Meteo 300  --  Introduction to the Atmospheric Sciences
Chapter 2 -- Second Law of Thermodynamics
 

The Second Law of Thermodynamics and entropy are probably the most misunderstood and most abused concepts in thermodynamics.  Hopefully this lecture will give you some insight into what it all means.

Carnot Cycle

We start with the idea of the Carnot cycle, which is simply a reversible process.  A cycle is a series of processes that eventually repeat themselves.  For a gas, which is the substance of great interest to the atmospheric sciences, these processes involve changes in state. such as pressure, temperature, and volume.  When the system goes through a cycle, it comes back to the state where it began.  Thus, the pressure, temperature, volume, and internal energy must all be the same as at the beginning of the cycle.  So, if some heat is added (Q1) and some given off (Q2), then the difference must be the amount of work done by the system (or on the systme, depending on the sign that you use for work).

The efficiency of the cycle is thus given by:              h= work done/heat energy supplied = (Q1 - Q2)/Q1 = (q1 - q2)/q1.

Carnot arrived at this conclusion for system efficiency by considering a cycle that uses four processes  -- two are isothermal and two are adiabatic.  The gas is compressed adiabatically on the first leg, resulting in a smaller volume, greater pressure and greater temperature.  Energy (heat ) is added on the second leg and the volume is allowed to expand isothermally.  On the third leg, the gas is allowed to expand adiabatically, thus lowering the temperatureand pressure.  Finally, the volume decreases isothermally, so that the volume decreases and the pressure increases.

We know that the work done by the gas is given by:  dW = p dV.  So by integrating the work done around the path, we can determine the total amount of work done.  We also know that in a cycle, that the work done is equal to the difference between the heat in and the heat out.

A result of the calculations on pages 89 and 90 is that:

Q1/Q2 = T1/T2, or that Q1/T1 = Q2/T2.

So, the efficiency of the cycle depends on the difference in the temperatures.

Entropy.

You have probably heard that entropy is the measure of disorder.  This state is correct only if we define disorder in a certain way.  Entropy is really a measure of the number of possible ways to rearrange the molecules a system and have the same energy.  The entropy of a system, like pressure and temperature, is a state variable.

Every time we add energy to a gas, we increase the entropy of that gas.  Entropy is denoted by the symbol S.  The reason is that we are giving the molecules in the gas more ways to arrange themselves for the greater energy.  This can be written mathematically as:

dS = dqrev/T

Thus, for a reversible process, the First Law of Thermodynamics can be written as:

T ds = du + p da

With a little bit of mathematical manipulation, we can see that for a reversible process, entropy is related to potential temperature.  Thus, processes that conserve potential temperature also conserve entropy.

ds = cp dJ/J

We keep talking about reversible processes.  These are processes for which the original state can be obtained by making small changes in pressure and temperature, like in the Carnot cycle.  Irreversible processes are ones where we cannot go back to the original state with small changes.  An example is a free expansion of gas through a small hole from one chamber to another.

The Second Law of Thermodynamics can be written:  The entropy of an isolated system can never decrease.