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<0 and ΔT<0. The latter term in ΔG, then, is positive. However, since both ice and salts melt, their entropy increase and ΔS>0. As a matter of fact, the increase in entropy is such that the absolute value of ΔU－TΔSis larger than SΔT and, in the end, ΔG<0.

When ΔG<0 during a process, the latter happens spontaneously, as we see below. The fact that the Gibbs function decreases can be interpreted as a tendency, for any physical system, to a state such that its Gibbs function reaches a minimum.

In summary, spreading salts on icy roads, lower the temperature of the system, however, it also favours a more disordered state, i.e., a state with higher entropy (here, the term “disordered” must be interpreted as usual in physics textbooks; there are good articles on Medium explaining that a common notion of order cannot represent correctly the notion of entropy). The increase in entropy is larger than the decrease in temperature and G=U－TS becomes smaller. Without the second law of thermodynamics, spreading salts on icy roads would have no effect, but to lower their temperature. The ice, in fact, melts, because a state with larger entropy is favoured (it is more probable) with respect to the initial state.

The Free Expansion of an Ideal Gas

As an example of a very simple process in which ΔG<0, consider the free expansion of an ideal gas (the so-called Joule’s experiment). A container is divided into two parts: one filled with an ideal gas and the other one empty. By opening a faucet on the wall that divides the two parts, the gas expands in the whole volume such that its volume increases, while its pressure decreases. Its temperature remains stable. Such an experiment allows us to establish a fundamental law: the internal energy of an ideal gas depends solely on its temperature.

As a result, U+pV remains constant during the process. The gas transformation happens spontaneously because G changes by －TS (i.e. ΔG<0). In fact, given that T remains constant, the entropy of the gas increases (the gas expands and its disorder grows), then ΔG=－TΔS, with both T and ΔS>0.

Why Systems Tend to a Minimum Value of G?

According to the second law of thermodynamics ΔS≥0 for isolated systems. The whole Universe is manifestly isolated, hence, ΔS(Univ)≥0. Given a system under observation, we can always write ΔS(Univ)=ΔS(sys)+ΔS(rest). Remembering that ΔS=ΔQ/T and considering that any heat absorbed (lost) by the system is lost (absorbed) by the Universe,

ΔS(rest)=ΔQ(rest)/T=－ΔQ(sys)/T

such that ΔS(Univ)=ΔS(sys)－ΔQ(sys)/T. By definition of enthalpy, and applying the first principle, ΔU=ΔQ－pΔV,

ΔH=ΔU+pΔV+VΔp=ΔQ－pΔV+pΔV+VΔp=ΔQ+VΔp.

From the latter we can write ΔQ as ΔH－VΔp and substitute in the expression of the entropy variation of the Universe; multiplying by －T we get

－TΔS(Univ)=ΔH(sys)－VΔp(sys)－TΔS(sys)

The left hand side is always negative (or, at most, zero), thus the right hand side is always negative, too. In our experiments, the pressure is constant (it’s the atmospheric pressure), hence ΔG=ΔH－TΔS<0. In other words, a decrease is the Gibbs function corresponds to an increase of the entropy of the Universe. Simply stated, ΔG<0 is yet another way of saying that the whole Universe tends to a more disordered state.