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Internal Energy - Is there a maximum value?

  1. Mar 23, 2009 #1

    I am going to teach a class of year 9 students on the topic "Thermal Properties of Matter" and the concept of Internal Energy is mentioned inside.
    As I was trying to anticipate the types of questions that will be asked by them, this question suddenly struck me: Is there a limit to the internal energy of a system?

    As far as the syllabus is concerned,
    Internal energy = Kinetic energy (KE) + Potential energy (PE).
    So, I am thinking: Once the matter reaches the gaseous state, can we continue to increase the KE and is there a limit to it?

    Thank you!

    P.S.: Feel free to add in extra information as they will be helpful for me to answer any questions from the students and improve my knowledge.
  2. jcsd
  3. Mar 23, 2009 #2
    Hello. To give a young student a very simple and intuitive answer, I would just let him imagine a free particle. You can ideally give it more and more KE but since it cannot overcome the speed of light, there you get an upper limit.
    Is that in line with your ideas in terms of explanations? Greetings.
  4. Mar 23, 2009 #3
    Generally no, there is no upper limit on say, molecules in a gas, but there are special cases where there might be.

    For instance, consider a system of a bunch of fixed magnetic dipoles that are free to rotate, but not free to translate. The lowest energy state would be all of the dipoles stationary and aligned with each other. Add energy, and they start to vibrate, and then they start to chaotically rotate, and then as you add more and more energy, it all goes into the potential energy of static dipoles again, but this time each anti-aligned with their neighbors. Once all the dipoles are static and antialigned with neighbors, there is no additional energy you can put into the system. It is interesting that such systems have negative temperatures, because T=dq/dS and as you add heat past a certain point, entropy decreases.

    Another case might be if you add so much heat to a self-gravitating system (such as a star) that the particles become relativistic. Then the addition of more and more relativistic pressure merely adds to the gravity, and the system can become a black hole.
  5. Mar 23, 2009 #4


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    If you keep adding energy to a gas, the molecules will eventually begin striking each other hard enough that an appreciable number of them will (1) dissociate into atoms, (2) lose electrons (i.e., become ionized, thus forming a plasma), (3) lose nucleons (i.e., fission), and so on. I suppose one could look at the Big Bang model and work backwards from the present day to see what happens when matter continues to gain energy density.
  6. Mar 24, 2009 #5
    Once that question also arised for me. E.g. you calculating Int. Energi, in thermodynamics you taking KE+PE. But what potential energy? It depens on task, you can add to PE of intermolecular enegry, energy of binding of electrons in atoms, then energy of nuclei, then energy of quark.... You can but it's not nessecary for thermodinamic calculations :)
  7. Mar 24, 2009 #6
    Hi, montecarlo. I'm afraid you have confused me a bit. Does the speed of light barrier really limit how much energy a particle can have? Perhaps the effective mass of such a particle continue to increase if I added energy, even its velocity grew only negligibly.
  8. Mar 24, 2009 #7
    Dear Cantab Morgan, I admit I could have been confusing. I was trying to give a quick and moreover the most straightforward and intuitive answer to youngsters (since wing81 is more concerned with didactics), but as a matter of fact I realize in this case relativity is somehow misleading...

    The definition of relativistic energy reads
    E = mc^2 = (p^2c^2+m0^2c^4)^0.5,
    where m is the relativistic or effective mass, p is momentum, m0 the rest mass. Therefore you can increase E by increasing the relativistic momentum (= classical momentum times gamma, which once again is a function of v/c), but you actually cannot reach the limit v=c where E would become infinite. Or maybe is that what you are saying, i.e. the limit is infinite?
  9. Mar 25, 2009 #8
    Thanks everyone!
    I love the input of answers as I have always been criticised by my mentoring teacher that I lack the subject mastery (as a matter of fact, I am criticised once more today...) and these answers really expose me to a higher realm of the physics world :!!)
  10. Mar 26, 2009 #9
    Yes, exactly. Thanks for the kind explanation and for inferring what I was unable to articulate.
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