Order and Disorder

In previous sections, we mentioned that heat energy is just the random kinetic motion of molecules, whose mean kinetic energy is a measure of their temperature. So Clausius statement just says that when two bodies are in contact, it is unlikely that the molecules in the hotter body will move even faster while colliding with the slower moving molecules of the colder body. In fact we naturally expect that due to the molecular collisons, the molecules in the colder body will speed up and those in the hotter body slow down, implying a heat transfer from the hot to cold body. Clausius's statement sounds reasonable, but how does one understand the limitations placed by the Kelvin statement? Again, note that heat energy is a disordered form of energy compared to ordered forms such as gravitational potential energy or bulk kinetic motion. While it is easy to create disorder out of order, the reverse is very difficult: Because the number ($\sim 10^{23}$) of molecules in any macroscopic piece of matter is very large, it is statistically improbable for disordered molecules to spontaneously act in concert to create order. One says that heat energy is of lower grade than ordered forms of energy and the Second Law places limitations of the conversion of low grades of energy into higher grades. To understand the the idea of order and disorder, consider the following example: An isolated box contains an equal number of two types of molecules, say 'white' and 'black'. Suppose one had the extreme case where initially all the white molecules were on the left and all the black molecules on the right. Of course this is an unnatural situation and soon, due to collisions, the molecules will totally mix. We say that the initial situation corresponds to a state of maximum order since there are fewer ways for the molecules to arrange themselves, while the final situation corresponds to large disorder since then there are many more ways for the molecules to arrange themselves(each molecule can now be anywhere in the box). We also know that if the number of molecules is very large, it is extremely unlikely that the gases will revert to the totally separated state at some future time. Thus the increase in disorder in the system is for all practical purposes irreversible. Note that the irreversibility is not due to the underlying fundamental laws (e.g. Newtons laws are reversible) but a result of the system going from an unlikely ordered state to a more probable disordered state, and the fact that for large systems (large number of molecules), the probability of the system reverting to the ordered state being negligible. In the above example, the probability of any one molecule being on the left half of the box is $1/2$. If there are $N$ molecules, the probability that all of them are on the left is $(1/2)^N$. Even for $N$ as small as $100$ this works out to be about $10^{-30}$, an infinitesimal quantity ! For macroscopic materials, $N$ is of the order of $10^{23}$, and so the resulting probability is even lower. A quantitative measure of disorder in a system is given by the entropy of the system, with the entropy of a disordered state much higher than the entropy of an ordered state. The above example and others like it lead one to the following equivalent form of the Second Law of Thermodynamics: The entropy of a thermally isolated system never decreases. Thermally isolated systems are also called closed systems: There is no exchange of energy or matter with the environment. As we have seen from the examples above, The Second Law defines the direction in which events in isolated systems can proceed: Isolated systems tend to decay and become more disordered, increasing their entropy (or keeping it constant but never decreasing it !). Here is another example: If sugar is placed in a cup of hot tea, it soon dissolves and one will not find the situation whereby the isolated system consisting of the tea and sugar spontaneously separates into its constituent parts. Similarly, if some ink is spilled in a glass water, it soon spreads and colours the whole glass. It is very important to note, from the argument given in the last subsection, that the Second Law is actually a statement about average behaviour that becomes overwhelmingly likely in a very large system, meaning that exceptions will be unobservable in all practical situations. Thus heat energy is of a higher entropy, or lower grade, than other forms of ordered energy like gravitational potential energy. The Kelvin statement places limitations on how much work a cyclical engine can extract out of disordered energy like heat - some of the input heat energy must be dumped into a lower temperature reservoir and is wasted. The limits placed by the Second Law are of practical concern and will be discussed below.