Word Statements of Second Law

When we introduced the first law of thermodynamics we claimed that it is a statement of repeated observation elevated to the status of a law. No one has ever been able to make a machine that produces work out of nothing (a perpetual motion machine of the first kind), so we assume that no such machine can be made.

We then write this statement in mathematical language and begin deriving the consequences of the statement. Ultimately, the validation of the law comes from the experimental verification of the consequences.

The second law of thermodynamics is also a statement of repeated observation (or perhaps better yet, a statement of some things that have never been observed).

Here are two things that have never been observed:

1. Heat has never been observed to move spontaneously from a cold body to a hot body.
2. Heat has never been observed to be converted entirely into work with no other result.

1. It is impossible for heat to move spontaneously from a cold body to a hot body with no other result.
2. It is impossible to convert heat quantitatively into work with no other result.

(A perpetual motion machine of the second kind is a machine that converts heat into work without doing anything else. Imagine an ocean liner that scoops up liquid water out of the ocean, pulls the heat out of the water and uses it to power the ship, and dumps the left-over ice cubes out the back of the ship.

Note that a perpetual motion machine of the second kind would not violate the first law. Energy would be conserved because any heat extracted would be converted into work.)

The second law is why automobiles have radiators. Someone might ask why we throw away all that energy that dissipates from the radiator. Why not capture the energy and use it do decrease our gas mileage? The answer is that if you don't dissipate the heat the engine burns up, as you would quickly find out if you bypassed the radiator with a hose or if you drained the coolant from the radiator.

First we define the "heat engine." A heat engine is a cyclic process that absorbs heat from a heat bath and converts it into work. We shall see that in the cyclic process the engine also dissipates some heat to a heat bath at a lower temperature.

A crucial feature of the heat engine is that it returns to its original state after each cycle. That means that for each cycle of the engine itself, ΔH = 0, ΔU = 0, ΔT = 0, and so on. Presumably, less heat is given back at the lower temperature than was absorbed at the upper temperature so that the difference can be used to supply work to the surroundings. (Otherwise we wouldn't have much of an engine.)

If you run the engine backwards by providing an external power source you get a heat pump (or a refrigerator), that is, a machine that absorbs heat from a lower temperature heat bath and gives it back to a heat bath at a higher temperature. But it takes work to do this.

Our procedure will be as follows:

1. Define and characterize a particular heat engine, the Carnot Cycle. The Carnot cycle is a heat engine operating between two heat baths, one at an upper temperature, which we shall call T_U and the other at a lower temperature, T_L. The Carnot cycle uses the expansion and compression of an ideal gas to convert heat into work.

2. Define the efficiency, e, of a Carnot cycle.

3. Assume that we can find a heat engine, operating between the same two temperatures, which has efficiency greater than a Carnot cycle efficiency and then show that this violates both of the word statements of the second law given above. This leads to the conclusion that no heat engine or cycle can have an efficiency greater than the efficiency of a Carnot cycle.

4. The conclusion that no cycle can have an efficiency greater than a Carnot cycle will lead us to the further conclusion that the integral of dq_rev/T is independent of path. Therefore, the differential dq_rev/T must be exact, which means that it is the differential of some state function which we will call, S. That is, dq_rev/T = dS.

5. Another cycle can have an efficiency less than the efficiency of a Carnot cycle. This will lead us to the conclusion that dq/T might be less than dS if there is some irreversibility in the process. Putting the two possibilities together we will conclude that, dq/T ≤ dS. This is the mathematical statement of the second law of thermodynamics.

Return to the local Table of Contents,

Return to the Table of Contents for the Dynamic Text, or

Return to the WRS Home Page.