"The entropy of the universe tends toward a maximum"

These words are a famous quote by Clausius in 1852. They are still true today. We will distinguish between the system and surroundings. While it is possible for the entropy of the system to be negative the sum of the system and surroundings must be positive for all spontaneous processes. Thus, entropy is a state function that defines the direction and magnitude of spontaneous change. We are most interested in chemical change, but entropy also applied to physical change as well.

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The adiabatic path

In order to illustrate the entropy and how it relates to the efficiency of thermodynamic processes we will use a thermodynamic cycle that consists of an expansion in two steps followed by a compression in two steps. The two steps are 1.) the isothermal path, seen previously, and 2.) the adiabatic path. The adiabatic path means that there is no heat transferred along that path. An adiabatic process is isolated. It is a process that occurs in a thermos bottle, so to speak. Of coruse, since dq = 0 along an adiabatic path dS = 0 as well. This follows from the definition of the entropy state function, dS = dq/T for a reversible process. In the Carnot cycle we need to relate the temperatures and volumes at the beginning and end of an adiabatic expansion or compression.

The Carnot cycle

The Carnot cycle provides a method for showing that the entropy can be defined as a function of the heat transferred divided by the temperature. Actually, in the early 1800s there were two applications that led to the insight that entropy is a state function that can inform us about the direction of spontaneous change. These were the steam engine and the cannon. While these may seem different they both consist of a cylinder and piston. In the steam engine, the piston moves as steam is heated inside the cylinder. The motion of the piston is converted into work. In a cannon the cannon ball acts as a piston. The explosion that occurs when gunpowder is ignited causes the "expansion", which is the exit of the cannonball from the cylinder. The common aspect that engineers wanted to understand was the efficiency of the process. By maximizing the efficiency one can also obtain the maximum work from the expansion in the steam engine. By the same reasoning, one can obtain the maximum range of the cannonball. In order to understand efficiency it is necessary to consider a cyclic process in which there is an expansion and then compression that returns the system to its initial state. One can ask the question, how much work is extracted for a given heat input? This is the efficiency. In the process of asking this question it turns out that one naturally encounters the definition of the entropy. Let's see what reasoning is employed in the Carnot cycle to obtain this result.