Salt on an Icy Road Lowers Its Temperature

In a previous post of mine, I illustrated a brief history of temperature scales in which I recalled that the reason for which 0 °F corresponds to －18 °C is that the latter was the lowest temperature that Daniel Gabriel Fahrenheit could achieve in his laboratory.

Such a temperature can be easily reached mixing water, ice and ammonium chloride. This was, in fact, a very common method to produce a frigorific mixture before the invention of the freezer. The melting of the salts in water, in fact, is a chemical reaction that requires heat to proceed. Indeed, as every Italian knows, to prepare a good pasta dish you have to throw it in plenty of boiling salted water. Salt is usually put in the water when it has already reached the boiling point, otherwise it would take longer to reach this condition. Putting cold salted water on the fire would, in fact, lower the temperature of the water even further, because the melting of the salt subtracts heat to the water, and this would require more heat to bring it to boiling temperature.

This observation is apparently in contrast with two simple experiments. The first consists of throwing a relatively large quantity of salt into boiling water in relatively small pot. You can easily observe that the boiling becomes more violent for a bit (this experiment must be done with care, because the reaction can even be almost explosive and can project hot water to large distances), as if more heat is injected into the water, rather than subtracted.

An even easier experiment consists in observing what happens during winter, when it snows. To prevent the formation of ice, salt is spread on the roads. Since the ice melts, in these conditions, one could believe that the melting of ice adds heat to the system such that the temperature of the ice grows, causing its melting. In fact, the ice become even colder, reaching a temperature close to －18 °C in certain cases.

This behaviour is usually explained as follows: mixing salt and water makes the freezing point of the mixture lower. In other words, if water freeze at 0 °C, a mixture of water and salts freezes at lower temperatures, hence, at the temperature of the melting ice, the mixture is liquid. The phenomenon can be described more formally using the so called thermodynamic potentials. Below, we briefly describe their definitions, then we discuss their role in melting ice.

Waring: as the article proceeds, it becomes more and more formal. Some math and physics skills are needed to proceed.

State Functions

A state function, or function of state, is a function of the state variables of a system. State variables, in turn, are those independent physical quantities that characterise the system under observation (see also my post on the importance of the description of the state in understanding quantum mechanics). State functions are usually introduced, as such, in thermodynamics, however they are, in fact, already defined in dynamics, when terms like Kinetic and Potential Energy are introduced.

The kinetic energy K is defined for a point-like particle of mass m as K=½mv²: it is a function of state because it depends solely on the state of the particle that, in turn, comprises both its velocity v and its position x. The same particle on Earth is said to have potential energy V=mgh, defined as the opposite of the work done by gravity to bring the particle from height h to height 0. Such a work depends only on the initial and final states (initial and final heights); not on the specific path along which the particle moves in the gravitational field. We often define other potential energies, such as the elastic potential energy and the electrostatic potential energy, in the same way, i.e., as the opposite of the work done by the corresponding forces to bring the particle from an initial state to a final one. Such works depend only on the initial and final states, and not on what happen to the particle between these points. The sum U=K+V is said to be the mechanical energy of a system (composed of at least a particle and something with which the particle interacts: other masses, springs, or electric charges) and is, in turn, a function of state. Any linear combination of state functions is, manifestly, a state function, in fact.

In thermodynamics, we deal with a very large number of particles and it is impossible to define U as above. Nevertheless, the constituents of a system must each have both kinetic and potential energy (the latter deriving from the interactions between them, that are null for an ideal gas, whose constituents do not interact, by definition). We then introduce the concept of internal energyU that is the same as above: the only difference being that we cannot compute it explicitly.

From the definition of work ΔW, one can see that, for an ideal gas at constant pressure p, ΔW=pΔV, ΔV being the variation of its volume V. Moreover, pV=NkT, k being the Boltzmann constant, N the number of its constituents and T=2E̅/3k, where E̅ represents the average kinetic energy of the gas constituents. As a matter of fact, the product pV has, thus, the same dimensions of the energy: mass times the square of a velocity. H=U+pV is thus another state function, called enthalpy.

Entropy S is defined such that its variation ΔS=ΔQ/T, where ΔQ is the heat exchanged during the transformation that leads to ΔS at temperature T. Since both T and S are functions of state, the product TS is a function of state and F=U－TS is a function of state, too. It is called the Helmholtz free energy.

The combination of enthalpy and Helmholtz free energy leads to the definition of the Gibbs function G=U+pV－TS=H－TS.

It is worth noting that, while ΔQ and ΔW are not, in general, state functions (they may depend on the particular transformation), their combination ΔU=ΔQ－ΔW is a state function.

Deicing a Road with Salts

When salts are spread on ice, part of them melt because water breaks electrostatic bonds between salts’ constituents. To break the bonds, some energy is required and the latter is subtracted to the water, whose temperature, as a result, decreases up to －18 °C. At the same time, salts and water pass from the ordered condition of solids into the less ordered condition of a solution of ions. This makes the entropy of the system to increase.

The process happens at constant pressure and does not involve a change in volume (in fact, there is a small change in V, however, we are going to neglect it, because it is small). The variation of G, then, has to be ascribed only to the variation of the internal energy of the system (water+salts) and to the product TS: ΔG = ΔU－TΔS－SΔT. When the temperature of the ice lowers, both T and U decrease, thus ΔU