Entropy was first defined in 1824 by a French military engineer, Sadi Carnot, in his very careful theoretical analysis of a steam engine. Carnot idealized the operation of the engine in four reversible steps using an ideal gas as the working fluid. However, his conclusions are independent of the working fluid. The four steps are

• isothermal expansion at T_hot (the temperature of the boiler)
• adiabatic expansion (during which the engine cools from T_hot to T_cold)
• isothermal compression at T_cold
• adiabatic compression to T_hot (return to the initial state of step 1).

A thermodynamic analysis of this cycle has fundamental implications for absolute efficiencies of cyclic heat engines and for how heat can be converted into work. Furthermore, if you run the cycle in reverse you get a refrigerator and so Carnot efficiencies apply to refrigerators too.

There are lots of animations on the web that show the Carnot cycle for a piston along with a plot of the relevant state variables, pressure (P) vs volume (V), for the gas along the cycle. This animation is pretty effective, but notice that it starts with the isothermal compression step so that you do not lose your place in the cycle. The one found here is very stylized, allowing a clear distinction to be made between the isothermal and adiabatic expansions in the P vs V plot, but suffering a bit of accuracy because of it.

Carnot performed a quantitative thermodynamic analysis of the work and heat produced and consumed during the Carnot cycle, and most chemical engineering and upper level undergraduate chemistry courses that study thermodynamics expect students to be able to repeat Carnot's analysis. You can see this done on Nichols' & Farris' slide show, but must follow slides 3-10 of the sequence for the entire Carnot cycle. If you are having a good time, go ahead and check out the Otto (automobile engine) and Diesel (diesel engine) cycles! Because these processes are cyclic you return to the same initial conditions at the end of a cycle and all state functions will be the same at the beginning and end of the cycle. Therefore, ΔU = 0 for a complete cycle and q = - w.

Carnot also found that his newly defined quantity, entropy, is a state function by integrating the changes that occur to S

as you go around a complete cycle:

Sadi Carnot did not set out to define a quantity that determined thermodynamic spontaneity. That is what entropy does and why it is so valuable to us in our study of thermodynamics. But he did not know it at the time. He did know that his analysis determined a number of things about heat engines (and refrigerators!). Among them are

• Cyclic heat engines require two heat reservoirs, one (the boiler or source of energy) at T_hot from which energy is extracted in the form of heat and one (the ambient air or radiator) at T_cold to which "waste" heat is sent. You cannot get by with a single heat reservoir.
• The maximum efficiency is 1 - (T_cold/T_hot). This efficiency assumes all the mechanical parts of the engine work perfectly and without friction.
• Heat flows from hot to cold spontaneously, but needs a net input of work to flow from cold to hot.
• The engine performs work equal to - w_system = q_hot-q_cold, where q_hot is the energy in the form of heat supplied by the hot temperature reservoir and q_cold is energy in the form of heat sent to the cold reservoir.
• The engine works most efficiently when it is working reversibly. All irreversible processes are less efficient in converting heat into work.

This last point can be written mathematically in the form known as the Clausius inequality

where dq_rev is the incremental transfer of heat for the cycle performed reversibly and dq is the actual transfer of heat that takes place whether or not the change is reversible. A little math turns this into the more familiar form of the second law of thermodynamics

To get this last step, I assumed that the heat gained by the system is the same as the heat lost to the surroundings, q_system= - q_surroundings and this is exactly the purpose of the surroundings in entropy calculations, to supply or remove heat.

Calculating the cyclic integral requires calculus, particularly if the temperature is changing. We, however, will be interested in applying thermodynamics to chemical reactions and phase changes. For chemical reactions, we can get ΔS_system by using thermodynamic tables and Hess's law. For the surroundings, we need to know something about the reaction conditions. Commonly we hold the temperature and pressure constant (for example as approximately the conditions in an open beaker in the lab). In this case, the laboratory air either absorbs heat from the reaction if it is exothermic or transfers heat to the reaction if it is endothermic to keep the temperature constant with the air. Yes, the lab temperature increases or decreases as it absorbs or transfers heat. But the lab is huge relative to the size of the beaker and the increase/decrease in temperature is negligible. Under these conditions, the entropy of the surroundings is merely the heat lost from the reaction divided by the lab temperature

To determine whether a reaction can proceed at constant T and P, we need merely test to see if

For a phase transition, we can find S_system by using thermodynamic tables and Hess's law, or we can make use of the fact that temperature does not change during a phase transition and

where "trans" refers to vaporization, boiling or some other first order phase transition.