User:Bgm2011/water (heat capacity)

Dulong-Petit and heat capacity of water: a contradiction?
The molar mass of water is (2*1.008+15.999)g/mol = 18,015 g/mol. In 1g of water are therefore 2*0.055509 mol H-atoms(!) and 0.055509 mol O-atoms.

The specific heat capacity is therefore -according to Dulong-Petit- max. 2*0.055509g/mol*3R + 0.055509g/mol*3R = 0.499958g/mol * 8.3145 J/molK =4.154 J/gK.

But according to textbooks and physical tables is c = 4,18 to 4,19 J/gK  (that's about 0,7% more)!

5 years ago, I've posted this problem:  (21:47, 31 October 2006 84.152.100.117) question&answers : http://en.wikipedia.org/w/index.php?title=Talk:Heat_capacity/Archive_2&action=edit

l'eau: capacité thermique molaire > 3R
-> plus de degrés de liberté ou la lois de Dulong et Petit n'est pas la limite supérieur (ou autre chose) ?

Le masse molaire de l'eau est (2*1,008+15,999)g/mol = 18,015 g/mol. Ainsi, en 1 g de l'eau, il y a 2*0,055509 mol H-atomes(!) et 0,055509 mol O-atomes.

-> Le maximum de la capacité thermique massique est -conformément à la loi de D. et P.-  2*0,055509g/mol*3*R +0,055509g/mol*3*R = 0,499958g/mol * 8,3145 J/molK =4,154 J/gK.

Mais la capacité thermique massique de l'eau est entre 4,18 J/gK et 4,19 J/gK (~20° C).

Qu' est-ce que la solution?

water: specific heat capacity > 3R
-> more degrees of freedom 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. It's 0,7% bigger!(not much but well above the error boundaries)

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


 * Water is a bit over 3R per mole of atoms, but nevermind water. Liquid bromine has a heat capacity of 3.5 R per mole of bromine atoms (not molecules, where it is of course twice as much, but that doesn't count). I've failed to get anybody how knows how molar heat capacities happen. In theory, the max is 3R per mole of atoms, and any kind of bonding between atoms only can cut that figure down, because it results in quantum barriers to equal partitioning into kinetic and potential storage modes. The only thing I can think of is that we're getting partitioning into electronic modes of excitation (rather as in gas phase NO), and that gives additional degrees of freedom which we're only beginning to see the tail of. S  B Harris 17:42, 9 December 2006 (UTC)
 * According to Herzberg, the lowest excited electronic state of Bromine is at 13814 cm-1, which is way too high to be thermally populated at room temperature. (In contrast, the first excited state of NO is 121 -1. A useful conversion factor to keep in mind for these sorts of comparisons is 300 K corresponds to 208 cm-1). So that can't be the explanation. I do wonder if you are trying to get too much out of the equipartition theorem by trying to use it to draw conclusions about liquid state heat capacities. Equipartition is really only useful if the potential energy is zero (free particles) or not too far from quadratic (crystals in the harmonic approximation.) If the potential energy is large but not anything close to harmonic, equipartion says basically nothing useful.--Rparson 22:57, 11 December 2006 (UTC)
 * I don't see why it shouldn't say something useful about MAXIMAL energy storage, which is what happens when you have an asymptotic approach to freedom from constraint in motion (as in free particles and particles near the bottom of quadratic potential wells where you can approximate the potential as square, and thus free). Again, if a atomic nucleus is free to move in 3 dimensions it should be able to store R per mole of nuclei per dimension. If things are screwed up by funny shapped potentials, all it can do is screw this up-- I can't think of any way it should be able to ADD to it. S B Harris 00:19, 12 December 2006 (UTC)
 * So what happens in the vicinity of the liquid-vapor critical point, where the heat capacity diverges to infinity? Yes, that's a mathematical singularity (it assumes an infinite number of particles) but it reflects a physical reality: the real, measureable heat capacity of a supercritical fluid in thhe immediate vicinity of the critical point is enormous. Ditto for the glass transition. It seems that when systems become large and "loose", so that small additions of kinetic energy can get dispersed into a wide variety of motions on all sorts of scales, heat capacities can get as large as one wishes.
 * Um, I was under the impression that the heat capacity of substances at glass transition or supercritical point was whatever they were for the substance on either side of the phase change, since the whole point of both of these states is that the enthalpy of transition goes to zero there. So why says the heat capacity of the stuff itself goes wild? I don't believe it. Some funny thing happen in liquid helium going from normal to superfluid, but that's due to the phase transition itself taking up energy and and heat capacity itself isn't high, just the CHANGE in heat capacity is high. S  B Harris 11:25, 26 December 2006 (UTC)
 * No. Check out the articles on critical phenomena, phase transition, etc. In the neighborhood of a 2nd order phase transition, the heat capacity generically diverges to infinity according to the power law |T-Tc|-ά, where the critical exponent ά depends upon a small number of "universal" parameters that characterize the type of phase transition. For the liquid-vapor critical point, ά is approximately 0.1 (www.nyu.edu/classes/tuckerman/stat.mech/lectures/postscript/lecture_25.ps), for normal-superfluid helium the measured value is 0.0127. As long as ά is less than 1 the singularity is integrable, so that the latent heat is zero, and with values like  0.1 or 0.01 you do have to get very close to the critical point to see the divergence, nevertheless it is there. Kenneth G. Wilson got the 1984 Nobel Prize for explaining how this comes about.Rparson 20:42, 26 December 2006 (UTC)
 * Nevertheless you do raise a puzzle, since I wouldn't expect liquid Bromine to be all that unusual. I do think that one should be careful because we don't have a whole lot of useful reference data here - the readily available data on heat capacities is mostly tabulated for elements and simple organics at standard temperature. I'd be interested to see what the heat capacity of, for example, liquid He or Ne is. It occurred to me that the value for Br2 might just be an error (it's been known to happen) but I traced it back to the NBS tables, which are about as authoritative as anything. —The preceding unsigned comment was added by Rparson (talk • contribs) 00:30, 14 December 2006 (UTC).
 * Perhaps the Dulong-Petit law does only apply to solids, where the structure doesn't change with temperature. (24 December 2006)

—The preceding unsigned comment was added by 84.152.105.34 (talk) 14:36, 24 December 2006 (UTC).


 * All: I originally used water as an example of the number of degrees of freedom because it was a substance familiar to all and is widely recognized for its high specific heat capacity. I also had visted several chemistry sites at universities that consistenly said water has six active degrees of freedom. Water apparently has more than six possible degrees of freedom but only about six or seven are active at 100 °C. It eventually developed that water was a poor choice to use in this article for illustrating the concept of degrees of freedom. By dividing the CvH of steam by that of the monatomic gases, one can see that water doesn't have a clean, interger number of degrees of freedom; it's more like 6.7 degrees of freedom. I don't know if there's some hydrogen bonding going on with steam at 100 °C or if a seventh, internal degree of freedom is active but is partially frozen out. Consequently, I substituted nitrogen in place of water. Nitrogen cleanly demonstrates the concept that the number of active degrees of freedom expresses themselves as a proportional increase in molar heat capacity under constant volume.  I also added a CvH column to the table to help in illustrating this concept.  Greg L 05:31, 25 December 2006 (UTC)


 * The Dulong-Petit law only gives a high temperature approximation for the mechanical atom movement. There are other ways (thermal) energy can be stored. In the water example, the structure of the liquide depends a little on temperature. So if water is heated, a part of the energy is used to make the structure a little more open. Near a critical point the structureal changes can get quite large an this way make the heat capacity very large.--91.3.124.78 (talk) 19:58, 9 September 2009 (UTC)