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Entropy and Heat Capacity have the same units. Connection? Redundancy?

  1. Jun 22, 2012 #1
    Given that heat capacity is a ratio of change of energy over change of temperature, while entropy is a change of energy over absolute temperature...

    I was wondering if there is any basis for the idea that energy will tend to flow from media having low heat capacity to media having high heat capacity, such that the imported energy becomes colder as a result of the higher heat capacity of the destination medium relative the medium departed.

    See a table of specific heat capacities for different materials:

    In other words, energy of an object could be thought of as getting hotter or colder due to heat capacity variations. As a result of such a energy transfer (=ΔE), the entropy would increase as result of ΔE/t increasing (i.e. 0 < ΔE/t_(cold, final) - ΔE/t_(hot, initial)), where ΔE specifically would refer to the energy that is transferred. In this sense, entropy is not so much a flow, but rather an intensity like heat capacity. And what if, in the end, if we account for entropy changes by such transfers of energy, couldn't that concept of pinning thermodynamic state variables to energy as opposed to pinning them to boundary-defined systems, as is almost always assumed in traditional teaching, make entropy and heat capacity one-and-the-same-thing?

    It is also known that there are different heat capacities for a given substance based on:
    * constant volume, as in an isochoric process
    * constant pressure, as in an isobaric process

    So depending on the present process occurring in a given thermodynamic cycle, the direction of energy flow that would increase entropy could change - or even reverse.

    Conversely, one could flip the perception around and imagine that entropy itself flows from systems of low heat capacity to those of high heat capacity, causing the heat capacity of the systems which they flow into to increase, while causing the heat capacity of systems they are leaving to decrease.

    Certain materials have different effects on the flow of heat, so couldn't understanding of that be aided by the idea that entropy and heat capacity are intimately connected? Could the ability to engineer extreme differences in heat capacity facilitate extraction of ambient thermal energy? If there can be a heat capacity for constant pressure and a heat capacity for constant volume, wouldn't the concept of pinning thermodynamic state variables to energy as opposed to pinning them to boundary-defined systems, as is almost always assumed in traditional teaching, suggest the existence of a temperature for constant pressure [dynamics] and a temperature for constant volume [statics], so maybe it is related to the relation between dynamic pressure vs. static pressure: https://www.physicsforums.com/showthread.php?t=169660?
    Last edited: Jun 22, 2012
  2. jcsd
  3. Jun 23, 2012 #2

    Simon Bridge

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    Entropy and Heat capacity are, indeed, related concepts.
    I think you want the statistical mechanics description of these things to understand your questions.

    What you don't want to do is think of the energy as having a temperature. An object may have thermal energy as well as temperature... two properties which are related.
  4. Jun 23, 2012 #3
    Why not?
  5. Jun 23, 2012 #4
    Well it is estimated that the average temperature of the world ocean is about 4°C whereas the average temperature the average temperature of the water in my kettle is 100°C. But which has more energy and entropy?

    Heat energy transferred in the process of phase change has nothing to do with heat capacity.

    Entropy (unlike energy) does not 'flow from one place to another'. In the above example the take up of latent heat on say melting goes into a rearrangement of the melting system such that it has greater entropy. The heat supplied to cause the melting may also result in greater entropy of the surroundings if it is the result of a say a chemical reaction such as burning a fuel.
    So the entropy does not flow from surroundings to system but increases in both. The overall entropy of the universe may increase by such a process.
  6. Jun 23, 2012 #5

    The ocean has more entropy because of two things:

    1: There is more heat energy in the ocean (Joules)

    2: The heat energy in the ocean is at lower temperature (Kelvins)
  7. Jun 23, 2012 #6
    Is that really a good reason why we should not think of energy as having a temperature?

    I can think of a van as having 1 passenger. Such a concept makes sense. I can state that, "A van has 1 passenger." That statement is not made incomprehensible or meaningless by the notion of a Mini Cooper which has three passengers on board. It means something, and that meaning makes sense. It is not unreasonable to think of such a case.

    How can you be so sure of that? Doesn't adding enough energy to 99 degree Celsius water at atmospheric pressure cause it to change its heat capacity as it transitions to the gaseous phase?


    How do you know that the entropy didn't already exist and that it didn't come from some atoms, causing the entropy within the atoms to fall and the entropy outside the atoms to increase?
    Last edited: Jun 23, 2012
  8. Jun 23, 2012 #7
    Do you think it makes sense for energy to have an entropy?

    If so, why is the same not the case for temperature?

    If not, then why does it make sense for a system to have entropy, while the same is not true for energy. Also, why does it make sense to talk about the temperature of a system, and the temperature of an object, but not the temperature of an energy?
  9. Jun 23, 2012 #8
    I will try one more time.

    Energy, entropy and temperature are different physical quantities or properties or variables.

    Temperature is an intensive property.

    Entropy and energy are extensive properties.

    (Pressure, temperature , molar volume, specific heat, refractive index are examples of intensive properties.

    Mass, volume, enthalpy, entropy, heat capacity, pressure_volume product are extensive examples.)

    The physical dimensions (units) of energy or entropy preclude them "having a temperature".
    An example property that does have a temperature is boiling point.

    The same goes for "energy having an entropy"

    Both are nonsensical notions.
  10. Jun 23, 2012 #9
    Like units of temperature, units of velocity "m/s" constitute an intensive property. Using the same logic that you just gave here, you would then say that the idea that a mass (kg) could have both velocity (m/s) and momentum (kg*m/s) would also be nonsensical. That does not follow, because mass with velocity and momentum is imaginable, and in fact I'm looking at one right now. So it's not imaginary.

    Using your contrived argument, you would also conclude that: "1 kg of water doesn't have a boiling point, because 1 kg of water is 1 kg of mass, 1 kg of mass is an extensive property, and an extensive property cannot have an intensive property such as a boiling point." That to me sounds a bit ludicrous.
    Last edited: Jun 23, 2012
  11. Jun 23, 2012 #10
    This is also not true.

    Take soldering iron (specific heat about 0.5) heat it and apply to solid solder (specific heat about 0.15).
    Heat flows from the iron to the solder and the solder melts, although the heat capacity of the iron is more than 3 times that of the solder.
  12. Jun 23, 2012 #11
    I'm not saying that heat cannot flow from media of high heat capacity to media of low heat capacity. I'm ask about the tendency for heat to flow from a material of low heat capacity to another of high heat capacity, versus the other way around.
  13. Jun 23, 2012 #12
    Heat flows (spontaneously) from a body with a higher temperature to one with a lower temperature.

    That is one version of the second law of thermodynamics, due to Clausius.
  14. Jun 23, 2012 #13
    Let's take another simple example.

    Liquid water has about 10 times the specific heat (5) of the iron in the previous example.

    Heat the iron to 180°C and dip it into the water at 25°C.
    Yes the heat will flow from the iron to the water, as you said.

    Now cool the iron to 5°C and again dip it into the water.

    Heat will now flow from the water to the iron.

    What's changed other than the direction of heat flow?

    Thermodynamics is quite clear about this. Temperature (difference) is the property which determines the direction of the heat flow.
    Nothing else.
  15. Jun 23, 2012 #14
    What I'm suggesting is that the temperature itself could be affected by the heat capacity. The temperature could be a secondary result. Think about iron, it has a high heat capacity, but compared to what? Many solids tend to have low heat capacity as a result of many factors, including their density for example. It takes less energy to increase the temperature of most solids relative to gases. Cause and effect could work like this:

    Energy -> Various heat capacities -> Temperature differences -> Heat flows
    Last edited: Jun 23, 2012
  16. Jun 23, 2012 #15
    No, it's the other way round.

    Heat capacity varies with temperature.

    When you have a better understanding of thermodynamics fundamentals you will be able to explore the reasons for this variation.

    But you need to come to terms with the basics first, rather than argue with the second law.
  17. Jun 23, 2012 #16
    "Vary" does not mean the same as "caused by".

    "Heat capacity varies with temperature" is equivalent to saying "temperature varies with heat capacity". "Varies" is an associative term.

    What explains the difference of heat capacity between lead and gold (i.e. what is it caused by)? It's obviously not the temperature.
  18. Jun 23, 2012 #17
    Heat capacity is actually an extrapolation of a more general statistical expression. From Hill's statistical thermodynamics I derived this expression for CV. If you have the book I used eqns [2-5], [2-1], and [1-11]. Of course, there are probably better definitions but this is as far as I've gotten in the text so far.

    [itex] C_V = ( E- (\sum_j E_{j}e^{-E_{j}/{kT}} /{ \sum_j e^{-E_{j}/{kT}} }) )^{2}/{kT^{2}} [/itex]

    where E is the energy of the system
    k is the boltzmann constant
    T is temperature
    Ej is the energy of an individual molecule in the system (a function of N,V for the canonical ensemble)

    So if you really wanted to you could sum over the energy this way, knowing the temperature distribution to solve for the heat capacity. DISCLAIMER: I am a novice at stat-thermo!
    Last edited: Jun 23, 2012
  19. Jun 23, 2012 #18
    Indeed correlation does not prove causation.

    But what I can do is take a block of material and set its temperature.
    Nature does not force a temperature on me.
    Nature, however does force a heat capacity on me, I cannot set a heat capacity, which is why I looked up the values I posted earlier in Kaye and Laby.
  20. Jun 23, 2012 #19
    Also, note that the entropy can be expressed statistically as:
    [itex] kT ({\partial Q} /{\partial T})_{V,N} + k ln{Q} [/itex]
    where Q is a partition function expressed by
    [itex]Q = \sum e^{ -E_{j}/kT }[/itex]

    equations [1-33] and [1-29] in Hill. Also note that this is for the canonical ensemble. I forgot to mention this earlier, but a canonical ensemble is a 'closed' thermodynamic system governed by the number of molecules, N, volume, V and temperature, T.

    So to answer your question, yes they are very much related and dependent on eachother!
    Last edited: Jun 23, 2012
  21. Jun 23, 2012 #20
    It's no use asking questions if you are not willing to listen to the answer, as you have done several times in this thread.

    So I will leave you for tonight with the suggestion that you get a good hold of the basics so that you do not come out with more ideas that contradict the basic laws as when you suggested that something other than temperature determines the direction of heat flow.

    It is useful and perhaps important to be able to have a good idea of what is happening in a simple example system such as my iron dipped in the water or solder before you delve into the statmech that Simon and Aero are offering. Otherwise it is easy to go hoplessly astray. You need to be aware that their approach has yielded some real successes but also some spectacular failures in explaining the behaviour and properties of real materials.
  22. Jun 23, 2012 #21
    It appears that when the association between object and "region" is made, not much is said, if anything at all, about the heat capacity of the imported thermal energy itself (e.g. thermal phonons). Perhaps the "heat capacity of a solid" is actually the "heat capacity of a solid plus any imported energies". This would appear to correspond to the fact that adding heat to a mass will tend to increase the (total) heat capacity. So maybe that increase can be thought of as an imported heat capacity coinciding with the import of energy (i.e. heat capacity and thermal energy could be cotransported).
  23. Jun 23, 2012 #22


    Staff: Mentor

    One kilogram doesn't have a boiling point. I.e. mass does not have a boiling point. Water can have mass and water can have a boiling point, so there is nothing wrong with saying that 1 kg of water boils at 100°C, but there is something wrong with saying that 1 kg boils at 100°C
  24. Jun 23, 2012 #23
    Right, and the reason has nothing to do with extensive vs. intensive properties, as was being claimed earlier. 1 kg of water is as "extensive" as 1 kg of mass itself. The former implies a boiling point (given that pressures are not extreme), while the latter does not.

    It is in this sense where I was suggesting that an energy could "have a temperature". Some people expressed some disagreement regarding that claim, saying that it makes no sense for an energy to have a temperature. However, if a mass can have a temperature, why not energy itself (whether in the form of large numbers of particles, quasiparticles, or waves)?

    What I'm questioning is the necessity of having a boundary-defined thermodynamic system be attributed with thermodynamic variables to explain thermodynamic phenomenon. I'm also questioning how such attribution could be distorting the relationship between entropy, energy, temperature, and other thermodynamic variables, and I'm thinking that these thermodynamic variables are better attributed to each other mutually, without regard to heat and mass flow to and from boundary-defined "systems". Such a change could lead to more accurate models, where no one talks about a "reservoir" having a temperature or about any such approach which seems to ignore inhomogeneities and anisotropies in thermodynamic systems.

    My concern is for systems in non-equilibrium thermodynamics, where a simple temperature function seems unobtainable, which might be due to the fact that the "heat reservoir" concept still sticks to some minds and which also manifests itself in the equations assumed by equilibrium thermodynamics, which might be shown to be inadequate.
    Last edited: Jun 23, 2012
  25. Jun 23, 2012 #24


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    Temperature can be thought of as energy per state. Doubling the mass, at the same temperature, doubles the energy and the number of states. In a gas mixture, monatomic gases have three states for storing energy (x, y, and z velocities) while diatomic gases have 5. Hence diatomic gases have a higher specific heat.
    If we add heat to a body mass M, raising its temperature from T0 to T1, and it has constant specific heat H, its gain in entropy is M H log(T1/T0). So you could think of s.h. as an entropy gain factor.
    One of the posts (considering oceans versus a kettle) implied that a colder body has more entropy than warmer one of the same mass etc. This is wrong. Its rate of increase of entropy as heat is added will be higher, but its total entropy will be lower.

    The whole discussion about intensive versus extensive can be resolved fairly simply: instead of suggesting that "energy has a temperature", try "energy per unit mass has a temperature". I'm not saying that's valid either, but it has a better chance.
  26. Jun 23, 2012 #25


    Staff: Mentor

    A mass cannot have a temperature. If you believe otherwise then what is the temperature of a mass of 1 kg?

    Similarly energy cannot have a temperature. If you believe otherwise, then what is the temperature of an energy of 1 J?

    I share your frustration with respect to non-equilibrium thermo, but I think this approach is a non-starter. I would look elsewhere.
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