Lecture

Introduction: what does molecular physics study?

A glass of water contains about $10^{25}$ molecules. Following the motion of each of them with Newton's laws is completely impossible, yet the temperature, pressure and density of the water can be measured and predicted. Molecular physics is the branch of physics that studies the macroscopic properties of matter (pressure, temperature, heat capacity, viscosity, surface tension, etc.) on the basis of the motion and interaction of its constituent particles. The central question: how do precise laws emerge from the chaotic motion of an enormous number of particles?

History of atomistic ideas

The idea that matter consists of indivisible particles — atoms — was put forward in the 5th century BC by Leucippus and Democritus and developed by Epicurus and Lucretius. For a long time it remained a philosophical guess. The scientific stage began in the 17th–19th centuries:

After Perrin's experiments no doubt remained about the real existence of atoms and molecules. In the 20th century it turned out that the atom itself has a complex structure (Rutherford, Bohr) and that its properties can be explained only by quantum mechanics.

Main postulates of molecular-kinetic theory

1) Every substance consists of particles — atoms and molecules. 2) These particles are in continuous chaotic (thermal) motion. 3) The particles interact with each other: they attract and repel.

The first postulate is confirmed by direct "imaging" of atoms in electron and tunnelling microscopes and by the laws of chemistry. The second is confirmed by diffusion and Brownian motion. The third is confirmed by the strength of solids and liquids, their resistance to compression, wetting and surface tension.

Amount of substance and the Avogadro constant

The number of particles is enormous, so it is convenient to count them in moles. The amount of substance is

where $N$ is the number of molecules, $N_A = 6.022\cdot10^{23}\ \text{mol}^{-1}$ is the Avogadro constant (the number of particles in 1 mole), $m$ is the mass of the substance and $M$ is the molar mass ($\text{kg/mol}$). The mass of one molecule and the number of molecules in a sample are

At standard conditions ($T_0 = 273.15\ \text{K}$, $p_0 = 101\,325\ \text{Pa}$) one mole of any gas occupies $V_m = 22.4\ \text{L}$ (Avogadro's law), and the number of molecules in 1 m³ of gas equals the Loschmidt number: $n_L = 2.69\cdot10^{25}\ \text{m}^{-3}$.

Size of molecules and their interaction

In a liquid or solid the molecules are densely packed, so the volume per molecule is $V_0 = M/(\rho N_A)$ and the molecular size is $d \approx \sqrt[3]{M/(\rho N_A)}$. For water $d \approx 3\cdot10^{-10}\ \text{m}$. In a gas at standard conditions the mean distance between molecules $\langle l\rangle \approx n^{-1/3}$ is $3.3\cdot10^{-9}\ \text{m}$ — about 10 times the molecular size.

Intermolecular interaction is electrical in nature. The potential energy as a function of distance, $W_p(r)$, has the shape of a "potential well": for $r > r_0$ attraction dominates, for $r < r_0$ strong repulsion, where $r_0$ is the equilibrium distance (of order $10^{-10}$ m). The Lennard-Jones potential is often used: $W_p = 4\varepsilon\left[(\sigma/r)^{12} - (\sigma/r)^{6}\right]$. The state of matter is determined by the ratio of the mean kinetic energy of thermal motion $\sim kT$ to the well depth $\varepsilon$: $kT \ll \varepsilon$ — solid, $kT \sim \varepsilon$ — liquid, $kT \gg \varepsilon$ — gas.

Brownian motion

A micron-sized particle in a liquid is struck by molecules from all sides; the numbers of impacts do not balance exactly, and the particle moves chaotically. Einstein showed that the mean square displacement of the particle along one axis is proportional to time:

where $D$ is the diffusion coefficient, $k = 1.38\cdot10^{-23}\ \text{J/K}$ is the Boltzmann constant, $\eta$ is the viscosity of the liquid and $r$ is the particle radius. By measuring $\langle x^2\rangle$, Perrin determined $k$ and $N_A = R/k$. In the interactive model, raise the temperature: how does the track of the large particle change? In the "Timeline" model, follow the sequence of the discoveries listed above.

Classical and quantum models of matter

In the classical model atoms and molecules are treated as point masses or hard spheres obeying Newton's laws; their energy takes continuous values. This model explains gas pressure, heat capacity and transport phenomena well. In the quantum model an atom consists of a nucleus and electrons, the electron energy takes discrete values, and particles also have wave properties (the de Broglie wave). The classical approximation is valid when the thermal de Broglie wavelength is much smaller than the distance between particles:

For example, at 300 K a nitrogen molecule has $\lambda_T \approx 1.9\cdot10^{-11}\ \text{m}$, i.e. 180 times smaller than the spacing — the gas is classical. For electrons in a metal $\lambda_T \approx 4.3\ \text{nm}$ while the interatomic distance is $\sim 0.25$ nm — the electron gas obeys quantum Fermi–Dirac statistics. Quantum effects explain the decrease of heat capacity at low temperatures, the superfluidity of liquid helium and other phenomena.

Example

How many molecules are there in 1 g of water? Estimate the mass and size of one molecule ($M = 0.018\ \text{kg/mol}$, $\rho = 1000\ \text{kg/m}^3$).

Solution. $N = \dfrac{m}{M}N_A = \dfrac{10^{-3}}{0.018}\cdot 6.022\cdot10^{23} \approx 3.35\cdot10^{22}$. $m_0 = M/N_A = 0.018/6.022\cdot10^{23} \approx 2.99\cdot10^{-26}\ \text{kg}$. $d \approx \sqrt[3]{M/(\rho N_A)} = \sqrt[3]{2.99\cdot10^{-29}\ \text{m}^3} \approx 3.1\cdot10^{-10}\ \text{m}$.