Anomalous expansion of water

Normal liquids vs. water. Most liquids get denser all the way down to their freezing point — cooling always means contracting. Water only behaves this way from higher temperatures down to 4°C. Below 4°C, right up to freezing at 0°C, it does the opposite: it expands as it cools. The temperature 4°C is therefore special — it is where water's volume is smallest and its density is largest for a given mass.

Volume and density move oppositely. For a fixed mass $m$ of water, volume $V$ and density $\rho$ are linked by

where: $\rho$ is density, $m$ is the (constant) mass of the water sample, and $V$ is its volume at a given temperature. Because $m$ never changes, whenever $V$ falls, $\rho$ must rise, and whenever $V$ rises, $\rho$ must fall. This is why the volume-vs-temperature graph and the density-vs-temperature graph for water between 0°C and 10°C are mirror images of each other: volume dips to a minimum at 4°C, while density rises to a maximum at 4°C.

Hope's experiment. This is the standard apparatus that demonstrates the effect directly. A tall vessel of water carries one thermometer near the top and one near the bottom, with a freezing mixture packed in a trough around its middle. The bottom thermometer settles first, at a steady 4°C, because that is the densest water and it sinks. The top thermometer keeps falling below 4°C and eventually reaches 0°C, where ice begins to form — because water below 4°C is less dense and stays near the surface instead of sinking.

Why it matters for ponds. As a pond's surface cools toward 4°C, that water becomes denser and sinks, and warmer water rises to take its place — ordinary convection, continuing until the whole pond is near 4°C. Once the surface cools past 4°C, it becomes lighter, so it stops sinking and stays on top, eventually freezing. The ice layer that forms insulates the still-liquid water below (held near 4°C, its densest and most stable state), letting aquatic life survive the winter instead of the pond freezing solid from the bottom up.

Why should water, alone among common liquids, expand while cooling? The reason lies in the shape of the hydrogen bonds between water molecules, not in anything a Class 9 numerical formula can capture — but the qualitative picture is worth having. Each water molecule can hydrogen-bond to up to four neighbours. Above 4°C, thermal jostling is still vigorous enough that these bonds constantly break and re-form, and the molecules pack together fairly efficiently, so cooling (which slows the jostling) lets them pack slightly tighter — normal contraction. Below 4°C, the molecules start locking into a more open, cage-like hydrogen-bonded arrangement that is a precursor to the hexagonal lattice of ice. This open cage takes up more space per molecule than the more randomly packed liquid does, so as cooling pushes more molecules into this arrangement, the sample's volume grows even though the temperature keeps falling. At 0°C the process completes: ice forms with a fully open hexagonal lattice, which is why solid ice is noticeably less dense (and therefore floats) than liquid water at 4°C. The single number 4°C is simply the temperature at which the two competing effects — ordinary thermal contraction, and the growth of the open ice-like cage structure — exactly balance, giving the smallest volume and hence the greatest density of all.

Water's density at its densest, 4°C, is $\rho_w = 1000\ \text{kg/m}^3$ (equivalently $1\ \text{g/cm}^3$). When it freezes into ice at 0°C, its density drops to $\rho_i = 917\ \text{kg/m}^3$ because of the open hexagonal lattice described above. Find the percentage increase in volume when 1 kg of water at 4°C freezes completely into ice.

Step 1 — Volume of the water before freezing. Using $\rho = m/V$, rearrange to $V = m/\rho$. For $m = 1\ \text{kg}$ of water at its densest, $V_w = \dfrac{1}{1000} = 0.001\ \text{m}^3 = 1000\ \text{cm}^3$.

Step 2 — Volume of the same mass as ice. The same 1 kg, now as ice, occupies $V_i = \dfrac{1}{917} = 0.00109\ \text{m}^3 \approx 1091\ \text{cm}^3$.

Step 3 — Percentage increase in volume. The volume grew by $V_i - V_w = 1091 - 1000 = 91\ \text{cm}^3$. As a percentage of the original volume, this is $\dfrac{91}{1000} \times 100 \approx 9.1\%$.

Answer: freezing 1 kg of water into ice increases its volume by about 9%, which is exactly why ice floats on water and why pipes can burst when the water inside them freezes.

• Assuming water contracts continuously as it cools all the way to 0°C, like most liquids — the reversal begins at 4°C, not at the freezing point.
• Swapping the two graphs: volume vs. temperature has a minimum at 4°C; density vs. temperature has a maximum at 4°C — never the other way round.
• Thinking the densest water sinks all the way to freeze at the bottom — the water at 4°C sinks and stays liquid there; it is the lighter, colder water below 4°C that stays on top and freezes.
• Treating Hope's experiment as a calorimetry setup — it demonstrates density behaviour only; no specific heat capacity or latent heat calculation belongs to it.

Ask the tutor: Want a walkthrough of how the volume and density graphs are actually plotted point by point between 0°C and 10°C, or a redo of the freezing-volume calculation with a different starting mass?