The atomic hypothesis
Richard Feynman called the discovery that matter is made of atoms the single most significant statement of science — the one fact he would choose to pass on if all other knowledge were lost. His phrasing is precise: all things are made of atoms, little particles that move around in perpetual motion, attracting each other when they are a little distance apart but repelling upon being squeezed together. Three ideas are packed into that sentence: particulate matter, ceaseless motion, and a force that is attractive at moderate range and repulsive at very short range.
Speculation that matter is not continuous is ancient — Kanada in India and Democritus in Greece both argued for indivisible constituents. But the scientific atomic theory belongs to John Dalton, about two centuries ago, who proposed atoms to explain the laws of definite and multiple proportions obeyed when elements combine into compounds. Since elements often exist as molecules, Dalton's atomic theory is equivalently the molecular theory of matter.
| Law / hypothesis | Statement | Role in the molecular picture |
|---|---|---|
| Law of definite proportions | A given compound has a fixed proportion by mass of its constituents | Implies a fixed small group of atoms per molecule |
| Law of multiple proportions | When two elements form more than one compound, for a fixed mass of one element the masses of the other are in ratios of small integers | Atoms combine in whole-number ratios |
| Dalton's atomic theory | The smallest constituent of an element is an atom; atoms of one element are identical but differ from other elements | Gives the laws above a mechanical basis |
| Gay-Lussac's law | When gases combine chemically, their volumes are in ratios of small integers | Hints that equal volumes hold equal counts |
| Avogadro's hypothesis | Equal volumes of all gases at the same temperature and pressure contain the same number of molecules | Combined with Dalton's theory, it explains Gay-Lussac's law |
Atomic theory is a beginning, not an end. We now know atoms are not indivisible: they contain a nucleus of protons and neutrons surrounded by electrons, and the nucleons are themselves built of quarks. For the kinetic theory of gases, however, this internal structure is irrelevant — the molecule is treated as the basic moving unit.
Evidence that matter is molecular
Two everyday observations give direct, visible support to the otherwise invisible molecular picture: Brownian motion and gas diffusion.
Brownian motion
The Scottish botanist Robert Brown, observing pollen grains suspended in water under a microscope, saw them tumbling and tossing along a zig-zag random path. The motion was first attributed to the grains being alive — until particles of dead matter such as mica and stone were seen to jiggle identically. The conclusion: the visible particle is being battered by the unbalanced impacts of fluid molecules striking it from all sides. Because the impacts are random and never cancel exactly, the particle is pushed first one way, then another. Brownian motion is therefore direct evidence for molecules in constant motion. The displacement is larger for smaller particles, increases with temperature, and decreases with the viscosity of the medium.
Diffusion of gases
When a gas is released in a corner of a room it spreads through the whole volume, mixing with the air. This is diffusion: molecules of one gas wandering among molecules of another. If matter were continuous, there would be nothing to wander. That a gas leaking from a cylinder takes appreciable time to reach the far corner — despite molecular speeds of the order of the speed of sound — tells us the molecules are not flying straight but are deflected by countless collisions along the way. Diffusion both confirms the molecular picture and, through its rate, yields estimates of molecular size.
Molecular spacing across the three states
What separates a solid, a liquid and a gas is not the molecules themselves but how far apart they sit and how tightly the interatomic force binds them. The size of an atom is about one angstrom, $1~\text{\AA} = 10^{-10}~\text{m}$. The spacing tells the rest of the story.
| State | Typical spacing | Interatomic force | Mobility & behaviour |
|---|---|---|---|
| Solid | A few angstroms (~2 Å) | Strong — atoms tightly held | Atoms vibrate about fixed positions; fixed shape and volume |
| Liquid | About the same as a solid (~2 Å) | Strong, but not rigidly fixed | Atoms can move around, so the liquid flows; fixed volume, no fixed shape |
| Gas | Tens of angstroms (factor of ~10 larger) | Negligible except during collisions | Molecules travel long distances freely; disperse if not enclosed |
The key contrast is solid/liquid versus gas. In solids and liquids the closeness keeps the interatomic force important; the atoms attract at a few angstroms but repel when squeezed closer. In a gas the molecules are an order of magnitude further apart and travel a long distance — the mean free path, of order thousands of angstroms — before colliding. This freedom is exactly why gas behaviour is simpler to model than that of condensed phases.
Avogadro's hypothesis and the mole
Bring the molecular idea into the experimental gas relation $PV = KT$, where $K$ is constant for a given sample but grows with the amount of gas. Writing $K = N k_B$, where $N$ is the number of molecules, observation shows the constant $k_B$ — the Boltzmann constant, $k_B = 1.38\times10^{-23}~\text{J K}^{-1}$ — is the same for all gases. It follows that
$$\frac{PV}{NT} = k_B = \text{constant for all gases.}$$
If $P$, $V$ and $T$ are the same, then $N$ is the same for every gas. This is Avogadro's hypothesis: the number of molecules per unit volume is the same for all gases at a fixed temperature and pressure. The number of molecules in 22.4 litres of any gas at STP (273 K, 1 atm) is the Avogadro number,
$$N_A \approx 6.022\times10^{23}.$$
The amount of substance containing $N_A$ molecules — whose mass equals the molecular weight in grams — is one mole. The bookkeeping that follows is what NEET tests:
| Quantity | Relation | Meaning |
|---|---|---|
| Number of molecules | $N = \mu\, N_A$ | Molecules in $\mu$ moles |
| Number of moles | $\mu = \dfrac{M}{M_0} = \dfrac{N}{N_A}$ | $M$ = sample mass, $M_0$ = molar mass |
| Ideal gas equation | $PV = \mu R T = k_B N T$ | $R = N_A k_B = 8.314~\text{J mol}^{-1}\text{K}^{-1}$ |
| Number density | $n = \dfrac{N}{V}$, $P = n k_B T$ | Molecules per unit volume |
Avogadro's hypothesis and $PV = \mu RT$ are unpacked in the experimental gas relations — see gas laws for Boyle, Charles and Dalton's law of partial pressures.
Estimating molecular size and spacing
The molecular picture is quantitative: from bulk data we can estimate the size of a single molecule and the spacing between molecules in different phases. The standard route uses water, following NCERT's worked examples.
Estimate the volume of a single water molecule, given the density of liquid water is $1000~\text{kg m}^{-3}$ and the molar mass of water is 18 g.
Mass of one molecule. One mole of water has mass $0.018~\text{kg}$ and contains $N_A \approx 6\times10^{23}$ molecules, so the mass of one molecule is $\dfrac{0.018}{6\times10^{23}} = 3\times10^{-26}~\text{kg}$.
Volume of one molecule. In the liquid the molecules are closely packed, so the density of one molecule is roughly that of bulk water. Volume $= \dfrac{3\times10^{-26}~\text{kg}}{1000~\text{kg m}^{-3}} = 3\times10^{-29}~\text{m}^3$.
Radius. Treating the molecule as a sphere, $\tfrac{4}{3}\pi r^3 = 3\times10^{-29}~\text{m}^3$ gives $r \approx 2\times10^{-10}~\text{m} = 2~\text{\AA}$ — consistent with the angstrom scale of atoms.
Extending the same data to the vapour phase shows how spacing differs between states. A given mass of water as vapour at 100°C and 1 atm occupies roughly $1.67\times10^3$ times the volume it had as liquid (NCERT Example 12.1 gives the molecular-to-total volume fraction of the vapour as about $6\times10^{-4}$). Because volume scales as the cube of the linear spacing, a thousand-fold volume increase means the spacing grows by $\sqrt[3]{10^3} = 10$. Starting from a molecular radius of 2 Å, the average interatomic distance in the vapour comes out near 40 Å — about ten times the spacing in the liquid, exactly the factor quoted for gases.
Why interatomic forces vanish for gases
The interatomic force is short-range: a long-range attraction that turns into a steep repulsion when atoms are pushed too close, both decaying quickly with separation. Atoms attract when a few angstroms apart and repel when squeezed nearer. In solids and liquids, where spacing is only a few angstroms, this force is decisive — it binds the condensed phase together. In a gas the molecules sit tens of angstroms apart, far beyond the range where the force has any appreciable strength.
The consequence is the central simplification of kinetic theory. Because the average separation is a factor of ten or more larger than the molecular size, the interaction between molecules is negligible, and a gas molecule moves freely in a straight line according to Newton's first law — until it occasionally comes close to another molecule, feels the force during a brief collision, and changes velocity. Between collisions, there is effectively no force. This is precisely the condition under which a real gas approaches ideal behaviour: low pressure and high temperature, where molecules are far apart and interactions are weak.
Negligible forces plus free straight-line motion is the starting assumption for deriving gas pressure — continue to kinetic theory of an ideal gas.
Molecular nature of matter in one breath
- Matter is made of molecules in perpetual motion — Dalton's atomic theory, equivalently the molecular theory.
- Evidence: Brownian motion (random impacts of molecules on a visible particle) and gas diffusion.
- Avogadro's hypothesis: equal volumes of all gases at the same $T$ and $P$ hold equal numbers of molecules.
- Avogadro number $N_A \approx 6.022\times10^{23}$ molecules per mole; $N = \mu N_A$, $\mu = M/M_0$.
- Atomic size ~1 Å. Spacing: solids and liquids ~2 Å; gases ~10× larger, so volume ~$10^3$ larger.
- Interatomic forces are short-range — strong in condensed phases, negligible in gases except during collisions.