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The efficiency of a heat engine is the work we can do divided by the heat we take out of the hot reservoir.

This quantity is always $ \le 1$.

The efficiency of a heat pump is the heat we can release divided by the amount of work we have to put in there and similarly for a fridge, it is the amount of heat that is taken from a reservoir divided by the work we have to put in.

Now the problem with the last two definitions is, I think, that the efficiency can be larger than one.

My question is: How can we make this plausible? I mean, somehow I guess that an efficiency should always be between 0 and 1 and define how good an energy of type $A$ is converted into energy of type $B$, but this seems to be different here. Thus, I have to be wrong here. So what does efficiency mean and when can we get efficiencies larger than one?

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It is a matter of definition. In the case of the heat engine, you are seeing how much of the heat can be converted into useful work - since something will be lost in the conversion process, a number between 0 and 1 makes sense.

When you have a heat pump, you are comparing the amount of work needed to move heat from one reservoir to another with the amount of heat that is actually added to the target reservoir - when the temperature difference is small, you only need a small amount of work to move a large amount of heat, so "efficiency" by that standard is indeed greater than one - because it's more efficient to use a heat pump than to create the same amount of heat by just converting work (e.g. by friction).

A more useful metric for the second case is "percentage of theoretical efficiency". If you say that an ideal (Carnot) engine would require an amount of work $W$ to move an amount of heat $Q$, then a "real" engine will always require more - say $W'$. You could now define the efficiency of the actual engine as

$$\eta = \frac{W}{W'}$$

which would once again be a number between zero and one. See also this earlier answer

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  • $\begingroup$ you wrote, ''you are comparing the amount of work needed to move heat from one reservoir to another'' . what do you mean by move heat. it is the energy that is transfered and the energy which is transfered due to temperature difference is called heat. $\endgroup$
    – Paul
    Commented Dec 29, 2014 at 12:13
  • $\begingroup$ @Paul - in a heat pump the objective is to warm up something (a building) or cool down something (a refrigerator). In either case, the question becomes "what did it cost to achieve the purpose" where cost = the work done by the heat pump (and thus, the energy you had to supply to the heat pump). Usually the heat sink's energy is considered "free", all you worry about is the work that has to be done to move the heat against the thermal gradient. $\endgroup$
    – Floris
    Commented Dec 29, 2014 at 13:14
  • $\begingroup$ Sorry but i didnt mean that. i am saying.''move the heat'' is not a good sentence. $\endgroup$
    – Paul
    Commented Dec 29, 2014 at 13:30
  • $\begingroup$ Its like saying ,''heat in an object''. which wrong.heat is energy in transit. $\endgroup$
    – Paul
    Commented Dec 29, 2014 at 13:33
  • $\begingroup$ @Paul it may just be that we are from a different educational background. Where I come from, "heat" is an acceptable term for thermal energy, and that is how I use it. When I remove thermal energy from one object and transfer it to another, I think I have moved heat. Do you have a good reference that discourages this usage? I am willing to adapt... $\endgroup$
    – Floris
    Commented Dec 29, 2014 at 15:00

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