Here was my argument against this, the second law of thermodynamics, in effect says that, there is no heat engine that can take all of some energy that was transferred to it by heat and do work on some object. So, if we can not take a 100% the thermal energy of an object, and use it to do work, what about the thermal energy that is rejected to the environment, can we use all of that energy to do work on an object? No, if energy is supposed to be the ability to do work, well that’s a contradiction.

  • $\begingroup$ There are also some other definitions of entropy. Check out the wiki article. Well, this definition macroscopically is certainly true. I'm not sure about the microscopic validity of this definition though. Don't worry: The second law is just a probabilistic law, and you haven't refuted it. $\endgroup$ – resgh Dec 5 '12 at 14:38
  • $\begingroup$ And by the way no, energy is not the ability to do work. Otherwise gravitation having a negative potential energy is troublesome. $\endgroup$ – resgh Dec 5 '12 at 14:39
  • $\begingroup$ I don’t understand the connection with entropy here. Please explain. $\endgroup$ – Kabelo Moiloa Dec 5 '12 at 14:47
  • $\begingroup$ Yes I agree with you, that was my argument that energy is not the ability to do work. $\endgroup$ – Kabelo Moiloa Dec 5 '12 at 14:49
  • $\begingroup$ You're studying the second law and don't know what entropy is!? Well, its... not easy to explain. Just go on the wiki page for it now. It basically represents the 'randomness' of a system. The second law states that the entropy of a system cannot decrease. $\endgroup$ – resgh Dec 5 '12 at 14:49

"The ability to do work" is certainly a lousy definition of energy.

Is it "merely" a lousy definition, or is it actually an incorrect definition? I think it could be either, depending on precisely how the word "ability" is interpreted. But if the words are interpreted as they would be in everyday speech and everyday life, I would say it's an incorrect definition.

UPDATE -- What is a definition of energy that is not lousy?

This is a tricky issue. Defining a thing that exists in the real world (like you do in physics) is quite different than defining a concept within an axiomatic framework (like you do in math).

For example, how do you "define" Mount Everest? Well, you don't exactly define it, you merely describe it! You describe where it is, you describe what it looks like, you describe how tall it is, etc. Since there is only one mountain that has all these properties, you wind up with a "definition".

Likewise, if I start describing energy (i.e. listing out various properties of energy), I will eventually wind up with a definition of energy (because nothing except energy has all these properties). Here goes:

  • The following are examples of energy: Kinetic energy, electric potential energy, gravitational potential energy, ...

  • The fundamental laws of physics are the same at every moment in time -- they were the same yesterday as they are today. This fact implies, by Noether's theorem, that there is a conserved quantity in our universe... This quantity is energy.

  • Special relativity relates energy to mass / inertia.

  • General Relativity relates energy to the curvature of spacetime.

  • In quantum mechanics, the energy of a system is its eigenvalue with respect to the Hamiltonian operator.

  • Whatever other things I'm forgetting or haven't learned...

All these properties are interrelated, and out of them bubbles a completely precise and unambiguous understanding of what energy is.

(I'm sure that some people will claim that one bullet point is the fundamental definition of energy, while the other bullet points are "merely" derived consequences. But you should know that this is a somewhat arbitrary decision. The same thing is true even in mathematics. What aspects of "differentiable manifold" are part of its definition, and what aspects are proven by theorems? Different textbooks will disagree.)

But can you boil that understanding of energy down into a one sentence "definition" that is technically correct and easy to understand? Well, I can't, and I doubt anyone on earth can.

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  • $\begingroup$ Oh, now I understand it is merely a misleading definition, so what is a good definition of energy then? $\endgroup$ – Kabelo Moiloa Dec 5 '12 at 17:35
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    $\begingroup$ In fact I would say that either energy is the quantity conserved by time translation invariance and Noether's Theorem, or it is the 'gravitational charge' in GR. $\endgroup$ – resgh Dec 6 '12 at 3:02
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    $\begingroup$ Well, why do they try to do it in high school then. It reminds me of the description of energy from Feynmann’s Lectures on Physics, it is an abstract thing that has certain properties that make it useful to scientists. $\endgroup$ – Kabelo Moiloa Dec 7 '12 at 9:13
  • $\begingroup$ As a question why are teachers in high school required to define energy in a one sentence form, that can cause confusion? $\endgroup$ – Kabelo Moiloa Dec 7 '12 at 9:15
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    $\begingroup$ @KabeloMoiloa -- You're asking "Why do high school physics teachers and physics textbooks occasionally say things that are incorrect?" I am not in a good position to answer that. Probably many factors are involved. Understanding the education system is even more difficult than understanding energy, in my opinion!! :-P $\endgroup$ – Steve Byrnes Dec 7 '12 at 16:09
  1. The 2nd Law, recasted (as you did) in terms of Carnot efficiency, just says the ideal scenario is all energy is converted to work while in reality there is a loss through some heating. So it doesn't contradict energy being the ability to do work.

  2. Your phrase "energy is the ability to do work" is justified by the Work-Energy theorem, i.e. $W=\triangle KE$. If you didn't start with kinetic energy, then use the Conservation of Energy Law first.

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  • $\begingroup$ Oh so you mean energy is the ability to do work then? $\endgroup$ – Kabelo Moiloa Dec 5 '12 at 16:18
  • $\begingroup$ what if the work to be done is purely thermal? Say, expanding a gas... $\endgroup$ – Vineet Menon Dec 5 '12 at 16:51
  • $\begingroup$ Moiloa: Look up that theorem/principle. @Menon: It can be recasted in terms of thermal or electrical or whatever you want. $\endgroup$ – Chris Gerig Dec 5 '12 at 21:42
  • $\begingroup$ Even ideally though, there is always some loss - a heat engine (assuming cyclic) can never convert all its input energy into work. $\endgroup$ – David Z Dec 6 '12 at 0:42
  • $\begingroup$ This statement of the work-energy theorem is only valid if a particle, or the center of mass of a multiparticle system, translates. There can be energy transfers within a system that don't give rise to net translation. If I stand on the floor and push against a wall with my hand and take my body to be the system, there is certainly no work done on me (because there's negligible displacement at the point of application of the wall's force on me) but energy is being expended because I get tired. $\endgroup$ – user11266 Dec 7 '12 at 18:09

I've always liked and used Feynman's definition of energy as articulated in The Feynman Lectures (don't have the specific reference in front of me, but it's in volume one in the chapter on conservation of energy). Feynman defines energy as a number that doesn't change as Nature undergoes her processes. Of course, there are quite a few such numbers, but nevertheless energy is one of those numbers. You may also find the book Energy, the Subtle Concept: The discovery of Feynman's blocks from Leibniz to Einstein by Jennifer Coopersmith a useful reference.

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Your statement of the Second Law is incorrect. Your version should be "there is no heat engine that can take all of some energy that was transferred to it by heat and do work on some object in a cyclic process." (My added words are in italics.)

It is certainly true that in a non-cyclic process all the heat can be converted to work. Think of the expansion of a gas in a cylinder with a movable piston raising a weight.

As for the definition of energy, defining it as the capacity to do work seems to be as good a definition as one can easily get.

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