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

1. kmarinas86

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:
http://en.wikipedia.org/wiki/Heat_capacity#Table_of_specific_heat_capacities

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. Simon Bridge

15,053
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.

Why not?

4. Studiot

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.

5. jartsa

566

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)

6. kmarinas86

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
7. kmarinas86

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?

8. Studiot

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.

9. kmarinas86

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
10. Studiot

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.

11. kmarinas86

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.

12. Studiot

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.

13. Studiot

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.

14. kmarinas86

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
15. Studiot

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.

16. kmarinas86

"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.

17. Aero51

546
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.

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

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
18. Studiot

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.

19. Aero51

546
Also, note that the entropy can be expressed statistically as:
$kT ({\partial Q} /{\partial T})_{V,N} + k ln{Q}$
where Q is a partition function expressed by
$Q = \sum e^{ -E_{j}/kT }$

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
20. Studiot

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.