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Tuesday, October 26, 2010

Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner

This post is part of a series, Nonsense and the Second Law of Thermodynamics. The previous post is entitled The Hydrogen Economy.

As of 10/26/2010, a survey on this site shows that 25% (Final result 21%) of the respondents thus far think that the second law of thermodynamics says that heat cannot be transferred from a cold body to a hot body.

Not only are these people mistaken, but they are also ignoring their own common experience of the world.

It is possible to transfer heat from a cold reservoir to a hot reservoir! 

Air conditioners do it all the time.  A typical air conditioner involves a cyclic process that transfers heat from a cold reservoir to a hot reservoir. That is not the only effect however.

In addition to transferring heat; the air conditioner must generate additional heat.

On a hot day in July, the outside air temperature is of course hot.  In an air conditioned room, the temperature is cooler.  To keep the temperature in the room cool, the air conditioner moves heat from a cold reservoir (the room) to a hot reservoir (the outside).

To do so, one must pay a price: more heat is released outside than is moved from inside (the energy comes from electricity of course).  Additionally, one should consider the heat from the generation and transmission of power to the house.

Yes, the air conditioner does indeed move heat from a cold reservoir to a hot reservoir, but it does not violate the the second law, because it makes the surroundings a little hotter.

Air conditioners contain a fluid  called a refrigerant.

Chlorinated fluorocarbons (CFCs) such as R-12 (dichlorodifluoromethane)  were commonly used as refrigerants, but because they are implicated in ozone depletion, they  been phased out in favor of  HCFCs (also implicated in ozone depletion, but less so), and hydrofluorocarbons (HFCs).

The key property of a refrigerant is that it has a boiling point below the target temperature, but not too far below, and a high heat of vaporization.

Liquid refrigerant is allowed to evaporate and expand into a gas.  This process requires heat that is supplied by the room that is to be cooled.  This is the same principle by which sweat cools.

As a liquid evaporates it requires heat.  This heat is called the heat of vaporization.  Some additional heat can be absorbed by the expanding gas, but most of the heat is absorbed from the liquid-gas phase transition.

The cool gas is allowed to heat from the room and is then pumped toward the outside.  Here a compressor applies pressure to that gas.  As the gas compresses, it must give off heat, which is released to the outside.

The gas becomes liquid in a condenser that releases heat ( [heat of condensation] = -[heat of vaporization]) to the outdoors.  The liquid can now be allowed to evaporate and cool the room again.

Refrigeration forms a cycle, and one valid formulation of the second law states:
It is impossible to carry out a cyclic process using an engine connected to two heat reservoirs that will have as its only effect the transfer of a quantity of heat from the low-temperature reservoir to the high-temperature reservoir.
The two heat reservoirs are outside (high-temperature) and the room (low-temperature).  Heat was moved from the low temperature reservoir to the high temperature reservoir, but there are other effects.

The air conditioner must do work to compress the refrigerant.

This work is not 100% efficient, and it must also release heat. 

Efficiency of an ideal air conditioner

In fact, the second law does limit how efficient an air conditioner can be.  Consider an ideal air conditioner that transfers heat, q1, from a cold reservoir at T1.

Consider the outside world as a reservoir at T2, and let q2 be the amount of heat transferred to the outside world.

By the definition of entropy, the amount of entropy decrease of the cold room is:


The entropy of the outside world must increase at least as much as the entropy of the room decreases, and for all irreversible processes, it must be greater.  For a reversible process:

     q2/T2  =  q1/T1

T2 is by definition greater than T1, and therefore q2 must be greater than q1.

The difference between the two is the work performed:

     w = q2 - q1

The efficiency is the work performed over the total heat transferred:

     efficiency = (q2 -q1)/q2 = 1 - q1/q2


     efficiency = 1 - T1/T2

Which is the same as the Carnot efficiency.

From this discussion, one might conclude that all methods of air conditioning must directly heat up the outside world more than they cool the indoors.

For a cyclic system, this conclusion would be correct, but some systems are a little more complicated to analyze.  The next post in this series is entitled the Second Law and Swamp Coolers.

  • 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.
  • How It Works: Air Conditioner 
  • How Stuff Works 
  • DuPont: Refrigerants
  • Wikipedia: Refrigerant 

  1. Introduction
  2. What the Second Law Does Not Say
  3. What the Second Law Does Say
  4. Entropy is Not a Measure of Disorder
  5. Reversible Processes
  6. The Carnot Cycle
  7. The Definition of Entropy
  8. Perpetual Motion
  9. The Hydrogen Economy
  10. Heat Can Be Transferred From a Cold Body to a Hot Body: The Air Conditioner
  11. The Second Law and Swamp Coolers
  12. Entropy and Statistical Thermodynamics
  13. Fluctuations
  14. Partition Functions
  15. Entropy and Information Theory
  16. The Second Law and Creationism
  17. Entropy as Religious, Spiritual, or Self-Help Metaphor
  18. Free Energy
  19. Spontaneous Change and Equilibrium
  20. The Second Law, Radiative Transfer, and Global Warming
  21. The Second Law, Microscopic Reversibility, and Small Systems
  22. The Arrow of Time
  23. The Heat Death of the Universe
  24. Gravity and Entropy
  25. The Second Law and Nietzsche's Eternal Recurrence
  26. Conclusion

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