CBSE Class 12 Physics Revision Notes: Chapter 12 Atoms

Early Models of the Atom: From Thomson to Rutherford

The journey to understanding the atom began with various models. J.J. Thomson's Plum Pudding Model (1898) proposed that an atom consisted of a uniformly positive sphere with electrons embedded in it, like seeds in a watermelon or plums in a pudding. This model could explain the overall neutrality of an atom but failed to explain the results of later experiments. Its major drawback was its inability to account for the scattering of alpha particles observed by Geiger and Marsden.

Rutherford's Nuclear Model (1911), based on the alpha-particle scattering experiment, revolutionized atomic theory. In this experiment, a beam of alpha particles was directed at a thin gold foil. The observations were crucial:

• Most alpha particles passed straight through the foil, indicating that most of the atom is empty space.
• A few particles were deflected by small angles, suggesting a positive charge concentration within the atom.
• Very few (about 1 in 8000) particles were deflected by more than 90°, and some even bounced back, implying a tiny, dense, positively charged core at the center, which Rutherford called the nucleus.

Based on these observations, Rutherford proposed that an atom has a tiny, massive, positively charged nucleus at its center, with electrons orbiting it, similar to planets around the sun. The size of the nucleus was estimated to be about $10^{-15} \text{ m}$, much smaller than the atomic size ($10^{-10} \text{ m}$). This model successfully explained the scattering experiment but had significant limitations:

1. Atomic Stability: According to classical electromagnetic theory, an accelerating electron (like one orbiting the nucleus) should continuously radiate energy and spiral into the nucleus, making the atom unstable. However, atoms are stable.
2. Line Spectra: It could not explain the characteristic line spectra emitted by atoms when excited. Instead, it predicted a continuous spectrum. These limitations paved the way for Bohr's quantum model.

Key Atomic Terminology

Atomic Number (Z)

The number of protons in the nucleus of an atom. It defines the element.

Mass Number (A)

The total number of protons and neutrons (nucleons) in an atomic nucleus.

Isotopes

Atoms of the same element (same Z) but with different numbers of neutrons (different A).

Isobars

Atoms of different elements (different Z) but with the same mass number (same A).

Excitation Energy

The energy required to move an electron from a lower energy level to a higher energy level within an atom.

Ionisation Energy

The minimum energy required to remove an electron from the ground state of an atom completely, making it an ion.

Rydberg Constant (R)

A physical constant relating to the atomic spectra, particularly for hydrogen. Its value is approximately $1.097 \times 10^7 \text{ m}^{-1}$.

Bohr's Model of the Hydrogen Atom

Hydrogen Spectrum and Spectral Series

When a hydrogen atom is excited, its electron jumps to higher energy levels. When it de-excites, it falls back to lower energy levels, emitting photons of specific energies, which correspond to specific wavelengths. These discrete wavelengths form the line spectrum of hydrogen, which was successfully explained by Bohr's model.

The Rydberg formula describes the wavelengths of the spectral lines for hydrogen:

$\frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

Where:

• $\lambda$ is the wavelength of the emitted photon.
• $R$ is the Rydberg constant ($1.097 \times 10^7 \text{ m}^{-1}$).
• $Z$ is the atomic number (for hydrogen, $Z=1$).
• $n_i$ is the initial higher energy level.
• $n_f$ is the final lower energy level.

Different series are formed depending on the final energy level ($n_f$) to which the electron transitions:

• Lyman Series ($n_f=1$): Transitions to the ground state ($n_i = 2, 3, 4, ...$). Lies in the ultraviolet (UV) region.
• Balmer Series ($n_f=2$): Transitions to the first excited state ($n_i = 3, 4, 5, ...$). The lines in this series lie in the visible region.
• Paschen Series ($n_f=3$): Transitions to the second excited state ($n_i = 4, 5, 6, ...$). Lies in the infrared (IR) region.
• Brackett Series ($n_f=4$): Transitions to the third excited state ($n_i = 5, 6, 7, ...$). Lies in the infrared (IR) region.
• Pfund Series ($n_f=5$): Transitions to the fourth excited state ($n_i = 6, 7, 8, ...$). Lies in the infrared (IR) region.

Series Limit: The shortest wavelength in any series corresponds to the transition from $n_i = \infty$ to the specific $n_f$ of that series. This is also called the ionization limit for that specific energy level.

Key Formulas and Must-Remember Points

• Rutherford's conclusions: Atom mostly empty space, positive charge concentrated in a tiny nucleus.
• Bohr's first postulate: Stable orbits where electrons don't radiate energy.
• Bohr's second postulate: Quantization of angular momentum $L = mvr = n\frac{h}{2\pi}$.
• Bohr's third postulate: Energy transition $\Delta E = E_f - E_i = h\nu$.
• Radius of $n^{th}$ orbit: $r_n = \frac{n^2}{Z} (0.529 \mathring{A})$.
• Energy of $n^{th}$ orbit: $E_n = - \frac{13.6 Z^2}{n^2} \text{ eV}$.
• Rydberg Formula: $\frac{1}{\lambda} = RZ^2 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$.
• Lyman Series ($n_f=1$) - UV region. Shortest wavelength (series limit) from $n_i=\infty$.
• Balmer Series ($n_f=2$) - Visible region. For $n_i=3,4,5,...$
• Paschen, Brackett, Pfund Series ($n_f=3,4,5$ respectively) - IR region.

Worked Example: Energy and Wavelength

• {"title":"Example 1: Calculate the energy of an electron in the second excited state of a hydrogen atom.","bodyMarkdown":"For a hydrogen atom, $Z=1$. The second excited state means $n=3$ (since $n=1$ is ground state, $n=2$ is first excited state).\nUsing the formula $E_n = - \\frac{13.6 Z^2}{n^2} \\text{ eV}$: \n$E_3 = - \\frac{13.6 \\times (1)^2}{(3)^2} \\text{ eV} = - \\frac{13.6}{9} \\text{ eV} \\approx -1.51 \\text{ eV}$.\nThus, the energy of the electron in the second excited state is approximately $-1.51 \\text{ eV}$."}
• {"title":"Example 2: Determine the shortest wavelength of the Lyman series of hydrogen atom.","bodyMarkdown":"For the Lyman series, the final energy level is $n_f=1$. The shortest wavelength corresponds to a transition from $n_i = \\infty$ to $n_f = 1$.\nUsing the Rydberg formula for hydrogen ($Z=1$):\n$\\frac{1}{\\lambda} = R (1)^2 \\left( \\frac{1}{1^2} - \\frac{1}{\\infty^2} \\right) = R (1 - 0) = R$\nSo, $\\lambda = \\frac{1}{R} = \\frac{1}{1.097 \\times 10^7 \\text{ m}^{-1}} \\approx 9.11 \\times 10^{-8} \\text{ m} = 91.1 \\text{ nm}$.\nThe shortest wavelength in the Lyman series is approximately $91.1 \\text{ nm}$, which is in the ultraviolet region."}

Exam Strategies: Avoid Common Traps

When tackling questions from the Atoms chapter, pay close attention to the details:

• Bohr's Postulates: Be able to state and explain all three postulates clearly. Often, questions test the application of the second postulate ($L = n h / 2\pi$) for angular momentum calculation.
• Derivations: Practice the derivation of the radius and energy of an electron in a Bohr orbit. While not always directly asked, understanding the derivation logic strengthens conceptual grasp.
• Spectral Series: Memorize the $n_f$ values for each series (Lyman $n_f=1$, Balmer $n_f=2$, Paschen $n_f=3$, etc.) and their respective spectral regions (UV, Visible, IR). Questions often ask for the longest or shortest wavelength within a specific series; remember that shortest wavelength corresponds to $n_i = \infty$ and longest to $n_i = n_f + 1$.
• Limitations of Models: Understand the limitations of both Rutherford's and Bohr's models. This is a common theoretical question.

Quick Check: Test Your Understanding

• Q: What were the two main limitations of Rutherford's atomic model? A: Rutherford's model failed to explain the stability of atoms (electron spiraling into nucleus) and could not account for the observed line spectra of atoms.
• Q: State Bohr's second postulate. What does it imply? A: Bohr's second postulate states that an electron can revolve only in those orbits for which its angular momentum is an integral multiple of $h/2\pi$. It implies that the angular momentum of an orbiting electron is quantized.
• Q: Which spectral series of the hydrogen atom lies in the visible region of the electromagnetic spectrum? To which final energy level do its transitions occur? A: The Balmer series lies in the visible region. Its transitions occur when electrons jump to the final energy level $n_f=2$.
• Q: An electron in a hydrogen atom jumps from $n=4$ to $n=2$. Calculate the energy of the emitted photon in eV. A: Using $E_n = -13.6/n^2 \text{ eV}$: $E_4 = -13.6/4^2 = -13.6/16 = -0.85 \text{ eV}$. $E_2 = -13.6/2^2 = -13.6/4 = -3.4 \text{ eV}$. Energy emitted $\Delta E = E_4 - E_2 = -0.85 - (-3.4) = 2.55 \text{ eV}$.

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