### The Modern Formulation

Some early formulations of the second law of thermodynamics, due to Clausius and Kelvin, were mentioned on the previous page. Those versions of the second law describe limitations on the flow of heat. But there are similar limitations on the flow of matter and other forms of energy. For this reason, the modern formulation of the second law doesn’t mention heat at all. Instead, it is expressed in terms of a quantity called entropy.

The term “entropy” was coined by Clausius, and the modern formulation of the second law is also credited to him.

Entropy is a quantity that increases as a system evolves toward a condition of equilibrium. In other words: the higher the entropy, the closer a system is to equilibrium. A more detailed description of entropy will be given below, and a slightly more technical explanation of the concept is given in appendix C [link coming soon]. And what is equilibrium? Equilibrium is a condition in which temperature and pressure are the same throughout a system, or at least throughout the parts of the system that aren’t isolated from each other. (In other words, the temperature and pressure is the same throughout any parts of the system that are able to exchange matter and energy.) A flat tire is in equilibrium with the surrounding atmosphere, for example, because the temperature and pressure are the same in both.

Pressure is the amount of force exerted against a surface, divided by the surface area. The standard unit of pressure is the pascal, symbolized ‘Pa.’ One pascal is equal to a newton of force per square meter: 1 Pa = 1 N/m2

A cup of hot coffee with ice cubes floating in it is not in equilibrium, because the ice and the coffee have different temperatures, and they aren’t isolated from each other. The entropy of the coffee cup increases as the ice melts, and its entropy reaches a maximum value when the cup is in equilibrium (i.e. when it contains lukewarm, diluted coffee and no ice). Similarly, air leaks out of a punctured bicycle tire until the pressure inside the tire is the same as the pressure of the surrounding air. The entropy of the tire-and-atmosphere system increases until it reaches equilibrium—the condition in which the air pressure inside the tire is the same as the air pressure outside.

The modern version of the second law of thermodynamics can be stated in numerous ways. One of the simplest formulations of the second law says that the entropy of a closed system does not decrease (though it may increase). In other words, closed systems do not evolve away from equilibrium; they only evolve toward equilibrium. Once a closed system reaches equilibrium, it stays in that maximum-entropy condition forever, unless it stops being a closed system—that is, unless it exchanges matter or energy with its environment. For example, if you drop an ice cube into a thermos of hot coffee and seal the lid, you’ve created a closed system in a low-entropy state. So long as the lid remains sealed, the system will evolve toward equilibrium and the ice will melt. And it will stay in equilibrium forever, unless someone opens the lid and disturbs the system (e.g. by adding more ice).

Since the universe as a whole is a closed system, the second law implies that the entropy of the universe never decreases. Therefore, whenever entropy decreases in some part of the universe, it must increase somewhere else by at least as much, so that the total entropy doesn’t go down. For this reason, the second law can also be expressed as follows: any process that lowers the entropy of a system also raises the entropy of the system’s environment by at least as much. In other words, the entropy of an open system can decrease, but only through a process that increases the entropy of the system’s environment. When an air conditioner transfers heat from the cooler air inside a building to the warmer air outside, for instance, the entropy of the universe increases somewhere else—namely, wherever the power is produced to run the air conditioner.

To summarize, the second law of thermodynamics says:

• The entropy of a closed system does not decrease.
• Any process that lowers the entropy of an open system also raises the entropy of the system’s environment by at least as much.