Talk:Heat capacity

If C = Cth and s = Cp, then surely 'Specific heat capacity' and 'Thermal capacitance' are one and the same? If so, then this article needs further restructuring. Ian Cairns 20:23, 17 Aug 2004 (UTC)

The notation I learned in thermodynamics was pretty straightforward and simple. We used $C_{v}$ as the constant volume heat capacity, and $C_{p}$ as the constant pressure heat capacity. These are the extrinsic heat capacities. We used $C_{m,v}$ as the molar constant volume heat capacity, and $C_{m,p}$ as the molar constant pressure heat capacity. In my opnion that's a great way to do it, systematic and unambiguous. IMO, Thermal capacitance sounds like an outdated term. Edsanville 23:24, 22 Aug 2004 (UTC)

Dimensionless heat capacity

Hi Gene. Why do you delete my addition on dimensionless heat capacity? Bo Jacoby 08:33, 17 October 2005 (UTC)

this part is inconsistent with other definitions in Wikipedia. For instance, see at the end of the article, as well as in specific heat capacity, the definition that engineers use all over the world. I'll redelete it. For something like what you define, I'd recommend using a term like molar hear capacity.ThorinMuglindir 00:59, 30 October 2005 (UTC)

Note that the ideal gas law says that PV/T=nR is the amount of matter measured in joule per kelvin. Heat capacity is also measured in joule per kelvin. Wikipedia is for everyone - not only for engineers who have grown accustomed to unnecessary complications. Bo Jacoby 08:10, 31 October 2005 (UTC)

Engineers do not like to make things complicated: keep in mind heat capacities are usually measured. For the engineer who does the measure, it is easier and less prone to mistakes to study one kilogram of matter than to study one mole of matter (in the latter case you have extra sources of mistake in the evaluation of the mole). For heat transfer models, the engineer who does the model generally has to multiply this value of specific heat by the value of density to get the volumetric heat capacity, which he uses directly in his calculation. Note that the density is also easier to obtain experimentally than the volume per mole. Engineers like to have it easy.ThorinMuglindir 09:47, 31 October 2005 (UTC)

Physicists working on statistical physics or condensed matter generally do not like to work on a per mole basis either, they prefer to work on a per molecule basis. Their formuilas involve $k_{B}$ rather than $R$ ThorinMuglindir 09:47, 31 October 2005 (UTC)

there remains chemists, whom I admit like to work with moles. ThorinMuglindir 09:47, 31 October 2005 (UTC)

But, keep in mind on this subject of heat capacity, we have:

• an article on heat capacity, an extensive quantity defined for a system.
• an article on specific heat capacity, an intensive quantity defined per unit of mass in the system.
• an article on volumetric heat capacity, an intensive quantity defined per unit of volume in the system. ThorinMuglindir 09:47, 31 October 2005 (UTC)

Given this, don't you think the quantity you have defined is best put in an article named molar heat capacity? Such an article could begin with molar heat capacity (which is most commonly used by chemists), then continue with your dimensionless molar heat capacity.ThorinMuglindir 09:47, 31 October 2005 (UTC)

Note that for the moment molar heat capacity is just a redirect to heat capacity ThorinMuglindir 09:51, 31 October 2005 (UTC)

There are many units of measurements of amount of matter. Macroscopic units: (kg, mol, liter, joule per kelvin) and microscopic units (AU, molecule). Each unit of measurement give a concept of specific heat: per kg, per mol, per liter, per joule per kelvin, per AU, per molecule. As densities of substances depend on pressure and temperature, the liter is no good for defining the amount of matter. Mass is not a thermodynamic concept, but rather a dynamical concept, so the kg is no good either. The mol depend on the mass. So the natural thermodynamic measure of amount of matter is the joule per kelvin of an ideal gas. That leads to the dimensionless heat capacity. The dimensionless heat capacity of a monatomic gas is 3/2, that of a diatomic gas is 5/2, that of a crystal is 3. This is simple. The gas constant R or the number of moles n represent unnecessary complications. Surely engineers do not like complications, but once they have learnt to deal with it, they don't object anymore. I don't think the dimensionless heat capacity requires an article of its own, unless small additions to existing articles make people delete it. Bo Jacoby 12:54, 31 October 2005 (UTC)

OK so you say that all intensive versions of heat capacity are specific. This might be a correct alternative way of defining them, but it don't think it is the most common, and in any case it is not the choice that has been done in this enclopedya (choice which is not mine: it largely predates my own first editions of articles in wiki). We have to remain consistent: if we decide that specific not only refers to mass, then we have to rewrite heat capacity articles, so that specific heat capacity becomes specific massic heat capacity, and volumetric heat capacity becomes specific volumetric heat capacity. That would be consistent but rather unusual and heavy notations, which is why I don't think many people would support such change. If now we stay consistent with those terms that are already defined, your quantity would be molar dimensionless heat capacity. I do realize that your quantity has some sympathetic properties, and I personally do not oppose it being put in an article or the other, it's basically the use of the word specific in its name that bothers me as possibily confusing readers.ThorinMuglindir 14:08, 31 October 2005 (UTC)

Thank you for your comment. I agree that the word specific need not be used as a synonym for intensive. I prefer the term dimensionless heat capacity. It is sufficiently descriptive without being too heavy. I don't like the word molar in this context, however, because it refers to the mol, which is not the point. On the contrary, I would like the mol to disappear from the SI. The systematic unit of measurement of an amount of matter is the joule per kelvin. This very important unit should have a name of its own. I suggest the clausius. Entropy is also measured in clausius, so intensive entropy (or specific entropy) is dimensionless. But that is another story. Bo Jacoby 15:01, 31 October 2005 (UTC)

dimensionless heat capacity seems better finally, yes, because that quantity is not dependent on the mole (I missed that at first). In order to show it, I suggest you define it is this way:

* $c_{v}={\frac {C_{v}}{nR}}={\frac {C_{v}}{Nk_{B}}}$,

where N is the total number of molecules in the gas.

YES exactly. Bo Jacoby 11:33, 2 November 2005 (UTC)

I don't understand why measuring amounts of matter in J/K makes more sense than measuring it in moles. Measuring it in J/K implies that the heat capacity of the same amount of any material would be the same, which is totally false, and only true for ideal gases which don't exist anyway. Ed Sanville 15:15, 2 November 2005 (UTC)

Hi Ed! If a substance consists of molecules, then a molecular mass can be determined, and then one can compute the number of mols by dividing the mass by the molecular mass. Some molecules are in the state of an ideal gas at sufficiently low pressures and not too low temperatures. The amount of ideal gas is pV/T, measured in J/K. The conversion factor between these two units of measurement is the gas constant, which is known with many significant digits, so ideal gases exist with that precision. That heat capacity C and amount of matter nR are measured in the same unit does not imply that they have the same value, but merely that their ratio C/nR is dimensionless. Using J/K instead of mol removes the R from the formulas. Bo Jacoby 08:14, 4 November 2005 (UTC)

PAR's edit

1. There is no such thing as heat content.

Right - That was a correction to the previous page that I neglected to make. I think its fixed.

2. Energy is not a unit. Bo Jacoby 13:06, 1 November 2005 (UTC)

Ok, I have changed it. Let me know how it sounds. Also, now that you mention it, I wonder about "the ability of matter to store heat". If you come up with a better statement, please fix it. PAR 14:55, 1 November 2005 (UTC)

I edited a little, hoping you agree. Bo Jacoby 08:31, 2 November 2005 (UTC)

Specific heat capacity at constant pressure

This section can be improved. The 'constant pressure' restriction is not used. The Dulong-Petit rule refers to molar specific heat, not to volumetric specific heat. I intend to edit when I get the time. Any objections out there? Bo Jacoby 11:33, 2 November 2005 (UTC)

I don't know if that is what you're referring to, but just this morning I noticed that the article speaks twice of Dulong-Petit law:

• near the end of the article there's a section dedicated to it, I believe you are its author, and the section is correct. BTW it makes (appropriate) use of your dimensionless heat capacity, so that it is worth defining it in this article.
• in the section about specific heat capacity, it also speaks of Dulong-Petit, but this time what it says is wrong... This part should be deleted. No need to replace it, because the subject is covered at the end of the article. Regarding the comparisons of the Dulong-Petit prediction to real values of Cp in solids, I guess it is probably not worthy correcting it and moving it to the end of the article. I'd just remove it: such a comparison is probably better placed in the article about Dulong-petit itself.

So no I don't object to you doing this editing. Just be sure to remove the wrong part and leave the right one.192.54.193.37 14:03, 2 November 2005 (UTC)

Hey that was me who forgot to log on again.ThorinMuglindir 14:09, 2 November 2005 (UTC)

Specific Heat

Do we really need a whole lot on specific heat capacity when there is a separate article on the subject? Also, I am reading this article: http://bayes.wustl.edu/etj/articles/gibbs.paradox.pdf which is really excellent, and I would like to incorporate some of the ideas in this article into some wikipedia articles. If you get a chance, take a look at it. PAR 19:21, 2 November 2005 (UTC)

this article only a short section about specific heat capacity. All it does is link it with the heat capacity, which is pretty much necessary to have. What I feel we should maybe be put in another article is the part about the diatomic ideal gases. I think we should have a specific page about the diatomic ideal gas (or does it exist?)

About your document, The the next thing I'm going to do now after I finish a couple small edits is to dive into the Microstate (thermodynamics) article where I'll explain how to count them in (quantum and classical) statistical mechanics. This is going to involve discernability, which apparently your document is all about. AFAIK the implications of the exclusion principle on thermodynamics are not covered at all in this encyclopedia, and I need them if I am to touch certain systems or quantities. Either we could work together on the microstate article, or we can separate that in two, one article that focuses on translating the exclusion priniple for stat mech, one article that focuses on applying this to counting microstates. Or, maybe you are planning to take it from a more historic angle... In fact I hardly only read the first page of your doc.

• I agree we need to mention the relationship to specific heat capacity, I just don't think an entire section needs to be devoted to it.

If you wish you can group all the intensive definitions into a single section, I have no problem with that.

• I don't know of an article on diatomic ideal gases.

if it doesn't yet exist once I have finished micro-states, and a couple other things, I'll make it. It yields good understanding of the passage from quantum to classical stat mechThorinMuglindir 22:14, 3 November 2005 (UTC)

• The exclusion principle effect on thermodynamics should be in the Fermi gas article, don't you think? This is an article I was thinking of adding to.

you can treat the exclusion principle exactly everytime you make a calculation in the grand-canonical ensemble (so that's thermodynamics of metals, semi-conductors, surely plenty of other systems that'd deserve to appear here. Fermi gas is only the simplest). Then, for fermions and bosons alike, and even in the classical limit there's also the effect of quantum symmetry of states for undiscernable particles. This is so general that it needs be adressed generally. For example, if you do a calculation in the canonical or micro-canonical ensemble, you also need to take the exclusion principle into account, otherwise your entropy is probably off. I say "probably" because to complicate things even more, there are cases where the exclusion principle sort of takes itself into account naturally (solids, or more generally systems with a network of fixed sites), so that adding those terms that you usually use to describe the exclusion principle leads to a faulty result. Add to that quantum symmetry of states is also by itself a complex concept. For statistical mechanics, you can simplify it slightly by saying that |psi1>|psi2> and |psi2>|psi1> are in fact one and the same state, rather than defining the ugly permutations. The microstates article will either speak of all this, or stay more focused on the practical aspects of counting microstates, which still necessarly involves speaking of the exclusion principle, because that's where it matters. Though I could use another article as a reference on a number of points.ThorinMuglindir 22:14, 3 November 2005 (UTC)

• I think one microstate article is best. PLEASE NOTE - I am going to change the microstate (thermodynamics) article to microstate (statistical mechanics) because microstate is not a thermodynamic concept, it is a statistical mechanics concept.PAR 01:53, 3 November 2005 (UTC)

good move.ThorinMuglindir 22:14, 3 November 2005 (UTC)

I don't mind having a short section on specific heat capacity. I made it refer to the main article. I re-instated the definition of the dimensionless heat capacity in analogy to the definition of the specific heat capacity. I'd like to simplify the next section on the heat capacity of monatomic and diatomic gases. Should this section be moved to the article on specific heat capacity or perhaps have an article of its own ? Bo Jacoby 07:54, 3 November 2005 (UTC)

I'm taking your question in a large sense - should the main article be "heat capacity" and have "specific heat capacity" defer to it, or vice versa? I am in favor of the "heat capacity" article being the main one, with all the main results there, and then "specific heat capacity" deferring to it. I also favor keeping everything in one article, absolute zero, gas phase monatomic, diatomic, solid phase, etc. Maybe in the future it will get unwieldy, but it seems ok now. PAR 12:06, 3 November 2005 (UTC)

Is Cx necessary ?

The heat capacity is undefined until it is specified which parameter is held constant, volume or pressure. So C means either C_V or C_p. So does C_x. Why not choose the shorter and simpler version ?

I changed P for pressure into p according to the article pressure, but it is not universally so.

We have to keep the x thermodynamic variable because the statement

$S(T_{f})=\int _{0}^{T_{f}}C(T)dT/T$

makes the following errors:

• Any thermo quantity (e.g. S and C) are generally a function of two variables, not one
• The above says that the integral of ANY C/T over temperature gives the same result, which, given that all C's are zero at T=0, means all C's are the same function, which is not true.

We have to have

$S(T_{f},x)=\int _{T=0}^{T_{f}}{\frac {\delta Q}{T}}=\int _{0}^{T_{f}}\left({\frac {\delta Q}{dT}}\right)_{x}{\frac {dT}{T}}=\int _{0}^{T_{f}}C_{x}(T,x)\,{\frac {dT}{T}}$

The expression δQ/T in the first integral means "over some reversible path". The limits of integration imply that one of the variables describing the path is temperature, but the other is left undefined. The expression $(\delta Q/dT)_{x}$ in the second integral specifies that the second parameter is x and is held constant over the path. The path is now fully defined. The third integral follows by definition of $C_{x}$. $C_{x}$ MUST be written as a function of T and x in the last integral, because x is being held constant while the integration is carried out. As a result, you get $S(T_{f},x)$. If you pick another x, you get a different $C_{x}(T,x)$ and therefore a different $S(T_{f},x)$.

The state of an incompressible body is a function of one variable only. Choose T or S or U or H as you like it. In this case the C is well defined. The state of a compressible body is a function of two variables. For the second variable, choose V or p. In this case C is not defined. The two-dimensional state space is restricted to a one-dimensional case by fixing some function of T,S,U,H,V,p. When this is done, you may proceed as before. The reader cannot reconstruct your subtle thoughts simply by reading the subscript x. The third law does not depend on the dimensionality of the state space, so all this is a little besides the point. Bo Jacoby 13:40, 3 November 2005 (UTC)

I don't understand your point. The section is meant to show that the specific heat with any appropriate thermo variable held constant is zero at absolute zero. No assumption of constant pressure. The previous statement was wrong. I'm trying to make it right. The reasoning may be subtle, but that is no reason to replace it with a falsehood. If the reasoning is too obscure, please, lets fix it.

This is a quibble, but it might explain why I don't act as if I understand "C" without a subscript. I am thinking that for any body, all Cx's are defined, and for an incompressible body, Cp=Cv. An incompressible body is a special case of the more general situation.

I don't understand the relevance of the statement "The third law does not depend on the dimensionality of the state space, so all this is a little besides the point." PAR 14:22, 3 November 2005 (UTC)

Each dimension of state space correspond to a path for energy to enter into the body. An incompressible stone has a onedimensional state space: dU=TdS. A spring has also a onedimensional state space: dU=-Fdx where F is force and x is length. A compressible body of gas inclosed in a cylinder by a piston has a twodimensional state space dU=TdS-pdV. It has heat capacity like a stone, and mechanical energy capacity like a spring. At any rate, the entropy is finite when the temperature tends to zero. That is the third law. It does not depend on the dimensionality of the state space. Is this helpful ? Bo Jacoby 14:50, 3 November 2005 (UTC)

Yes, I understand the meaning of the statement - but I thought you had an objection to the way the "Heat capacity at absolute zero" section was expressed. PAR 17:16, 3 November 2005 (UTC)

Why not move your section on "Heat capacity at absolute zero" up before the section on compressible bodies, and then remove the x in order to improve readability? Happy editing ! Bo Jacoby 07:17, 4 November 2005 (UTC)

Ok, I don't know how to explain this any better. Please read the following carefully, I have tried not to make it too messy. If its clear, but wrong, then correct me. If its not clear, ask me to clarify it. If its right, then lets fix the article.

Explanation 1 (Quick) - Since we are working with compressible substances, everything is a function of two parameters. Suppose we have the two specific heats, both expressed as functions of temperature and pressure: $C_{V}(T,p)$ and $C_{p}(T,p)$. If we plug them into a "no-x" equation the way you want, we have:

$S(T,p)=\int _{0}^{T_{f}}{\frac {C_{V}(T,p)}{T}}dT=\int _{0}^{T_{f}}{\frac {C_{p}(T,p)}{T}}dT$

This must hold at ANY $T_{f}$. Do I need to go through the proof that this will imply that $C_{V}(T,p)=C_{p}(T,p)$? The above is obviously false.

Explanation 2 (Not so quick) You have to think in terms of the PATH OF INTEGRATION through the two dimensional (T,p) space. The statement

$S(T,p)=\int _{T=0}^{T_{f}}dS=\int _{T=0}^{T_{f}}{\frac {\delta Q}{T}}$

HAS MEANING because $dS$ is an exact differential dS, and only depends on the endpoints, which are specified by 0 and Tf. Expanding dS gives:

$S(T,p)=\int _{T=0}^{T_{f}}\left({\frac {\partial S}{\partial T}}\right)_{V}\,dT+\int _{T=0}^{T_{f}}\left({\frac {\partial S}{\partial V}}\right)_{T}\,dV$

Each integral now DOES NOT HAVE MEANING because they are not exact differentials when separated. They are now path integrals, and much each be integrated over the same path. So lets specify the path. Lets say that the path is such that the volume remains unchanged. If we think of things in (T,P) space, then the path is some squiggly line (an isochore, a path of constant volume) from (0,0) to (Tf,pf), where we have to figure p_f out from the equation of state. Note that if we were in (T,V) space the path would be much simpler, a straight line from (0,V) to (Tf,V) where V is the known volume. But we have chosen (T,P) space, so lets stick it out. Having specified the path it is clear that the first integral can be written in terms of Cv and the second is zero. However the following statement is WRONG

$S(T,p)=\int _{0}^{T_{f}}{\frac {C_{V}(T,p)}{T}}dT$

Because it implies that the path of integration holds p constant. We must hold v constant, because then it will be along the v=constant path. We need to integrate

$S(T,p)=\int _{0}^{T_{f}}{\frac {C_{V}(T,p(V,T))}{T}}dT$

while holding V constant. Another way of saying it is to simply choose our two variables to be T and V to begin with and say

$S(T,p(V,T))=\int _{0}^{T_{f}}{\frac {C_{V}(T,V)}{T}}dT$

and finally, lets just say that S(T,p(V,T)) is just S(T,V) so we finally have:

$S(T,V)=\int _{0}^{T_{f}}{\frac {C_{V}(T,V)}{T}}dT$

where the integration is carried out holding V constant. The subscript on the C must match the second argument of the C and the second argument of the S. Thats why I wrote C_x(T,x) and S(T,x). PAR 17:50, 4 November 2005 (UTC)

A few problems I have with this article as it stands now

Neither the word equilibrium, nor the word reversible appears in the article. It needs to mention that all definitions involving heat are true only for a reversible transformation, probably voicing it more than the small addition I had left last time, and which was removed. The other definition involving S is true in the differential sense as long as a transformation goes from an equilibrium state to another (not necessarly pasing through a series of equilibrium states), which is the "normal" situation as far as thermodynamics are concerned, so that extra care isn't, I think, needed in this case. Another option is to mention delta Q not as the amount of heat exchanged in an imaginary reversible tranformaiton, rather than the amount of heat actually exchanged by the system, but I don't find this very satisfactorly.

The formula at the start should also mention that this is for incompressible bodies. We can either say incompressible, or say "for a liquids and a solids," at the start, introducing incompressibility later. That should remain quite simple.

Otherwise C can be derived as a partial derivative of the internal energy.

I agree we should be clear about the importance of reversibility.

With regard to compressible vs. incompressible - I think this article should be about the thermodynamic concept of heat capacity. It should NOT be about the special case of heat capacity for incompressible bodies. On the other hand, to start out with complete generality may not be the best idea, because it can be difficult for a beginner to grasp. I think the introduction could start out by introducing the concept of heat capacity without the idea of holding N-1 parameters constant where N is the dimension of the thermodynamic space you are working in. Then, the introduction could point out the difficulties encountered with the simplistic view when dealing with compressible bodies, and introduce Cp and Cv. Then finally, it could point out the fact that N-1 parameters must be held constant.

where to speak of this is in the "thermodynamics equations" article

I don't think it is too much of a problem to then write the subsequent sections in terms of Cp and Cv as a compromise between simplicity and generality. A one-dimensional thermodynamic space is too simple, it leaves out too much. Almost all the wikipedia thermodynamics articles are written assuming a 2-dimensional space (P-V or P-T or whatever) and we should do the same. Incompressible bodies have a separate section. A final section could introduce heat capacities for spaces with more dimensions as the final generalization.

Another option is to make the final generalization in the thermodynamics equations article. One I'll edit, but first I have others to finish.Just mention in this article that for the expressions are different for a body who exchanges work with the outside in another form than pressure work.ThorinMuglindir 18:45, 6 November 2005 (UTC)

I have a question, since this has become the last place where one speaks: do van der waals effects play a big role for hydrogen, oxygen and nitrogen? Or, are they just half-classical half-quantum ideal gas (depending on the degree of freedoms you are looking at)? We'll need to get that info for the diatomic ideal gas article, so I wondered if someones already knows. It's easy to write bullshit on such questions.ThorinMuglindir 18:45, 6 November 2005 (UTC)