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Atomic Structure

CSCA Atomic Structure study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Tutorial Content

Modern Physics: Atomic Structure

1. Evolution of Atomic Models

Understanding the atom has been a progressive journey. For the CSCA exam, you must distinguish between these three historical models:

1. **Thomson's "Plum Pudding Model" (1897)**:

* Envisioned the atom as a sphere of positive charge with electrons embedded like raisins in a pudding.

* **Flaw**: Could not explain the large-angle deflection in the $\alpha$-particle scattering experiment.

2. **Rutherford's "Nuclear Model" (1911)**:

* Based on the **$\alpha$-particle Scattering Experiment** (Gold Foil Experiment).

* **Core Conclusion**: A tiny, massive, positively charged **Nucleus** exists at the center, with electrons orbiting it.

* **Flaw**: Could not explain atomic stability (orbiting electrons should radiate energy and crash) and discrete spectra.

3. **Bohr Model (1913)**:

* **Status**: The core of calculation problems. Introduced quantum theory to successfully explain the Hydrogen spectrum.

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2. Core Topic: Bohr's Hydrogen Theory

Bohr proposed three postulates that broke away from classical physics. Memorize these with the help of diagrams:

(1) Stationary States

Electrons can only move in specific, **discrete** orbits. In these orbits, electrons orbit without radiating energy.

(2) Quantization of Orbits

The orbital radius is not arbitrary but satisfies specific conditions. For Hydrogen:

$$ r_n = n^2 r_1 $$

* $n = 1, 2, 3...$ is the **Principal Quantum Number**.

* $r_1 = 0.53 \mathring{A}$ (or $0.053 \text{nm}$) is the **Bohr Radius** (Ground State).

* **Rule**: Radii increase in the ratio $1:4:9$.

(3) Energy Levels & Transitions

Atomic energy is also discrete. The energy of the $n$-th level in Hydrogen is:

$$ E_n = \frac{E_1}{n^2} $$

* $E_1 = -13.6 \text{eV}$ (**Ground State Energy**, memorize this).

* $E_{\infty} = 0$ (Ionized State).

* **Rule**: As $n$ increases, energy increases (gets closer to 0), and levels become closer together.

**Frequency Condition (Transition Formula)**:

When an electron jumps between levels, it absorbs or emits a photon with energy equal to the difference:

$$ h\nu = |E_m - E_n| $$

* $E_{\text{high}} \to E_{\text{low}}$: **Emits** photon (Light).

* $E_{\text{low}} \to E_{\text{high}}$: **Absorbs** photon (Requires external energy).

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3. Hydrogen Spectrum & Energy Level Diagram

Mapping the quantized energies creates an Energy Level Diagram. This is your map for solving problems.

* **Lyman Series**: Transitions down to $n=1$. Emits **Ultraviolet** (High Energy).

* **Balmer Series**: Transitions down to $n=2$. Emits **Visible Light**.

* **Paschen Series**: Transitions down to $n=3$. Emits **Infrared**.

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4. Problem-Solving Strategies

(1) Counting Transitions

For a group of atoms in state $n$, how many different photon frequencies can be emitted when falling to the ground state?

* **Formula**: Combination $C_n^2 = \frac{n(n-1)}{2}$.

* **Example**: For $n=4$, calculation is $\frac{4 \times 3}{2} = 6$ types.

(2) The 13.6 eV Rule

* **Ionization Energy**: Energy needed to remove an electron from the ground state ($n=1$) to infinity ($n \to \infty$) is $13.6 \text{eV}$.

* **Photon vs. Particle Collision**:

* **Photon Absorption**: Photon energy must **exactly equal** the energy difference (unless ionizing), or it passes through.

* **Electron Collision**: External electron energy just needs to be **greater** than the difference; excess becomes kinetic energy.

(3) Typical Example

**Problem**: Hydrogen atom transitions from $n=3$ to $n=2$.

**Solution**:

1. Find Energy Difference: $\Delta E = E_3 - E_2 = \frac{-13.6}{9} - \frac{-13.6}{4} \approx -1.51 - (-3.40) = 1.89 \text{eV}$.

2. Find Wavelength: $\lambda = \frac{hc}{\Delta E} \approx \frac{1240 \text{eV}\cdot\text{nm}}{1.89 \text{eV}} \approx 656 \text{nm}$.

3. Conclusion: This is red light, part of the Balmer series.