Why Sugar Lowers the Freezing Point

Advanced Chapter 2 handled the power of sugars to lower the freezing point as a relative coefficient, PAC. Dextrose stands at about 1.7 times sucrose, fructose at about 1.9 — why those values? The answer lies in the colligative properties (proprietà colligative) of solutions. This piece takes that convenient coefficient apart, down to its grounds.

Freezing point depression is settled by number

Pure water freezes at 0 °C. Dissolve sugar in it, and the temperature at which freezing begins comes down below zero. This is freezing point depression.

Seen at the scale of molecules, the reason is plain. As ice grows, water molecules climb onto the lattice one at a time. A dissolved sugar molecule has no place on that lattice. Having no place, it takes up the spot where the next water molecule would climb. So the water freezes less readily, and the temperature at which freezing begins comes down. Salt scattered on a winter road works the same way.

And here is the decisive fact. The size of the depression is settled by the number of dissolved particles, not by their kind. That is the heart of the colligative properties. With the same number of particles, sucrose and dextrose lower the freezing point equally. Taste plays no part in it, nor does the shape of the molecule. What counts is the head count of strangers mixed in among the water molecules.

There is a reason for saying particles and not molecules. With sugar the two words agree: a dissolved molecule behaves as one particle. Salt is different. Put into water, salt stops being a molecule and breaks into two ions, sodium and chlorine. What has to be counted is the head count after the break, so one unit of salt does the work of two. About twice what molecular weight alone would predict — a little under twice, in truth, because the parted ions still pull on one another.

Drop that distinction and the account is half told. Some of the literature explains the greater power of salt by its small molecular weight alone, and that is not enough. Salt is strong because it is small and because it breaks in two.

In a thin solution, the depression ΔT is written like this.

The mole fraction x is the number of solute molecules divided by the number of molecules in the whole system, water and solute together. The equation says one thing only — what counts is the head count.

Why PAC is settled by molecular weight

From here the identity of PAC follows. The head count of molecules — the number of moles — comes apart like this.

Dissolve the same weight, and a sugar of smaller molecular weight gives more heads and a larger depression. That is why PAC runs roughly inverse to molecular weight. The relation is not confined to sugars. One gram of salt (formula weight 58.5) lowers the freezing point far more than one gram of sucrose (molecular weight 342). The reason is the same — with the extra from breaking in two, as seen above. Take sucrose as the base: anhydrous dextrose, at 180, carries 342 ÷ 180 ≒ 1.9 times the moles, and its PAC comes near 190.

The table in Advanced Chapter 2 gave dextrose a PAC of 173 because it took the monohydrate (molecular weight 198), the form common in gelato. 342 ÷ 198 ≒ 1.73 — the water of crystallization raises the effective molecular weight, and PAC falls from 190 to 173. The number in that table rests on this one line of division.

Why gelato has no melting point

So far this has been the ideal case of a thin solution. A gelato mix is neither thin nor ideal. And it is that departure which gives gelato its physical character.

For a pure substance the freezing point is one point. Cool a sugar solution, though, and at some temperature water alone comes out as ice. Sugar has no place in the ice lattice, so it is left behind in the solution that has not frozen — the unfrozen phase. The sugar there grows more concentrated, and by the colligative properties its freezing point falls further. The next ice needs a lower temperature still. Repeat that, and freezing happens across a range of temperature, never at a point.

This process — ice drawing water out so that what remains grows stronger — is freeze concentration (crioconcentrazione). It is why gelato has no single melting point, and why it stiffens a little more with every degree taken away. The unfrozen phase of Advanced Chapter 1 is nothing but where freeze concentration arrives.

All of it can be drawn as the ice curve. Put temperature along one axis and the amount of ice up the other. Past the freezing point the ice rises sharply, and from there it goes on rising gently. Take a measured example for ice cream. At around −5 °C, leaving the freezer, ice makes up a little over three tenths of the weight of the whole mix. At a storage temperature of −18 °C it passes five tenths.

The unit is worth pausing on. What that axis carries is the weight of ice against the whole mix, not the share of the water that has frozen. Count it the other way: what share of the water is ice. At the serving temperature of about −14 °C, the figure comes to some eight tenths. That is the count used in Advanced Chapters 1 and 7. The two counts give quite different numbers, and they point at one and the same curve. Before any talk of how much is frozen, the denominator has to be settled.

It does not set all at once at a single melting point. It freezes slowly across a range. The reason gelato can be cold and still soft is gathered into this one curve.

The phase diagram settles how hard it is

Seen this way, the working meaning of PAC becomes clear. At a given serving temperature, what share of the water is frozen — that is how hard the gelato is. Where the liquidus is steep, less water freezes at the same temperature, more unfrozen phase remains, and the gelato is soft. A sugar of high PAC pushes the liquidus down and lowers the share of water frozen at the serving temperature. That is the physical substance of "raise the PAC and it softens".

And this is not a matter of concepts. Given the phase diagram, the amount of ice can be calculated. Take a 30 percent sucrose solution, cooled to −10 °C. Read the liquidus at that temperature and the unfrozen phase stands at 57 percent sucrose. A kilogram of the solution holds 300 g of sucrose. Concentrated to 57 percent, the solution that remains is only 300 ÷ 0.57 ≒ 530 g. The difference — 470 g, that is 47 percent of the whole, has turned to ice. This one line of division is how the amount of ice is drawn out of a phase diagram.

Repeat the calculation at each temperature and the ice curve appears. And that is what makes it a tool for the design of a formulation. It asks whether the amount of ice at the serving temperature comes out as intended. A phase diagram is not a picture to be looked at. It is an instrument for calculating hardness.

Advanced Chapter 5 added total PAC as Σ (amount of each sugar × its PAC coefficient), in a straight line. That is this phase diagram, made linear for practice. Strictly, once the sugar concentration passes about one tenth, the depression begins to depart visibly from the simple proportion above. Solutes act on one another, and the assumption of an ideal solution gives way. Think of the sugar concentration of a real mix, and every formulation on the floor sits in that region of departure. The PAC sum still works only because, within the working range of temperature, the departed curve too can be taken as near enough straight. PAC is a practical tangent drawn to the curve of the phase diagram. Move away from the point of contact and it departs, as tangents do.

Lower still — the glass transition

Freeze concentration has an end. The road to it, for sugar, is a little unusual.

On a salt–water diagram, the liquidus runs down to a eutectic point. There the salt itself comes out as crystal, and the story ends. Sucrose is slow to crystallize. The theoretical eutectic — about 63 percent, −13.7 °C — arrives, and the sugar does not turn to crystal. It goes on down the liquidus, supersaturated. Dissolved past what will dissolve — this state of strain has the same shape as the supersaturated lactose of Approfondimento 6. Lactose, though, does crystallize, where sucrose holds out. That holding out is what keeps gelato from turning to sand.

The solution that has passed the eutectic grows stronger still and thicker still, until at last the molecules can no longer slip past one another. This is setting into the glassy state. Unlike a crystal, the molecules do not line up on a lattice. They stop as if frozen in the disordered arrangement of a liquid — the same kind of solid as window glass.

The temperature at which this transition happens is the glass transition temperature. In ice cream it depends on the formulation, broadly −30 to −40 °C, and in a typical one it shows around −30 °C. It is a property settled by the composition of the unfrozen phase, and the amount of air does not move it. Below this temperature the unfrozen phase is too thick to flow, and the solute molecules lose their motion. Recrystallization, on any practical scale of time, effectively stops. Gelato held below the glass transition temperature does not, in principle, deteriorate.

One difference of character is worth holding on to. The glass transition is not a true phase transition, as freezing is. It is not a matter of equilibrium settled by thermodynamics but a kinetic one — whether the molecules can move. So strictly, a diagram with the glass transition line drawn into it is no longer a phase diagram. It should be called a state diagram. In practice the point of maximum freeze concentration is not easily reached: before it arrives, the motion of the molecules has grown too slow.

Not below it — near it is enough

So should gelato be stored at −40 °C? In practice it cannot be. But there is a mercy here.

Without going below the glass transition temperature, coming near it is enough to tell. Above the transition, the speed of change is set by the gap between the storage temperature and the glass transition temperature. The smaller the gap, the slower the change. Not a cliff, but a slope. So the cold stores of industry are held at −25 °C or below. Not in order to get under the glass transition, but because coming near enough will do. When Advanced Chapter 7 asks for a low storage temperature and for stability, it is asking for a walk down this slope.

One measure, pulling both ways

Then why not raise the glass transition temperature itself? If nearness is the aim, the target can be drawn nearer instead.

The move exists. Use sugars of large molecular weight — the powdered starch syrups and the corn syrups — and the glass transition temperature rises. Against storage, the gelato grows stronger.

But here the law this piece has followed all along bares its teeth. The depression ran inverse to molecular weight. A sugar of large molecular weight is, by that very size, weak at lowering the freezing point. Add high-molecular-weight sugars to raise the glass transition, and by the same stroke the depression thins and the gelato turns hard. On top of that, the flavor is not always welcome.

One measure — molecular weight — pulls two demands in opposite directions. Large sugars for stability, small sugars for softness. Take one and the other is lost. When Advanced Chapter 2 set POD and PAC as two coordinates for telling sugars apart, it was to stand on this tug of war. The design of sugars comes down, in the end, to deciding where along that pull to drive the stake.

The deep reason why hardening and very low storage guard the structure lies here. The detail is left to Approfondimento 2 and Approfondimento 3.