Talk:Heat capacity/Archive 1

Thermal capacitance

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 IMO, Thermal capacitance sounds like an outdated term.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)

• "The Dulong-Petit limit results from the equipartition theorem, and as such is only valid in the classical limit of a microstate continuum, which is a high temperature limit."

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) Secific heat is also called as Heat capacity. —Preceding unsigned comment added by 218.248.9.222 (talk) 03:31, 21 June 2010 (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

Explanation 1 (Quick) - Since we are working with compressible substances, everything is a function of two parameters.

No. The article is about heat capacity. Heat capacity is a property of a 'heat capacitor', a system (or a 'body' or a 'device') that is capable of receiving and delivering some heat energy on the expense of raising and lowering the temperature a little. It is the analog to an electric capacitor that is capable of receiving and delivering some electric energy on the expense of raising and lowering the voltage a little. The heat capacitor may be made of an incompressible substance, and in that case there is no complication from pressure or volume. It may also be made of an compressible substance such as a gas. That can be done in several ways. Either enclose the body in a strong container to make the volume constant, or in an open container to keep the pressure constant, or some other device such as a cylinder with a piston loaded by a spring, to keep some function of volume and pressure constant depending on the equation of state of the spring. Once the device is specified, the heat capacity of that device is defined. A mixture of ice and water constitutes a heat capacitor with infinite heat capacity, because it can be heated without rising the temperature. This is analog to an electric element which can be charged without rising the voltage.

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.

No, the formula for the entropy at temperature $T_{f}$ of a heat capacitor is simply

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

There is no second variable in a heat capacitor. From this the third law of finite entropy implies that the heat capacity is zero at zero temperature, for any heat capacitor. The complications of the detailed design of the heat capacitor does not enter into this result, nor should it confuse the explanation. You are not wrong, but you are confusing things. Bo Jacoby 11:31, 7 November 2005 (UTC)

The heat capacity of a gas at constant pressure is a function of that pressure which is being held constant, and the temperature. Change either the temperature or the pressure or both, and you have a different heat capacitor with different properties. The heat capacity Cp is not a function of temperature alone, nor is the entropy, which is why I say the above equation is misleading at best. PAR 16:13, 7 November 2005 (UTC)

A heat capacitor depend on many parameters: amount of substance, chemical constitution, phase, pressure or volume, and temperature. These parameters must be known before the heat capacitor is specified and its heat capacity defined. There is nothing special about pressure and volume in this respect. One heat capacitor may consist of an amount of gas at constant volume. A second heat capacitor may consist of the same gas at constant pressure. A third heat capacitor may consist of a mixture of steam and water at constant volume. At any rate the entropy of a heat capacitor is the integral of (C/T)dT. Bo Jacoby 14:58, 8 November 2005 (UTC)

Yes - with all those other parameters held constant during the process of integration. We both understand that. I say we have to make that point very clear. Our disagreement is notational (sound familiar ?:). If someone is dealing with a monatomic gas, all their concepts are in two dimensions. When you say "integral" they say "over what path?". We need to be clear that the answer is "the path where whatever constraint you temporarily chose upon C is maintained". I understand that once such a constraint is placed upon a system, we have a one dimensional system and that your notation is not wrong. But the constraint may be a temporary conceptual device. FOR EXAMPLE, how would you write an expression illustrating the difference in entropy when integrating Cp versus Cv? I'm not hung up on the "x" notation, but I am hung up on the idea that a person dealing in two or more dimensions may need to make use of Cp or Cv without permanently reducing the dimension of their thermodynamic space to one. They need to have a notation which allows them to stay in two dimensions. The x-notation provides that, yours does not. Again - I'm not defending the x-notation, just the need to make above points notationally clear. If you have a better way of doing that, lets do it. PAR 18:48, 8 November 2005 (UTC)

Yes. Heat capacity is defined for one-dimensional systems only. Consequently, a WP-article on heat capacity should focus on one-dimensional systems. Entropy is then defined for one-dimensional systems as the well-known integral. You cannot talk about the heat capacity of two-dimensional systems. You cannot even talk about the heat content of a two dimensional system, because heat can be transformed into work by a cyclic process. However, entropy is generalized to two-dimensional systems by the requirement that a reversible and adiabatic compression (or expansion) does not change the entropy. It is true that the value of entropy differences does not depend on the path of integration, but this two-dimensional fact is not related to the one-dimensional fact that the entropy of a system is finite at zero temperature. These two facts should not be mixed up. Bo Jacoby 10:59, 9 November 2005 (UTC)

What are you proposing? That no mention be made of any subscripted C? Because the minute you agree to introduce a subscripted C, you contradict yourself. What are you proposing? PAR 17:18, 9 November 2005 (UTC)

I regret having been inconsistent. I think that this article should be very elementary. Heat capacity is a simple concept, very analogous to the capacities of reservoirs for gas, electricity or water, where pressure or voltage is increased when the reservoir is filled, and where the capacity is the amount per pressure increase. No reference to two-dimensional systems should to be made in this article. It belongs to the article on specific heat. If this article can be happily understood by the newcomer, then we are succesful. Bo Jacoby 15:40, 16 November 2005 (UTC)

Bo - I strongly strongly disagree. Confining the concept of specific heat to one dimensional systems is far too constrictive. You are basically denying the use of the partial derivative in thermodynamics! If this article were about driving a car, then you are saying we should avoid discussing the steering wheel, and any concept of what a right or left hand turn is. Thats how strongly I object to your proposal. We don't need to confuse the beginner with partial derivatives and path integrals at the beginning, but we need to ultimately include them or else this article is for kindergarden. I have never heard of any textbook discussing the specific heat this way.

There is a difference between a textbook and an encyclopedia. Reading a textbook on thermodynamics you expect to find all the important information on thermodynamics. Reading an encyclopedic article on heat capacity you merely expect the answer to the question: "What is heat capacity and what is the use of it?" but not all of thermodynamics. If you want to know more just follow the links. Having read the article you should understand the meaning of 'heat capacity'. If you do not understand the article, the author was unsuccesful. The article on heat capacity is for those who do not already know what heat capacity is, (maybe including kids in kindergarden). 'Heat capacity' make sense for one-dimensional systems only. Two dimensional systems must be restricted to one dimension (by keeping one variable constant) before the concept applies. Partial derivatives are important to thermodynamics, but in some other article. Bo Jacoby 08:17, 17 November 2005 (UTC)

I think we have stated our points of view clearly, and now its time to get a consensus, because I think our differences on this subject are fundamental, but a matter of style and not substance. Lets get User:ThorinMuglindir and User:Karol Langner to weigh in, they have been recent contributors. PAR 15:45, 17 November 2005 (UTC)

We don't need to agree. It is all right if I did not convince you. I'm not reverting your edit. The pedagogical sequence of facts is: 1. Measurement of (relative) temperature by means of a mercury column. 2. Concept of heat - the stuff that makes temperature rise. 3. Heat capacity - the ratio of heat to temperature increment. 4. Defining the specific heat of water to be one calorie per degree per gram. 5. Identification of heat with mechanical energy by Rumford and Joule and abandonment of the calorie. All this is done with incompressible, one-dimensional systems, so this is all you need to know in order to understand heat capacity. Later developments are: 6. Concept of entropy and absolute temperature. 7. The reversible conversion of heat into work by a cyclical process using compressible (gas) bodies where mechanical and thermodynamical degrees of freedom are connected. That is where the distinction between constant volume and constant pressure specific heats belongs. Bo Jacoby 13:18, 18 November 2005 (UTC)

Like I say, we disagree. Specific heats are experimentally defined as the heat capacity of a homogeneous system divided by the number of particles. With respect to your remark below, yes, that system has to contain a single phase, but it doesn't have to be a gas. Solids and liquids have heat capacities which must be defined at constant volume or pressure as well, its just that in most cases the difference can be ignored.

Secondly, the specific heat article has no more claim to being n-dimensional than the heat capacity article. They are the exact same concept, except specific heat is divided by the number of molecules. PAR 16:35, 16 November 2005 (UTC)

Well, heat capacity is a property of a heat reservoir, while specific heats are properties of the material that constitutes the heat reservoir. You may construct heat reservoirs from the same material in many ways, such as constant volume or constant pressure. Bo Jacoby 08:17, 17 November 2005 (UTC)

No - its not one way for reservoirs and one way for materials. Specific heats are material properties, but the material has to be held under certain conditions for these specific heats to have any meaning. Heat capacities are reservoir properties, but the reservoir has to be held under certain conditions for these heat capacities to have any meaning. Its totally analogous. PAR 15:45, 17 November 2005 (UTC)

A closed bottle containing liquid and vapour is a heat reservoir at constant volume, but the specific heat make no sense. Bo Jacoby 13:18, 18 November 2005 (UTC)

Of course not, its not defined in terms of a multi-phase system, but the specific heat at constant volume of the liquid does make sense (no real physical material is incompressible). So does the specific heat at constant volume of the vapor. They make sense because they are defined in terms of systems which are completely liquid and completely gas. PAR 15:40, 18 November 2005 (UTC)

Yes, and you need to know the latent heat of evaporation too, and the vapour pressure as function of temperature. Neither the pressure, nor the volume, nor the amount of matter in the gas phase is constant. So the heat capacity, which is conceptually a very simple measurement, depend in quite complicated ways on several properties of the components of the system. The article on heat capacity need not deal with such complications, I think. Bo Jacoby 11:47, 21 November 2005 (UTC)

Explanation 2

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)

Heat capacity without an index

I'm sorry, I don't have time right now to read the whole discussion above. I understand the question is whether to leave C without and index in the definition (I'm not sure if I understand your problem)? If so, it certainly should! Heat capacity can be measured for any system at any conditions, because it is a phenomenological quantity. Karol 12:49, 19 November 2005 (UTC)

Thanks for your response. The question is whether the article should draw the distinction between specific heat at constant pressure and specific heat at constant volume.

It definately should. It shouldn't however, use specific heat too extensively, because that's another page. Karol 17:50, 19 November 2005 (UTC)

Suppose I have an ideal gas at equilibrium with a fixed temperature, pressure, volume, number of particles, etc. It is my contention that it has a particular value of heat capacity at constant volume (3Nk/2), and another value for its heat capacity at constant pressure (5Nk/2). It seems to me that if someone asks "what is the heat capacity of this system", that it is wrong to say "you have not specified the system completely until you have not only given its thermo parameters, but have also specified which of them shall be held constant in any future thermodynamic process". Thats not how systems are defined is it? PAR 16:30, 19 November 2005 (UTC)

Well, yes and no. Heat capacity is not a unique, inherent property of a thermo system, it is a quantity defined by the heat provided to the system and by the effects of that heat - so we can't really talk about a system's measured heat capacity without an accompanying process taking place. Since in physics we like to predict things, however, we say that a system's capacity is so-and-so if the pressure is constant, or if the volume is constant, and so forth - but the is should be replaced formally by would be. So the system has a number of different heat capacities (defined its possible histories) that we can use to theoretize about its behavior when placed in various external conditions. Alternatively, we can imagine two identical systems, one held at constant pressure, the other at constant volume, and conclude about the differences between them. The nice thing is, once you know these two cases (constant pressure and volume), you can, for some systems, predict the heat capacity for all possible processes (or paths in the "phase space of thermodynamic paramters" if you like :D). Karol 17:50, 19 November 2005 (UTC)

I'm sorry, but this is driving me up a wall. No - my inability to explain myself is what is driving me up a wall. But I will keep trying.

I just reverted the statment that, in effect, a system has to have its state of motion through the space of thermo parameters to be defined in order for a heat capacity to be defined.

Yes, that does soudn quite akward. Karol 17:50, 19 November 2005 (UTC)

The heat capacity Cx of a system is a state function, on a par with other extensive quantities like volume, entropy, and number of particles

I don't need to ask "what kind of process is this system involved in?" in order to specify its volume, or its entropy, or the number of particles, or the heat capacity at constant volume, or the heat capacity at constant pressure. A system does not have to be constrained to constant pressure in order for the value of its heat capacity at constant pressure to be defined. It DOES have to be so confined for that value to be measured, however. PAR 17:11, 19 November 2005 (UTC)

When you say that pressure is constant, you are already narrowing the class of possible processes taking place, namely to the class of isobaric processes. And sure, you can define quantities related to any hypothetical processes without that process actually taking place. Karol 17:50, 19 November 2005 (UTC)

I could define the volume as the change in enthalpy with respect to pressure at constant entropy (which is true). But I don't then say that volume is defined only for isentropic processes. I see now that it all boils down to the statment that the specific heat at constant volume (or constant whatever) is a state function. Once you accept that, then you have accepted that its value is in no way related to the particular path (process) the system is taking through the thermo parameter space. By definition. If you require that the specific heat at constant whatever be defined only for certain paths (processes), then you deny that it is a state function. And I'm quite certain its a state function. I will try to find a reference though. PAR 18:42, 19 November 2005 (UTC)

Yes, heat capacities are function of state. Karol 11:31, 20 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)

Hydrogen, oxygen and nitrogen can be liquefied and solidified. Ideal gases can't. So the nonideal effects play a big role at low temperature. Bo Jacoby 11:40, 7 November 2005 (UTC)

Specific heat 2

I don't understand why there's such a lengthy discussion about specific heat here, since there's a separate page on it? Karol 13:40, 19 November 2005 (UTC)

Not enough for the layman and too much emphasis placed on theoretical models

I think this article requires cleanup but not for any of the reasons described above. Concepts of reversibility and the nature of processes and systems should not be emphasized in an article about a property of matter. But "heat capacity" isn't like many other concepts in thermodynamics ("Gibbs free energy" and "Carnot cycle" come to mind) because heat capacity is something that a layman can intuitively understand by the words themselves, and those words correspond with the technical description too. The article needs to go from that level of understanding to the thermo much more gently by including applications and more content for the general reader (and, I think, less content for people who already understand heat capacity). I added the tag to this talk page that says not to remove technical details, but I would support the removal of some technical details. I suggest leaving the take-home message of the final section and moving the content itself elsewhere, perhaps as an application of the theories or models. I liked this content and it belongs in wikipedia. I just didn't think it belongs here in this level of detail, and I'm not knowledgeable enough to move it myself. Flying Jazz 17:02, 7 January 2006 (UTC)

Agree, the first equation in the article does not even explain the parameters in the equation. The term entropy is introduced without linking to the article on the subject. --Renier Maritz 10:46, 18 August 2006 (UTC)

Use of the word "specific"

This applies to the two questions above. Yes, there's a lengthy discussion of specific heat on the Wiki page for that, BUT the problem is that this is a sloppy term and should be fixed up. "Specific heat capacity" really is a short and sloppy term for mass-specific heat capacity.

"Specific" in science and engineering is much the same kind of term as "intensive" (as somebody above points out), but it's a bit more broad. What it really means is that your quantity has been divided at some point by a reference standard, so that your answer comes out as a sort of comparison. Thus, the original idea behind "specific" gravity of water is that you divide by the density of water to give you a dimensionless ratio of density of what you're interested in, to that of water. Similarly, dividing by mass gives "mass-specific" quantities, such as "mass-specific metabolic rate" (this is larger for a mouse than an elephant, even though the elephant obviously has a much larger gross or plain "metabolic rate"-- burning more calories per elephant per time). So mass-specific terms aren't confined to engineering. And it's NOT always dimensionless. In fact, usually it's not.

"Volume-specific" has the same caveat. It just means you divide by the volume. It generally implies that this volume is constant, else which value would you divide BY? However, it's usually used for solids and liquids where expansion due to heat is not much of an issue.

Both "mass-specific heat capacity" ($C_{m}$) and "volume-specific heat capacity" ($C_{v}$) are used extensively in engineering, as terms for solids and liquids (diffusivity = conductivity/($C_{v}$)= K/(density*($C_{m}$), for example). However, note that this ($C_{v}$) isn't the same as the used as the constant-volume heat capacity for a gas, which is abbreviated often the same ($C_{v}$) but doesn't become an intensive property until further divided by some measure of material (mass or moles or molecules or whatever). This is what the previous poster (Edsanville) was talking about in using $C_{v,M}$ as the molar constant volume heat capacity, and $C_{p,M}$ as the molar constant pressure heat capacity (the analogous mass-specific terms could simply use m instead of M). Again it is possible to talk about either volume-specific (or mass-specific) constant-volume heat capacity of a gas. Simply saying the constant-volume heat capacity vaguely implies that it's an intensive quantity, but not till you say for sure. It might not be! You can talk about the constant-volume heat capacity of a whole building full of air, for example.

The same goes for $C_{p}$ which isn't an intensive property, unless divided by a material measure, i.e. $C_{p,m}$. The $C_{p}$ for a building full of air is larger than for a cup full of air.

"Specific gravity" is actually a sort of "water-specific density". But you'll never see that term used, as the original is too archaic. So it's usually simply defined directly.

BTW, The poster who is pushing this C* idea is wrong about mass not being a thermodynamic quantity. Mass is the same as energy, and energy is what the first law is about. The statement that mass is "dynamical" is wrong, and is a confusion of the old conflict in what term to use in special relativity for rest-mass (invariant mass) and relativistic mass. See Mass in special relativity. In the end, rest-mass won out as the synonym for "mass" in this debate, and that's now what we *mean* (by accepted definition) when we say "mass." It's invarient. It observer-INDEPENDENT. It's the same as total energy (just multiply by c^2). Thus, it's a perfectly good thermodynamic quantity. You can measure it in kg or J/c^2 or whatever your favorite SI units are.

The "dimensionless" heat capacity is sort of "R-specific" or (if you will) $k_{B}$-specific per particle. It gives heat capacity per particle in "natural units," if you wish Bolzmann's constant $k_{B}$ to be your natural unit. But this is not very satisfying to people used to working in M.K.S. It's like expressing speeds in terms of the speed of light. In those units, the speed of light is a unitless 1, and mass = energy with no proportionality constant. Only relativists like this; it leaves other scientists and engineers cold.

Anyway, the answer to Flying Jazz is that perhaps these points should be addressed in the "upfront" section the heat capacity article (to make it accessible), and more technical stuff including partial diff equations, put in later on. Sbharris 21:22, 9 March 2006 (UTC)

Not using italic variables makes your treatise look amateurish. Gene Nygaard 21:49, 9 March 2006 (UTC)

Only if the reader is unable to distinguish between physics and typesetting. But I've added a few just for you. Sbharris 19:39, 10 March 2006 (UTC)

Sbharris, The dimensionless heat capacity is neither "R-specific" nor "$k_{B}$-specific". It gives heat capacity per amount of matter. The SI-unit of heat capacity is joule per kelvin, and the SI-unit of amount of ideal gas is the same. So the dimensionless heat capacity is truly dimensionless. This is very satisfying to people used to working in M.K.S. It's not at all like expressing speeds in terms of the speed of light. Bo Jacoby 10:23, 22 March 2006 (UTC)

COMMENT:

As you've defined the "dimensionless heat capacity" in this article it's not dependent on the amount of matter, and thus is an intensive quantity. As such, it doesn't really belong in this article *at all*, but probably should be moved to the article on specific heat. This article talks about a quantity of a system or body called "heat capacity," which by definition gets larger as you increase the amount of matter.

As for heat capacity being "dimensionless," it's dimensionless only if you choose to measure it in units which you've defined in some system (which isn't S.I.) as having no dimension. That's fine, but it's not S.I., as you note. Measuring heat capacity in dimensionless units derived by dividing by number of moles and also by R, amounts to actually talking about specific heat in a new system where R has been arbitratily given the unitless value of 1. That would make sense only if we could measure R to greater precision than we can now measure energy, and the scale-points in the scale which we use for temperature, but such is not the case. Thus, I fail to see the point of such a system. We don't need a new temperature scale right now, and we certainly don't need a new system of units based on R.

R has the value of joules/kelvin in the S.I. system for the very good reason that we don't want to give up either joules or kelvins, as we presently define them using mass for joules and triple point of water for kelvin. FYI, we actually can't measure R (or $N*k_{B}$) directly-- instead various thermodynamic thermometers measure RT. The most accurate of these are actually not gas thermometers (PV = nRT) but rather acoustic thermometers (speed of sound measured) or shot-noise themometers in which thermal noise of a voltage drop across a resistor is measured. If you want a T scale, you can either arbitrarily fix R at some exact definitional value (as we've done with the speed of light) and just go ahead and use the resulting T scale; OR else you can measure RT at some T which you define by physical conditions (like the triple point of water) and arbitratily fix THIS point (as we do in the kelvin scale), then use that scale to MEASURE R (in terms of the other scale). We presently use the latter method, because T at the triple point of water is reproducable to at least 3 parts in 10 million accuracy, whereas absolute RT as measured by various thermodynamic thermometers (and thus by implication the magnitude of R if you were to choose this value to make your T scale) is not reproducable to better than 2 parts per million (given the various best methods). We've chosen to construct our T scale the latter way, therefore, and that's why R (and $k_{B}$) are good in value only to 2 parts per million or so. As thermometers get better this will improve, but meanwhile we keep the kg and the triple point of water as our references, which is the way we want things. It doesn't make sense to use R for our basic reference and scale (as your "unitless" heat capacity system does) until we can reproduce some measure of it more accurately that we now reproduce the references we use to set up our energy and kelvin scales. Until then, heat capacity is going to be in joules/kelvin. And better so. Sbharris 02:35, 24 March 2006 (UTC)

Sbharris. Your information on state-of-the-art technology for measuring temperature is interesting and might be included in the article on temperature. I agree that this article on heat capacity should not be about specific heat, but PAR didn't agree and he got his way. I think that you misunderstood the dimensionless heat capacity. I am not giving up either joule nor kelvin, but I'd like to give up the mole (unit). The SI has two units of measurement of amount of substance, one being the joule per kelvin and the other being the mole. The conversion factor between these two units is the gas constant R. The dimension of R is not joule per kelvin, but rather joule per kelvin per mole. From a logical or pedagogical point of view there is no need for two units of measurement for the same quantity. The appearance of R in the formulae is a confusing complication. So the logic goes like this: Define joule (based on MKS) and kelvin (based on the triple point of water). Define pascal = joule per m³. Define amount of ideal gas, pV/T, in joule per kelvin. Define heat capacity in joule per kelvin. Define dimensionless heat capacity in (joule per kelvin)per(joule per kelvin). Forget the mole and the gas constant. Bo Jacoby 08:46, 24 March 2006 (UTC)

COMMENT:

I don't know where you came up with the bizarre and just plain wrong notion that the S.I. uses more than one unit for amount of substance. It does not. The only S.I. unit for amount of substance is the mole which is the number of carbon-12 atoms in 0.012 kg of carbon-12 at rest, unbound, in ground-state energy. End of definition and end of discussion. That's the way S.I. defines it, and if you want to argue, argue with them. Even Wiki has gotten it more or less right.

It would be silly to try to use joules/kelvin as a definition of amount of substance, because a mole of substance A may have a very different heat capacity (joules/kelvin) than a mole of substance B. That's the whole point of specific heats, don't you know. Even specifying ideal gas behavior and subsuming the 3/2 factor into your value of "R" just gets us back into the above discussion of the difficulty in measuring R, as compared with joules, kelvins and (now) moles. We know the value of R to less than 6 places, but we know Avogadro's number to at least seven places. So it's (again) not smart to make the first number do the work of the second. And S.I. doesn't. So stop advocating this idea. Nobody agrees with you. Wiki is not the place to push your private ideas for codifying the science of measurement. Even if they were better ideas than are accepted in the field, which they aren't. Sbharris 19:52, 24 March 2006 (UTC)

Why are you so angry? Be nice. The joule J is an SI unit and the kelvin K is an SI unit, so the J/K is an SI unit too. The ideal gas law tells that the expression pV/T is proportional to the amount of substance. The SI unit of that expression is J/K. So J/K is an SI unit of amount of substance. However SI also defines the mole as a unit of measurement of amount of substance. That gives two SI-units of measurement of amount of substance. This is not an argument but a matter of counting to two. Everybody can do that. Surely different substances do have different dimensionless heat capacities. Bo Jacoby 15:45, 26 March 2006 (UTC)

I'm angry because you're making statements about the SI system which are wrong, and which you could easily correct if you'd spend 5 minutes reading about the SI system instead of arguing on this page with myself (and also the many, many posters above who've tried to teach you about this). You say: "The joule J is an SI unit and the kelvin K is an SI unit, so the J/K is an SI unit too." NO! There is NOTHING in the SI system which says that any unit made from two other SI units is an SI unit. In fact, there a dozen examples of older units which can be expressed in SI units which are explicitly NOT SI units. The poise, often used as a unit of viscosity, is 0.1 pascal*seconds (both SI units), but the poise is not an SI unit. For more examples see: [[1]].

J/K is not an SI unit of amount of substance. For reasons carefully explained already, J/K would not be a good unit even to use as an alternate for amount of substance, since it would require a better measure of R than we currently have, a redefined scale in terms of R. Finally, your assumption that the ideal gas law is to be taken as an exact relationship amounts to the assumption that gas thermometers are the more accurate of ways to measure R (or N$k_{B}$) . This has not been proven, and in fact there are many reasons to think it is not the case, as I've said. The ideal gas law is an appoximation to an ideal limit under certain circumstances, like Newton's "law" of gravity. Using it as a definition and thus forcing it to be true, only forces you to measure G or R in certain ways, which physicists may (or may not) prefer to. In any case, SI is what it is. Live with it. If you don't like it, argue with the SI people. Don't put falsehoods about the SI system on Wikipedia, even after many people have tried to tell you that you need to go back to your basic reading here. Okay? Sbharris 17:15, 27 March 2006 (UTC)

The rule for SI-units is simpel: "The derived units are obtained by multiplication and division of base units". [[2]]. The J/K is obtained by multiplication and division of base units. So, J/K is an SI unit according to the rule. The poise=0.1 Pa·s is not an SI-unit because of the factor 0.1, but the Pa·s is an SI-unit according to the rule. The precision of measurement is simply not the point. Bo Jacoby 10:52, 28 March 2006 (UTC)

P or p for pressure ?

The article on pressure says that the symbol for pressure is p, (not P). The article on Power (physics) says that the symbol for power is P, (not p). Why change here from p to P for pressure ? Please defend or revert. Bo Jacoby 11:16, 2 May 2006 (UTC)

I believe it's dependent on the field. The symbol P may be used for power, sure, but the symbol p is used for momentum so there's that line of argument gone nowhere. In physics I believe pressure is usually P and this is physics. JIMp _talk·cont 20:14, 7 July 2010 (UTC)

water: mol-specific heat capacity > 3R ?

-> more degrees of freedom(electrons) or Dulong-Petit not the upper boundary or else ?

The molar mass of water is (2*1,008+15,999)g/mol = 18,015 g/mol. In 1g water are therefore 2*0,055509 mol H-atoms(!) und 0,055509 mol O-atoms.

The maximum value -according Dulong-Petit law- of the specific heat capacity of liquid water is therefore 2*0,055509g/mol*3R +0,055509g/mol*3R = 0,499958g/mol * 8,3145 J/molK =4,154 J/gK. But the real value is 4,18-4,19 J/gK. This is 0,7% bigger than the theoretical maximum!

What is the explanation of this? (31 October 2006)

Wrong units?

It appears to me that the units for the heat capacity of water are wrong, and should be kJ/(kg.K) instead of kJ/(g.K) for the same value. Knuffels (talk) 12:18, 5 August 2010 (UTC)

more wrong units

What's with J·cm-3·K−1 as units? J·m-3·K−1 is the SI unit, or MJ·m-3·K−1 if there's some reason to keep the decimal point in the same place. 91.84.95.81 (talk) 23:17, 7 December 2010 (UTC)

Even more on the units

The section called "Extensive and intensive quantities" correctly states that the units for specific heat capacity are J/(kg.K). By "correct" I mean that this is the heat capacity expressed in the base SI units. These units should be used throughout the article. However, in the tables further down in the article the units J/(g.K) are used. I think we should switch to J/(kg.K) and if no-one objects I'll make the change in a couple of days. O. Prytz (talk) 08:36, 21 December 2010 (UTC)

definitions

There are undefined terms in this equation in the article. Can someone please define all the terms below the equation in the article? Thanks. Rtdrury 09:34, 19 May 2007 (UTC)

$C_{V}=\left({\frac {\delta Q}{dT}}\right)_{V}=T\left({\frac {\partial S}{\partial T}}\right)_{V}$

$C_{p}=\left({\frac {\delta Q}{dT}}\right)_{p}=T\left({\frac {\partial S}{\partial T}}\right)_{p}$