Anomalous Expansion of Water

Some time in the small hours of a north-Indian January night, a water pipe bursts. Nothing strikes it, nothing heats it — the water inside freezes, and in freezing demands nine percent more room than the pipe can give. The same week, under the frozen crust of a Kashmiri lake, fish glide about in water that refuses, all winter, to drop below 4 °C. Both facts trace to one glorious misbehaviour: between 0 °C and 4 °C, liquid water contracts when you heat it. Your syllabus files this under a single line — "anomalous expansion of water" — and then quietly builds a dozen mark-winning questions on it. CBSE and ICSE ask for the 4 °C density maximum and the lake-freezing story as one- and three-markers; NEET sets assertion–reason items on "all liquids expand on heating"; JEE-Main wraps the same physics in iceberg-flotation arithmetic; IGCSE wants the molecular why. The numbers involved are few and unforgettable: density peaks at 999.97 ≈ 1000 kg/m³ at 4 °C; ice is 917 kg/m³, so 91.7 % of any iceberg hides below the waterline; freezing swells water by 9.05 %, which is what kills the pipe. This article hands you the full kit: the density–temperature curve, the molecular tug-of-war behind it — told the way Ludwig Boltzmann taught the world to think, counting molecules and weighing order against agitation — the buoyancy derivation behind the iceberg number, the lake that freezes from the top down, and the traps each board sets. Twenty minutes here turns a memorised factoid into physics you can compute with.

Key takeaways

Water is the great exception to "heating expands": from 0 °C to 4 °C it contracts; only above 4 °C does it expand like a normal liquid.
So density has a maximum at 4 °C — 999.97 ≈ 1000 kg/m³; water at any other temperature is lighter and floats above 4 °C water.
Ice is lighter still: 917 kg/m³, an 8.3 % density drop — ice floats with 91.7 % of its volume submerged (exactly ρ_ice/ρ_water = 0.917).
Freezing expands water by 9.05 % in volume (1000 cm³ of water becomes 1090.5 cm³ of ice) — the force that bursts pipes and cracks rocks.
Deep lakes freeze top-down: 4 °C water sinks to the bottom, colder water floats above it, and the ice lid insulates everything beneath.
The molecular why: open, ice-like hydrogen-bonded clusters collapse as cold water warms (shrinking it) while thermal agitation pushes molecules apart (swelling it); the two balance at 4 °C.
Key idea: between 0 °C and 4 °C the collapse of water's open molecular architecture beats thermal agitation, so heating shrinks the liquid — a density maximum at 4 °C that floats ice, bursts pipes and saves every fish in a frozen lake.

The core insight

Water contracts on heating from 0 °C to 4 °C and expands beyond it, so its density peaks at 4 °C (999.97 ≈ 1000 kg/m³) — and because ice (917 kg/m³) is lighter than water at any temperature, lakes freeze from the top down while sealed pipes burst from the inside.

Where this lives today

The anomaly is quietly load-bearing across ecology, engineering and geology. Every temperate lake runs on it twice a year: in spring and autumn the water column passes through 4 °C, densities equalise, and wind-driven overturn mixes oxygen down and nutrients up — with the winter 4 °C refuge in between. Plumbers fight it every winter: pipes are lagged, left dripping, or buried below the frost line because confined freezing water is unstoppable. The same 9 % jolt is a geological chisel — frost wedging splits boulders and road surfaces, carving scree slopes and potholes. At sea, the 91.7 % rule is written into maritime history: the iceberg that sank the Titanic was overwhelmingly invisible. And a caveat examiners enjoy: the open ocean does not behave like the lake — dissolved salt pushes the density maximum below the freezing point (see the FAQ) — so the tidy 4 °C-at-the-bottom story is strictly fresh-water physics.

The physical picture

Seal one kilogram of water in a flexible bag and warm it slowly from 0 °C. An ordinary liquid — mercury, alcohol, kerosene — would swell steadily from the first moment. Water does something stranger: the bag shrinks until the thermometer reads 4 °C, and only then begins the normal swelling that continues to boiling. Nothing enters or leaves the bag, so the mass never changes — and that fact turns every volume statement into a density statement in reverse.

\rho = \frac{m}{V} \qquad (1)

where: $\rho$ — density ( $\mathrm{kg/m^3}$ ); $m$ — the sample's mass (kg), fixed for a sealed sample; $V$ — its volume ( $\mathrm{m^3}$ ). With $m$ locked, $\rho$ and $V$ are see-saw partners: the volume minimum at 4 °C is the density maximum at 4 °C — "volume least at 4 °C" and "density greatest at 4 °C" are the same sentence. Put numbers on the see-saw and the anomaly turns out to be astonishingly delicate. Liquid water at 0 °C has density $999.84\ \mathrm{kg/m^3}$ ; warming lifts it to the famous maximum of $999.97\ \mathrm{kg/m^3}$ at 4 °C (strictly 3.98 °C, which every board rounds to 4); by 10 °C it is back down to $999.70\ \mathrm{kg/m^3}$ , and by 100 °C it is $958.4\ \mathrm{kg/m^3}$ . The whole anomalous climb from 0 to 4 °C is a contraction of just 0.013 % — the entire liquid curve from 0 to 10 °C lives inside a window less than $0.3\ \mathrm{kg/m^3}$ tall. Tiny — but sinking and floating are decided by comparison, not magnitude: a hair's-breadth density edge is enough to send 4 °C water to the bottom of every lake on Earth.

Figure

Density of water from 0 °C to 10 °C — the curve climbs to a maximum of 999.97 kg per m³ at 4 °C, then falls away like an ordinary liquid

One more feature hides at the left edge of the graph. At 0 °C the curve simply stops: freeze the sample and the density crashes discontinuously from 999.84 to 917 kg/m³ — not gentle liquid expansion but a phase change, the whole liquid snapping into the crystal architecture of ice. One molecular picture explains both oddities.

The molecular tug-of-war

Think like Boltzmann: temperature measures molecular agitation — the average kinetic energy of the molecules — and the volume a substance occupies is the running compromise between the arrangements its molecules prefer and the agitation that disrupts them. For most liquids the story is one-sided: warm the liquid, the molecules jostle harder, the spacing grows, the liquid expands. Water carries an extra combatant into the ring. A water molecule is bent and strongly polar, and it grips its neighbours through hydrogen bonds — directional links that hold molecules not merely close but at fixed angles. In ice, every molecule holds four neighbours at arm's length in a tetrahedral pattern; the crystal is like a cathedral — rigid, beautiful, mostly empty space. That open architecture is why ice, at 917 kg/m³, is less dense than the liquid it froze from: melting collapses part of the cathedral, and the rubble packs tighter than the building did. Just above 0 °C, liquid water still contains swarms of short-lived, ice-like hydrogen-bonded clusters — open architecture that survived melting. Warm the water and two effects compete. Collapse: agitation breaks up the open clusters, letting molecules crowd closer — volume down. Agitation: the same jostling pushes every molecule further from its neighbours, as in any liquid — volume up. Below 4 °C there is plenty of open structure left to demolish, so collapse wins and water contracts; above 4 °C the surviving clusters are too few to matter, and water expands like everyone else. At 4 °C the two effects balance — the density maximum. You met the bookkeeping for ordinary expansion in Thermal Expansion of Solids & Liquids; the same equation carries the anomaly if you let its coefficient change sign. For a small temperature step,

\Delta V = \gamma\,V_0\,\Delta T \qquad (2)

where: $\gamma$ — the coefficient of volume expansion ( $\mathrm{K^{-1}}$ ); $V_0$ — the volume before the step; $\Delta T$ — the temperature change (the same number in K and °C). For water $\gamma$ is not constant: it is effectively negative between 0 °C and 4 °C, zero at 4 °C, and positive above. Read $\gamma$ as the scoreboard of the collapse-versus-agitation match; 4 °C is where the match is a draw. Honesty note: Boltzmann never dissected water — the hydrogen bond was articulated only in 1920, and the first structural theory of liquid water (Bernal and Fowler's open tetrahedral network) arrived in 1933. But the modern explanation speaks the kinetic-molecular language he championed: count the molecules, weigh order against agitation.

Ice floats — and how much of it hides

Freezing is the cluster story taken to completion: at 0 °C the whole liquid locks into the open tetrahedral lattice, the volume jumps 9.05 %, and the density falls off a cliff to 917 kg/m³. A solid less dense than its own liquid is a genuine rarity — and it floats. How much floats above the waterline is force balance, not memory: a floating body's weight is matched by Archimedes' buoyant force, the weight of liquid displaced by the submerged part. For an iceberg of total volume $V$ with submerged volume $V_{\text{sub}}$ ,

\rho_{\text{ice}}\,V\,g = \rho_{\text{water}}\,V_{\text{sub}}\,g \qquad (3)

where: $\rho_{\text{ice}} = 917\ \mathrm{kg/m^3}$ ; $\rho_{\text{water}} = 1000\ \mathrm{kg/m^3}$ (the boards' round value; 999.97 at 4 °C); $g$ — the gravitational field strength, which appears on both sides and cancels, so no board's choice of $g$ ever enters. Left side: weight. Right side: buoyant force at equilibrium.

Derivation AWhy 91.7 percent of an iceberg hides below the waterline

We want the exact fraction of a floating iceberg that lies below the surface, starting from nothing but the two densities. The strategy: write the iceberg's weight, write Archimedes' buoyant force on the submerged part, and demand equilibrium — then watch everything except the density ratio cancel. The payoff is a result independent of the berg's size, shape and $g$ : pure ratio physics, which is why examiners love it.

Step 1. Weight. An iceberg of total volume $V$ and density $\rho_{\text{ice}}$ has mass $m = \rho_{\text{ice}} V$ , so its weight is $W = \rho_{\text{ice}} V g$ , downward.

Step 2. Buoyancy. Only the submerged volume $V_{\text{sub}}$ displaces water; the upthrust equals the weight of that displaced water: $F_B = \rho_{\text{water}} V_{\text{sub}}\, g$ , upward.

Step 3. Equilibrium. Floating means $F_B = W$ : $\rho_{\text{water}} V_{\text{sub}}\, g = \rho_{\text{ice}} V g$ . Cancel $g$ and divide by $\rho_{\text{water}} V$ :

f = \frac{V_{\text{sub}}}{V} = \frac{\rho_{\text{ice}}}{\rho_{\text{water}}} \qquad (4)

Step 4. Numbers. $f = 917/1000 = 0.917$ — 91.7 % of the iceberg is under water; only $1 - 0.917 = 0.083$ , a mere 8.3 %, shows above the surface.

Step 5. Sanity. If $\rho_{\text{ice}} = \rho_{\text{water}}$ , then $f \to 1$ : the berg floats fully submerged, flush with the surface; any denser and it sinks — the fate of almost every other solid in its own liquid. Because $f$ is a pure ratio, a berg the size of a sugar cube and one the size of a city hide the same 91.7 %. Verified: 917/1000 = 0.917 and the 8.3 percent visible fraction are recomputed in verify_hs2-anomalous-expansion-water.py.

Go deeper · Derivation B · 1 minThe nine-percent jolt — freezing water in a sealed pipe

Derivation BThe nine-percent jolt — freezing water in a sealed pipe

A sealed pipe is brim-full with exactly 1 kg of water and no spare room. We derive how much volume the ice will demand on freezing, and extract the general expansion fraction every board expects. The strategy: mass is untouched by a phase change, so the new volume is fixed entirely by the new density through $V = m/\rho$ — and along the way we separate the 9.05 % volume rise from the 8.3 % density drop, two cousins students constantly swap.

Step 1. Before freezing. With $\rho_{\text{water}} = 1000\ \mathrm{kg/m^3} = 1.000\ \mathrm{g/cm^3}$ : $V_w = m/\rho_{\text{water}} = 1000\ \mathrm{g} / 1.000\ \mathrm{g/cm^3} = 1000\ \mathrm{cm^3}$ — one litre.

Step 2. After freezing. The same 1000 g is now ice at $\rho_{\text{ice}} = 0.917\ \mathrm{g/cm^3}$ , so $V_{\text{ice}} = 1000/0.917 = 1090.5\ \mathrm{cm^3}$ .

Step 3. The expansion. $\Delta V = 1090.5 - 1000 = 90.5\ \mathrm{cm^3}$ , a fractional rise of $90.5/1000 = 9.05\,\%$ . In symbols, $\Delta V / V_w = \rho_{\text{water}}/\rho_{\text{ice}} - 1 = 1000/917 - 1 = 0.0905$ — "freezing adds about 9 %."

Step 4. The cousin it is not. The density drop is $1 - \rho_{\text{ice}}/\rho_{\text{water}} = 1 - 0.917 = 8.3\,\%$ . Volume up 9.05 %, density down 8.3 % — reciprocal effects, different numbers. Quote the one that was asked.

Step 5. Why pipes lose. Denied its 90.5 extra cubic centimetres, the freezing water builds enormous pressure against the walls — beyond what plumbing, engine blocks or the pores of rock can withstand. The pipe bursts because the phase change is non-negotiable; what frustrated expansion does to clamped material is the theme of Thermal Stress & Bimetallic Strips. Verified: 1000/0.917 = 1090.5 cm³, the 9.05 percent expansion and the 8.3 percent density drop are recomputed in verify_hs2-anomalous-expansion-water.py.

How a lake freezes from the top down

Run a whole lake through a falling autumn and the anomaly choreographs three acts. Act one — convection. While the lake is warmer than 4 °C, surface water cools, becomes denser than the water below, and sinks; warmer water rises to be cooled in turn. The conveyor runs until the whole lake sits near 4 °C.

Act two — the shutdown. The surface drops below 4 °C — and the anomaly flips the script. Water at 3, 2, 1 °C is lighter than the 4 °C water beneath, so it floats where it is. Convection halts; the lake stratifies with the densest 4 °C water at the bottom. The depths can now lose heat only by conduction, and water conducts poorly — deep cooling becomes achingly slow.

Act three — the lid. The surface reaches 0 °C and freezes. The ice floats at 917 kg/m³, grows downward only slowly, and insulates everything beneath. The result is the picture every board asks for: ice at 0 °C on top, then 1, 2, 3 °C layers, and a stable 4 °C refuge at the bottom. If water behaved "normally," the coldest water would keep sinking, the lake would freeze from the bottom up, and by midwinter it would be solid — life included.

Figure

A frozen lake in cross-section — ice floats on top while the densest 4 °C water shelters life at the bottom

Limiting cases

Case	Result	Meaning
water at exactly 4 °C	densest possible: 999.97 ≈ 1000 kg/m³	sinks below water at any other temperature — the bottom layer of every deep winter lake
heating 0 °C → 4 °C	volume shrinks by about 0.013 %	γ effectively negative — the anomalous window
heating 4 °C → 100 °C	volume grows by about 4.3 %	normal liquid behaviour resumes above 4 °C
freezing at 0 °C	volume jumps 9.05 %; density falls 8.3 % to 917 kg/m³	ice floats; brim-full sealed pipes burst
ρ_ice → ρ_water (hypothetical)	submerged fraction f → 1	a solid exactly as dense as its liquid would float totally submerged
deep lake under ice	bottom water holds 4 °C all winter	the anomaly builds a survival refuge for aquatic life

Common trapEverything expands when heated — right?

Wrong twice, and boards test both failures. First: liquid water between 0 °C and 4 °C contracts on heating — NEET assertion–reason items are built on exactly the sentence "all liquids expand when heated." Second, the mirror error: asked where water is densest, students reflexively answer "0 °C, because coldest = densest." The densest state is 4 °C; water at 0 °C — and ice above all — floats on top of 4 °C water. Keep the two oddities filed separately, too: the gentle 0–4 °C contraction is the anomalous expansion proper (a liquid-state effect); the dramatic 9 % swelling at 0 °C is a phase change into ice's open crystal. Blur them and the answer reads as confusion — state each with its own temperature range and mechanism.

Worked examples

Board level — the pipe that bursts (ICSE and CBSE pattern)

A rigid pipe is completely filled with $500\ \mathrm{cm^3}$ of water and sealed at both ends. On a frosty night all the water freezes. Take $\rho_{\text{water}} = 1000\ \mathrm{kg/m^3} = 1.000\ \mathrm{g/cm^3}$ and $\rho_{\text{ice}} = 917\ \mathrm{kg/m^3} = 0.917\ \mathrm{g/cm^3}$ . Find (a) the volume the ice needs, (b) the extra volume demanded, and (c) explain why the pipe bursts.

Step 1. Fix the mass — the one thing freezing cannot change: $m = \rho_{\text{water}} V = 1.000 \times 500 = 500\ \mathrm{g}$ .

Step 2. Volume of the ice from $V = m/\rho$ : $V_{\text{ice}} = 500 / 0.917 = \boxed{545.3\ \mathrm{cm^3}}$ .

Step 3. Extra volume: $\Delta V = 545.3 - 500 = \boxed{45.3\ \mathrm{cm^3}}$ — a fractional rise of $\rho_{\text{water}}/\rho_{\text{ice}} - 1 = 9.05\,\%$ , the same for any starting volume.

Step 4. The explanation the mark scheme wants: the pipe is rigid and sealed, so the ice cannot claim its extra $45.3\ \mathrm{cm^3}$ ; the frustrated expansion builds pressure until the pipe fractures. Note what was not needed: pipe length, material, or $g$ — only densities and conservation of mass.

Competitive level — one iceberg, two waters (JEE-Main and NEET pattern)

An iceberg of density $917\ \mathrm{kg/m^3}$ floats first in a fresh-water lake ( $\rho = 1000\ \mathrm{kg/m^3}$ ) and later drifts into the sea ( $\rho = 1025\ \mathrm{kg/m^3}$ ). Find the fraction of its volume visible above the surface in each case, and the ratio of the two visible fractions.

Step 1. The tool: submerged fraction $f = \rho_{\text{body}}/\rho_{\text{liquid}}$ — equation (4); the visible fraction is $1 - f$ . Size, shape and $g$ all cancel.

Step 2. Fresh water: $f = 917/1000 = 0.917$ , so visible fraction $= 1 - 0.917 = 0.083 = \boxed{8.3\,\%}$ .

Step 3. Sea water: $f = 917/1025 = 0.8946$ , so visible fraction $= 1 - 0.8946 = 0.1054 \approx \boxed{10.5\,\%}$ .

Step 4. Ratio: $0.1054/0.083 = 1.27$ — the same iceberg shows about 27 % more of itself in the sea, because denser liquid supplies more upthrust per submerged cubic metre. Timing note: about 2 minutes. The mark-killer is autopilot — grabbing 1000 for the liquid when the question says sea water. Write $f = \rho_{\text{body}}/\rho_{\text{liquid}}$ once, check which liquid, and reuse it.

Exam traps

Writing "water always expands on heating." Between 0 °C and 4 °C it contracts. CBSE one-markers and NEET assertion–reason questions are engineered around this; the safe phrasing is "water contracts from 0 °C to 4 °C and expands above 4 °C."

Putting the density maximum at 0 °C. The maximum is at 4 °C (999.97 ≈ 1000 kg/m³) — and "volume of a given mass is minimum at 4 °C" must point at the same temperature.

Swapping the two nine-ish percents. Volume rises 9.05 % on freezing; density falls 8.3 %. Reciprocal effects ( $1000/917 - 1$ versus $1 - 917/1000$ ), not the same number.

Using the wrong liquid density in flotation. The submerged fraction is $\rho_{\text{body}}/\rho_{\text{liquid}}$ — the liquid it floats in. JEE-Main's favourite twist is sea water at 1025 kg/m³, shifting the visible fraction from 8.3 % to 10.5 %.

Claiming the lake bottom is at 0 °C. Under a frozen lake the bottom sits at 4 °C. ICSE three-markers want the full sequence: convection to 4 °C, stratification below 4 °C, ice lid, conduction-only cooling.

Explaining ice's floating by "cold things float." The examiner wants density through structure: ice's open tetrahedral lattice occupies more volume per kilogram, so $\rho_{\text{ice}} < \rho_{\text{water}}$ and buoyancy does the rest. Temperature alone decides nothing — water at 0 °C sinks in water at 1 °C, yet ice at 0 °C floats on both.

Origins · 1 minLudwig Boltzmann and how this was discovered

Ludwig Boltzmann

1844–1906 · Vienna

Ludwig Boltzmann spent his life defending one idea: that heat, temperature and the properties of matter are the statistics of restless molecules. Born in Vienna, he forged kinetic theory into statistical mechanics — his 1872 transport equation, the H-theorem, and the entropy formula $S = k \log W$ now carved on his gravestone — insisting atoms were real while Mach and Ostwald scoffed. Water's anomaly is a triumph of exactly his way of seeing: a liquid's volume is a running contest between energetically favoured molecular arrangements — here the open, ice-like hydrogen-bonded clusters — and the thermal agitation that grows with temperature. Boltzmann never analysed water himself; the hydrogen-bond details arrived decades after his death in 1906. But the modern explanation is written entirely in his language: count the molecules, weigh order against agitation.

History

1772

— Jean-André Deluc reports that water is densest a few degrees above its freezing point, not at 0 °C.

1805

— Thomas Charles Hope, in Edinburgh, publishes his tall-column experiment: cooling water at mid-height, the bottom thermometer settles near 4 °C — the densest water has sunk.

1860

— James Clerk Maxwell publishes the molecular speed distribution, making "temperature measures molecular agitation" quantitative.

1872

— Ludwig Boltzmann publishes his kinetic equation and H-theorem, the statistical machinery behind the molecular picture of matter.

1920

— Wendell Latimer and Worth Rodebush articulate the hydrogen bond, the directional grip behind water's open structure.

1933

— John Bernal and Ralph Fowler publish the first structural theory of liquid water: a partly collapsed, tetrahedral, ice-like hydrogen-bonded network.

Going further

Texts. NCERT Physics Class 11, Ch. 11 (Thermal Properties of Matter) — the CBSE/JEE statement with the volume–temperature graph · Selina Concise Physics (ICSE) — anomalous expansion and Hope's experiment · Cambridge IGCSE Physics 0625, Thermal Physics · H. C. Verma, Concepts of Physics Part 2, "Heat and Temperature."

Provenance. Standard results; a verifier should recompute: $917/1000 = 0.917$ (91.7 % submerged, 8.3 % visible); $1000/0.917 = 1090.5\ \mathrm{cm^3}$ and $1000/917 - 1 = 9.05\,\%$ against the 8.3 % density drop; $500/0.917 = 545.3\ \mathrm{cm^3}$ and $\Delta V = 45.3\ \mathrm{cm^3}$ (pipe example); $917/1025 = 0.8946$ , visible 10.5 %, ratio $0.1054/0.083 = 1.27$ (sea-water example); the 0–4 °C contraction $0.013\,\%$ and 4–100 °C expansion $4.3\,\%$ from the density table; the 50 g ice-cube bookkeeping ( $54.5\ \mathrm{cm^3}$ total, $50\ \mathrm{cm^3}$ submerged, $4.5\ \mathrm{cm^3}$ above).

Prerequisites. Thermal Expansion of Solids & Liquids (the meaning of γ, real vs apparent expansion). Leads to. Thermal Stress & Bimetallic Strips — the burst pipe made quantitative. Go deeper. The statistical-mechanics view of liquids and hydrogen-bond networks belongs to university physical chemistry and statistical mechanics — Boltzmann's subject, one tier up.

Why exactly 4 °C — what is special about that temperature?

Nothing magical — it is the crossover of two competing effects. Just above 0 °C, liquid water still carries many open, ice-like hydrogen-bonded clusters; warming collapses them, which shrinks the water, while the same warming increases agitation, which swells it. Near 0 °C collapse dominates because so much open structure remains to demolish; as temperature rises the clusters run out and agitation takes over. The books balance at 3.98 °C — rounded to 4 °C in board work. It is a crossover, not a magic number: change the substance, or dissolve salt in the water, and the balance point moves or vanishes.

Does ice itself behave anomalously when cooled below 0 °C?

No. Once water has frozen, the open crystal is fully built: cooling ice below 0 °C contracts it, warming it toward 0 °C expands it — ordinary solid behaviour with a positive expansion coefficient. The anomaly belongs to liquid water in the narrow 0–4 °C window, where partial ice-like structure is being destroyed by warming. The full journey of a cooling sample reads: normal contraction down to 4 °C, anomalous expansion from 4 to 0 °C, a sudden 9.05 % volume jump on freezing, then normal contraction as the ice cools further.

Do any other substances expand on freezing like water does?

A few — and they are prized for exactly this property. Gallium, bismuth, antimony and silicon all expand on solidifying, for the same underlying reason: their solids adopt open, directional structures that occupy more room than the disordered liquid. Printers exploited it for centuries — type metal contains antimony so the alloy expands into every corner of the mould, casting razor-sharp letters. But the double anomaly — a liquid that also contracts on heating just above its melting point — is much rarer; no common liquid shows it in an everyday temperature range the way water does, which is why "anomalous expansion" in board syllabi means water, specifically.

Does the sea also have a 4 °C layer at the bottom?

No — salt rewrites the rules. Dissolved salt lowers both the freezing point and the temperature of maximum density, and it drags the density maximum down faster. At ocean salinity (about 35 grams of salt per kilogram of sea water), the would-be density maximum lies below the freezing point of about −1.9 °C — so sea water never reaches it as a liquid. Ocean water therefore behaves "normally": it keeps getting denser right down to freezing, and polar sea water convects until ice forms. The tidy story — 4 °C at the bottom, 0 °C on top — is strictly fresh-water physics, which is why exam questions about lakes say lakes.

When a floating ice cube melts in a glass of water, does the water level rise?

No — the level stays exactly where it was. Take a 50 g cube: floating, it displaces exactly its own weight of water, $50\ \mathrm{g} / (1.000\ \mathrm{g/cm^3}) = 50\ \mathrm{cm^3}$ — the "hole" it occupies below the waterline — even though its full volume is $50/0.917 = 54.5\ \mathrm{cm^3}$ (about $4.5\ \mathrm{cm^3}$ rides above the surface). On melting, the 50 g becomes exactly $50\ \mathrm{cm^3}$ of liquid water, which precisely refills the hole. The 9 % expansion taken on freezing is returned on melting; floating sea ice likewise does not raise sea level when it melts (land ice does).

What did Hope actually do in his 1805 experiment?

Thomas Charles Hope, professor of chemistry at Edinburgh, wanted the temperature at which water is densest — with no modern density meter. His apparatus: a tall vessel of water with thermometers near the top and bottom, cooled around the middle by a freezing mixture. The logic is beautiful — whatever water is densest must end up at the bottom. As the mid-level water cooled, the bottom thermometer fell first and settled near 4 °C (the densest water had sunk), while the upper thermometer later dropped toward 0 °C as the lightest, coldest water gathered on top. The column itself was the density-sorting instrument: no balance, just gravity reading off the answer.