6.8.2: Entropy, Life, the Universe, and Everything

Learning Objective

• Apply the Second Law of Thermodynamics to a variety of situations.

Let us consider another example of the second law. Suppose we mix equal masses of water originally at two different temperatures, say \(20.0^{\circ} \mathrm{C}\) and \(40.0^{\circ} \mathrm{C}\). The result is water at an intermediate temperature of \(30.0^{\circ} \mathrm{C}\). Three outcomes have resulted: entropy has increased, some energy has become unavailable to do work, and the system has become less orderly. Let us think about each of these results.

First, entropy has increased for the same reason that it did in the example above. Mixing the two bodies of water has the same effect as heat transfer from the hot one and the same heat transfer into the cold one. The mixing decreases the entropy of the hot water but increases the entropy of the cold water by a greater amount, producing an overall increase in entropy.

Second, once the two masses of water are mixed, there is only one temperature—you cannot run a heat engine with them. The energy that could have been used to run a heat engine is now unavailable to do work.

Third, the mixture is less orderly, or to use another term, less structured. Rather than having two masses at different temperatures and with different distributions of molecular speeds, we now have a single mass with a uniform temperature.

These three results—entropy, unavailability of energy, and disorder—are not only related but are in fact essentially equivalent.

To summarize, we had several statements of the Second Law of Thermodynamics:

THE SECOND LAW OF THERMODYNAMICS (FIRST EXPRESSION)

Heat transfer occurs spontaneously from higher- to lower-temperature bodies but never spontaneously in the reverse direction.

THE SECOND LAW OF THERMODYNAMICS (SECOND EXPRESSION)

It is impossible in any system for heat transfer from a reservoir to completely convert to work in a cyclical process in which the system returns to its initial state.

THE SECOND LAW OF THERMODYNAMICS (THIRD EXPRESSION)

A Carnot engine operating between two given temperatures has the greatest possible efficiency of any heat engine operating between these two temperatures. Furthermore, all engines employing only reversible processes have this same maximum efficiency when operating between the same given temperatures.

THE SECOND LAW OF THERMODYNAMICS (FOURTH EXPRESSION)

The total entropy of the universe increases or remains constant in any process; it never decreases.

As we shall see with the ensuing examples, this fourth expression is the most generalizable, as it applies to systems which involve flows of energy beyond just heat.

Additionally, the reason why processes work in this way can be explained by the Boltzmann equation, which relates the number of possible arrangements of a thermodynamic system with its entropy. the Boltzmann equation is the mathematical description of the disorder that we understand entropy to be a measure of. These results are numerically equivalent for an ideal gas, and more advanced mathematical techniques can result in an equivalent relationship for more complicated systems.

BOLTZMANN EQUATION FOR RELATIONSHIP BETWEEN ENTROPY AND STATISTICAL ANALYSIS OF THE SYSTEM

\[S=k \ln W \nonumber \]

We can see the connection between these different statements of The Second Law of Thermodynamics and our example of the mixed water at the start of this chapter.

It turns out that the Second Law of Thermodynamics is not just true for heat engines that we have manufactured, but for every process in the universe. However, it might be beneficial to start with another manufactured item before moving on to natural processes: computers. At their most basic level, computers are devices that organize information: whether that is a database or an aesthetically appealing video game, the information that we see on a computer screen is of low probability compared to all of the possible ways in which the pixels on the screen might be arranged. In order for your computer to organize this information, it needs a source of energy. It also must release some of this energy to the surrounding environment through the cooling system (generally a fan). We can think of your computer like a heat engine: energy in the form of electricity flows into it via the plug, performs work by organizing data (thereby lowering the entropy of those data), and then emitting heat to the environment (thereby increasing the entropy of the air surrounding the computer). (Strictly speaking, a computer is not a heat engine, because the electricity coming into it is of a much more useful form than heat, but we can see the same limits to what is possible for energy through this example.)

Turning to biological systems, the same ideas apply but with more complexity. As adult human beings, we are much more organized than the zygotes we existed as before birth. How is this possible if entropy is increasing? It is possible because we have eaten a lot of food and breathed a lot of air throughout our lives. The food we eat and the air we breathe contain energy that we extract from it via chemical reactions in our bodies. (We measure the amount of energy extracted in units of Calories, as labeled on our food containers.) These reactions provide the energy for us to grow. Part of this growth includes the organization of our body parts, including our brain. We also have a variety of waste matter that leaves our body. This waste matter does have significant potential energy stored within its chemical bonds, but it is a much lower amount than that stored in the bonds of the matter we take in through food and air. Although the energy entering and leaving our bodies is in the form of chemical energy stored in bonds, we see the same principle as we do with heat engines: some energy comes into a system and does useful work, but some amount of that energy leaves the system without doing any work. If we stopped eating, we would eventually die because the system of our body is much more ordered than our surroundings.

Finally, we can consider the Earth itself. Almost all of the energy on the Earth comes from the Sun. Most of this energy is eventually radiated back into space, but some of it powers chemical reactions which result in a more ordered environment. One of the most abundant such reactions is called photosynthesis, which turns low energy carbon dioxide into higher energy oxygen and carbon. The carbon becomes a part of the ordered structure of the organism which produced the photosynthesis, while the oxygen provides a readily available fuel for other organisms in the environment to use in the ordering of their own structures. If there were no sunlight, life on Earth would not be possible.

You might be wondering about all of the other planets where life does not exist (at least that we know of). Why does the application of the Second Law allow for life on Earth, but there is no life in those other places? The reason is because useful work was extracted on Earth but not in those other places. If we think of the example of mixing different temperature waters at the start of this section, that was another possibility for useful work to be extracted that didn't occur. If the situation had been a little bit different, useful work could have been extracted from it. The situations on Earth so happen to be those which allow for the extraction of useful work in the flow of energy and those conditions do not exist on those other planets. Just because it is possible to extract useful work does not mean that it always happens. (Consider all of the waste food in the United States, for example.) We are very fortunate to exist on a planet where life is possible!

Section Summary

• The Second Law of Thermodynamics is generalizable beyond what we have defined for heat engines.
• Some other applications include computers, biological life forms, and environments that support such life forms.