Second Law of Thermodynamics Summary

• Heat Engines

  A heat engine is a cyclic device that takes heat $Q_H$ in from a hot reservoir, converts some of it to work $W$, and rejects the rest of it $Q_C$ to a cold reservoir so that at the end of a cycle it is in the same state (and has the same internal energy) with which it began. The net work done per cycle is (recall) the area inside the $PV$ curve.

  The efficiency of a heat engine is defined to be
  

  $$\epsilon = \frac{W}{Q_H} = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}$$ (249)

  
  

• Kelvin-Planck statement of the Second Law of Thermodynamics

  It is impossible to construct a cyclic heat engine that produces no other effect but the absorption of energy from a hot reservoir and the production of an equal amount of work.

• Refrigerators (and Heat Pumps)

  A refrigerator is basically a cyclic heat engine run backwards. In a cycle it takes heat $Q_C$ in from a cold reservoir, does work $W$ on it, and rejects a heat $Q_H$ to a hot reservoir. Its net effect is thus to make the cold reservoir colder (refrigeration) by removing heat from inside it to the warmer warm reservoir (warming it still further, e.g. as a heat pump). Both of these functions have practical applications - cooling our homes in summer, heating our homes in winter.

  The coefficient of performance of a refrigerator is defined to be
  

  $${\rm COP} = \frac{Q_C}{W}$$ (250)

  
  
  It is not uncommon for heat pumps to have a COP of 3-5 (depending on the temperature differential) giving them a significant economic advantage over resistive heating. The bad side is that they don't work terribly well when the temperature difference is large in degrees K.

• Clausius Statement of the Second Law of Thermodynamics

  It is impossible to construct a cyclic refrigerator whose sole effect is the transfer of energy from a cold reservoir to a warm reservoir without the input of energy by work.

• bf Reversible Processes Reversible processes are ones where no friction or turbulence or dissipative forces are present that represent an additional source of energy loss or gain for a given system. For the purposes of this book, both adiabatic and isothermal processes are reversible. Irreversible processes include the transfer of heat energy from a hot to a cold reservoir in general - heat engines and refrigerators can be constructed whose steps in a cycle are all reversible, but the overall effect of transferring heat one way or the other is irreversible.

• Carnot Engine

  The Carnot Cycle is the archetypical reversible cycle, and a Carnot Cycle-based heat engine is one that does not dissipate any energy internally and uses only reversible steps. Carnot's Theorem states that no real heat engine operating between a hot reservoir at temperature $T_H$ and a cold reservoir at temperature $T_C$ can be more efficient than a Carnot engine operating between those two reservoirs.

  The Carnot efficiency is easy to compute (see text and lecture example). A Carnot Cycle consists of four steps:

  1. Isothermal expansion (in contact with the heat reservoir)
  2. Adiabatic expansion (after the heat reservoir is removed)
  3. Isothermal compression (in contact with the cold reservoir)
  4. Adiabatic compression (after the cold reservoir is removed)
  The efficiency of a Carnot Engine is:
  

  $$\epsilon_{\rm Carnot} = 1 - \frac{T_C}{T_H}$$ (251)

  
  

• Entropy

  Entropy $S$ is a measure of disorder. The change in entropy of a system can be evaluated by integrating:
  

  $$dS = \frac{dQ}{T}$$ (252)

  
  
  between successive infinitesimally separated equilibrium states (the weasel language is necessary because temperature should be constant in equilibrium, but systems in equilibrium have constant entropy). Thus:
  

  $$\Delta S = \int_{T_i}{T_f} \frac{dQ}{T}$$ (253)

  
  
  has limited utility except for particularly simple processes (like the cooling of a hot piece of metal in a body of cold water.

  We extend our definition of reversible processes. A reversible process is one where the entropy of the system does not change. An irreversible process increases the entropy of the system and its surroundings.

• Entropy Statement of the Second Law of Thermodynamics

  The entropy of the Universe never decreases. It either increases (for irreversible processes) or remains the same (for reversible processes).