## Friday, October 8, 2010

### The Carnot Cycle

This post is part of a series, Nonsense and the Second Law of Thermodynamics.  The previous post is entitled:  Reversible Processes.

In 1824, Nicolas Léonard Sadi Carnot tried to explain how heat could be converted into useful work. He came up with a four-step cycle that is known as the Carnot cycle.

There are two conditions used in the Carnot cycle: 1) constant temperature processes , 2) adiabatic processes.

The condition  of constant temperature is straightforward: the temperature is maintained throughout the process.  Assume that there are two heat reservoirs, one at T2 and another at T1. To simplify matters, the reservoirs are so big that the heat extracted or added to the reservoirs is negligible, and the reservoir temperatures do not change.  In real life, one could account for such changes, but to keep things simple, we simply assume that the reservoirs are very big.

In an adiabatic process, no heat is transferred between the system and its surroundings: for a real gas, the temperature will change, if it is allowed to expand, or else is compressed adiabatically.

In the Carnot cycle, all steps are conducted reversibly, resulting in the most efficient way of carrying out the cycle.
1. Expand a gas at constant temperature, T2
2. Expand the gas adiabatically (no heat transferred to or from the gas) to T1
3. Compress the gas a constant temperature, T1
4. Adiabatically compress the gas back to T2
Heat, q  is withdrawn from the hot reservoir and is added to the cold reservoir.  Readers unfamiliar with the difference between heat and temperature may want to review that topic.  In the future, I hope to write a short blog post to help explain that difference.

As heat is transferred from the hot reservoir to the cold reservoir, some of that heat could be used converted to do useful work.  The efficiency of the cycle is defined as the maximum work extracted divided by the heat transferred from the hot reservoir.  The next post discusses this efficiency in more depth.

Note that  T2 is larger than T1, and that therefore q/Tis less than q/T1. This fact will become important as I go on to define entropy and discuss the efficiency of the Carnot cycle. The next post is entitled the Definition of Entropy. It also discusses the efficiency of the Carnot cycle.

Sources
• Atkins, P. W. Physical Chemistry, W. H. Freeman and Company, New York, 3rd edition, 1986
• McQuarrie, Donal d A., Statistical Thermodynamics,  University Science Books, Mill Valley, CA, 1973
• Bromberg, J. Philip, Physical Chemistry, Allan and Bacon, Inc., Boston, 2nd Edition, 1984
• Anderson, H.C., Stanford University, Lectures on Statistical Thermodynamics, ca. 1990.
• Wikipedia: Nicolas Léonard Sadi Carnot