Entropy

Entropy, S, measure of the disorderliness with which energy is stored in a system: the greater the disorder, the greater the entropy. At the absolute zero of temperature (when there is no thermal motion) a perfect crystal (in which there is no spatial disorder) has zero entropy. Raising the temperature stimulates thermal motion and the entropy increases. Entropy also increases when a solid melts and when a liquid vaporizes, because in each case the spatial disorder of the system increases.

There are two formal definitions of entropy. The thermodynamic definition, which is due to Sadi Carnot and Rudolf Clausius, and arose from the 19th-century interest in improving the efficiency of steam engines, equates a change in entropy (S) to the energy (Q) transferred to a system divided by the temperature (T) at which the transfer takes place (dS = dQ_rev/T). (The subscript “rev” indicates that the transfer must take place reversibly, which in practice means that the temperature of the “heater” should be only infinitesimally greater than that of the system.) Heat transferred at high temperature results in a smaller change in entropy than the same quantity of heat transferred at a lower temperature. An analogy is that a sneeze in a busy street (where there is already a lot of disorder) creates less disorder than the same sneeze in a quiet library. The statistical definition, which is due to Ludwig Boltzmann and was formulated when scientists tried to explain bulk properties in terms of atoms and molecules, expresses the entropy in terms of the number of ways (W) in which the system can be prepared yet have the same total energy (S = k in W, where k is Boltzmann’s constant). The more ways there are of arranging molecules to achieve the same total energy, then the greater is the statistical entropy. The two definitions turn out to be equivalent and are used interchangeably. The thermodynamic definition allows us to use thermodynamic measurements, particularly measurements of heat capacity, to determine the entropies of substances. The statistical definition allows us to calculate entropies from the spectroscopic determination of energy levels.

The second law of thermodynamics makes use of the concept of entropy. It states that in an isolated system any change is accompanied by an increase in entropy. That is, the only changes that can occur naturally (the technical term is “spontaneously”) are accompanied by an increase in overall disorder. In other words, the natural direction of change in the universe is towards greater disorder overall. However, that does not mean that order cannot arise spontaneously, as one part of an isolated system may decrease in disorder, provided that another part undergoes a greater increase in disorder.

Some examples will help to clarify these concepts. One common natural change is the cooling of a hot object to the temperature of its surroundings. The opposite change, the sudden spontaneous increase in temperature of a body above the temperature of its surroundings, has never been observed. When the hot body loses a certain amount of energy, its entropy decreases by a small amount. When the cooler surroundings accept that energy, their entropy increases by a larger amount (because their temperature is lower). The overall effect is an increase in entropy, which is consistent with cooling being a natural, spontaneous process. When the temperature of the object is the same as that of its surroundings, the increase in entropy of the surroundings is equal to the decrease in entropy of the object, so there is no net change in total entropy and the object and its surroundings undergo no further net change. We say that they are then at thermal equilibrium.

Another common change is the natural expansion of a gas to fill the volume available to it. Suppose the expansion is isothermal (occurring without change of temperature). From a statistical viewpoint, expansion corresponds to an increase in the number of locations that the molecules can be found, so W increases and therefore the entropy increases too. From a thermodynamic viewpoint, the isothermal expansion of a gas is accompanied by an influx of energy as heat to make up for the energy lost as the gas does work while it expands (such as pushing back a piston). That influx raises the entropy of the gas, and we conclude that a gas has higher entropy after it has expanded to fill the available volume than when it occupied a smaller volume.

All spontaneous (natural) processes can be analysed similarly, and it is invariably the case that the total entropy of the system and its surroundings increases if the change is spontaneous. Thus, entropy provides an “arrow of time” which points towards the future: events take place in the direction of the arrow; they do not take place spontaneously in the opposite direction.

The concept of entropy imposes certain restraints on the efficiencies of heat engines, such as steam engines and internal combustion engines. A heat engine consists of a hot source, a cold sink, and a device for converting the flow of energy as heat into work. If the engine is to operate spontaneously (as it must—an engine that must be driven is useless), then overall there must be an increase in entropy as it runs. The removal of heat from the hot source results in a lowering of entropy. When the same amount of heat is deposited in the cold sink, there is an increase in entropy that is greater than the reduction in entropy of the hot source, so that flow of heat is natural. However, we can still achieve an overall increase in entropy if we do not deposit all the heat into the cold sink. Some heat can be converted into work, and the remainder deposited into the cold sink can still, because of the latter’s lower temperature, achieve an increase in entropy overall. Thus, we see that some heat must be discarded into the cold sink for the engine to work, so no engine can be 100 per cent efficient. However, less heat need be discarded to achieve an overall increase in entropy when the temperature of the cold sink is very low. It turns out that the maximum efficiency of a heat engine depends on the temperatures of the hot and cold reservoirs as:

regardless of the mode of construction or operation of the engine. Thus, the greatest efficiency is found when the cold reservoir is very cold and the hot source very hot.

The arrow of time points to the future, but what happens when the entropy of the universe has reached a maximum value? At that speculative stage, no natural processes can occur and the universe will have achieved its “heat death”. The prospect, however, remains controversial. If, as now seems likely, the universe will expand forever, there appears to be no reason why the entropy should not continue to increase, even as matter decays into radiation, so an arrow of time and a distinction between past and future will persist forever.