left( \frac{nh}{2\pi mr} \right)^2 = \frac{Ze^2}{4\pi\epsilon_0 r}$$
$$\frac{m n^2 h^2}{4\pi^2 m^2 r^2} = \frac{Ze^2}{4\pi\epsilon_0 r}$$
Solving for $r$, we get the radius of the $n^{th}$ orbit:

B. Derivation of Velocity of Electron ($v_n$)

Substitute the derived value of $r_n$ back into the angular momentum equation ($v = \frac{nh}{2\pi mr}$):

$$v_n = \frac{nh}{2\pi m \left( \frac{n^2 h^2 \epsilon_0}{\pi m Z e^2} \right)}$$

3. Energy Levels in Hydrogen Atom

To understand energy transitions, we must find the total energy of the electron in the $n^{th}$ orbit.

• Kinetic Energy ($K$): We know from Eq 1 that $mv^2 = \frac{Ze^2}{4\pi\epsilon_0 r}$. Therefore:
  $$K = \frac{1}{2}mv^2 = \mathbf{\frac{Ze^2}{8\pi\epsilon_0 r}}$$ (Always positive).
• Potential Energy ($U$): The electrostatic potential energy between nucleus ($+Ze$) and electron ($-e$):
  $$U = \frac{1}{4\pi\epsilon_0} \frac{(+Ze)(-e)}{r} = \mathbf{-\frac{Ze^2}{4\pi\epsilon_0 r}}$$ (Negative sign indicates bound state).
• Total Energy ($E$): The sum of kinetic and potential energy.
  $$E = K + U = \frac{Ze^2}{8\pi\epsilon_0 r} - \frac{Ze^2}{4\pi\epsilon_0 r} = \mathbf{-\frac{Ze^2}{8\pi\epsilon_0 r}}$$

Now, substituting the radius formula $r_n$ into the Total Energy expression gives us the exact energy for any orbit:

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

Substituting all the massive constants ($m, e, \epsilon_0, h$) yields the energy in Joules, which we convert to Electron-volts (eV):

Energy Definitions

• Ground State: The lowest energy state ($n=1$). For Hydrogen, $E_1 = -13.6 \text{ eV}$.
• Excited States: Any state where $n > 1$. ($n=2$ is the 1st excited state, $n=3$ is the 2nd excited state).
• Excitation Energy: Energy required to bump an electron from ground state to an excited state.
  Example: Energy to reach 1st excited state $= E_2 - E_1 = -3.4 - (-13.6) = +10.2 \text{ eV}$.
• Ionization Energy: The energy required to completely remove an electron from its current orbit to infinity ($n=\infty$).
  Example: Ionization energy for H ground state $= E_\infty - E_1 = 0 - (-13.6) = +13.6 \text{ eV}$.

4. The Line Spectra of Hydrogen Atom

When an excited electron (at orbit $n_2$) spontaneously falls to a lower, more stable orbit ($n_1$), it must shed the excess energy as an electromagnetic photon. The energy of this photon is $\Delta E = E_{n2} - E_{n1}$.

Using the energy formula, $\Delta E = 13.6 Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \text{ eV}$.

Converting this to wavelength using $E = \frac{hc}{\lambda}$ yields the empirical Rydberg Formula:

Spectral Series of Hydrogen (High Board Priority)

Depending on which lower orbit ($n_1$) the electron ultimately crashes into, the emitted photons fall into distinct named series:

5. De Broglie’s Explanation of Bohr’s Second Postulate

Bohr's quantization of angular momentum ($mvr = nh/2\pi$) worked perfectly, but it was essentially a lucky guess—he had no theoretical proof for why it had to be $h/2\pi$. Ten years later, Louis de Broglie provided the brilliant missing link using matter waves.

Concept: De Broglie argued that a revolving electron acts as a wave. For an orbit to be stable (stationary state), the electron wave must not destructively interfere with itself as it goes around. It must form a perfect Standing (Stationary) Wave.

Condition: A standing wave can only form in a circular orbit if the total circumference is an exact integral multiple of the electron's wavelength.

$$2\pi r = n\lambda$$ (where $n = 1, 2, 3 \dots$)

6. Limitations of the Bohr Model

While historically monumental, the Bohr model is an incomplete semiclassical model. It eventually gave way to the full Quantum Mechanical Model (Schrödinger). Its flaws are:

• Single-Electron Constraint: It is mathematically valid ONLY for hydrogenic (single-electron) atoms/ions ($\text{H}, \text{He}^+, \text{Li}^{2+}, \text{Be}^{3+}$). It utterly fails to predict the spectra of multi-electron atoms (like Helium) because it ignores electron-electron repulsion.
• Relative Intensities: In hydrogen spectra, some lines are much brighter than others. The Bohr model predicts wavelengths perfectly, but cannot predict the probability/intensity of these transitions.
• Fine Structure: When viewed under highly sensitive spectrometers, a single spectral line (like the $H_\alpha$ line) is actually composed of multiple closely packed fine lines. Bohr's circular orbits cannot explain this (requiring Sommerfeld's elliptical orbits and electron spin).
• Magnetic and Electric Fields: Bohr's model provides no explanation for the Zeeman Effect (the splitting of spectral lines when the gas is placed in a strong magnetic field) or the Stark Effect (splitting in an electric field).
• Violation of Uncertainty: Bohr assigns an exact radius and an exact velocity to an electron simultaneously, which directly violates Heisenberg's Uncertainty Principle.