You might find the webpage by MF Chaplin to be quite comprehensive in dealing with the complexity of water--which is a huge field. In particular, for thermodynamic anomalies, check this http://www1.lsbu.ac.uk/water/explan4.html

You might find the webpage by MF Chaplin to be quite comprehensive in dealing with the complexity of water--which is a huge field. In particular, for thermodynamic anomalies, check this http://www1.lsbu.ac.uk/water/explan4.html

The heat capacity of water is not so special. Ammonia has a higher heat capacity for example. The problem is to find a way comparing heat capacities of different materials. The best way to do it, is to normalise it to one atom. A monatomic gas has a heat capacity of 3/2 R (gas constant). This is coming from the degree of freedom (3 translational). In a diatomic gas, two degree of freedom (rotational) and one degree of freedom (vibrational) are added and so on. However, some degree of freedom may be frozen in at a given temperature (vibrational).

Water has a heat capacity of 3 R.

I think the widespread opinion of the high heat capacity of water is coming from the large capacity of storing heat in the sea (when compared to air and rocks), however, here we compare apples with oranges (air has much lower atoms per volume compared to water and rocks has no convections...).

A water molecule has anyway more internal degrees of freedom (vibrations and rotations) then a simple liquid. Second, the hydrogenbonding network introduces other degrees of freedom. The heat capacity increases with the number of internal degrees of freedom of a material/liquid/gas.

Thanks all ... I need a more detailed molecular level reason ...

Simple calculation of degrees of freedom works only in gas. For liquid water the heat capacity is much higher than 3R. This reflects very complicated molecular motions present in liquid water.

I do not think, that the heat capacity of water normalised to one atom is much higher than 3 R. Water has a heat capacity of 4.2 J/g = 75.6 J/mol. Normalised to one atom, this is 25.2 J/mol/atom and this is 3 R.

Some vibrational modes of water are low in frequency (hydrogen bridge bonds for example), which are fully excited at room temperatures. This behviour contributes to the heat capacity. However, the O-H stretching modes for example have high frequencies and the probability of exciting such modes are low, so that these modes do not contribute to the heat capacity of water.

The heat capacity of GAS water is close to 3R, if we count per number of molecules. All 6 degrees of freedom come from kinetic energy.

The heat capacity of LIQUID water is indeed apparently close to 3R but if we count per number of atoms. In liquid water, potential degrees of freedom should play a role. How can we end up with 3R per atom then?

Also, this scaling - from number of molecules to number of atoms - is not found in other liquids of similar size (methanol etc).

Ok let's look at the definition of heat capacity: it means the thermal energy needed to increase the temperature one degree (per unit volume or mole). In the case of water the large portion of given thermal energy was spent for breaking or weakening the large and strong hydrogen bonding network so this portion of thermal energy can not increase the temperature.

Thank you all ...

I think hydrogen bondings are responsible for the high heat capacity of liquid water. The heat supplied to increase the temperature of one mole of a substance through 1 0C is the molar heat capacity. In the presence of strong H-bonding more heat is required to increase the kinetic energy of molecules. If more degree of freedom were responsible for it then CH4 might have been more heat capacity than water which is not the case. only H-bondings are responsible for it.

"In the presence of strong H-bonding more heat is required to increase the kinetic energy of molecules."

I don't think so. Independent degerees of freedom can be treated independently and velocity does not enter any potential. Hence, classic kinetic energy distributions do not depend on the interactions.

Instead, when dealing with liquids collective degrees of freedom (phonons) have to be considered and might partly account for the differences.

Water is a molecular substance. The center of mass kinetic energy is only a modest part of the total specific heat. Water molecules in a liquid do not rotate; they librate. Intermolecular forces replace free rotation with back and forth rocking motions.The three rotational modes then all act as harmonic oscillators (the same is true of the translational modes. While the three internal vibrational modes are all at fairly large frequencies, and are substantially frozen out of making a specific heat contribution, there is the interesting issue that the hydrogen atoms in a hydrogen bond may exchange back and forth between the two oxygens they are bonding, which gives an additional specific heat contribution. This gives us 3R for the translational motions plus 3R for the rotational motions plus something substantial for the hydrogen bonds, giving us 9R or so for the molar specific heat of liquid water, which if I recall from 45 years ago and a Physics General exam is reasonably close to correct.

Equipartition tells us Cv per particle should approach 3R, and it generally falls short for most liquids because vibrational modes are frozen out. The interesting fact about liquid water is that there is, apparently, close to the full vibrational contribution even though the single-molecule vibrational energy level spacing would suggest they should be. However, water is a remarkably structured liquid, meaning it looks locally a lot like a solid. That strong and highly directional interaction between water molecules (via H bonding) presumably gives liquid water an unusual (for a liquid) number of low-energy collective vibrational modes, more or less transverse phonons. So the vibrational contribution in water is not nearly as frozen out as a look at just the single molecule would suggest, and as is the case in liquids with weaker and/or less directional interactions between molecules.

Not to disagree, but the actual vibrational modes are two OH stretch modes (symmetric and antisymmetric) near iirc 3600 cm^-1 and an HOH bend mode near 1400 cm^-1, which are fairly thoroughly frozen out.

Those are the single-molecule modes, George. The point is when you have strong intermolecular interactions you have to rediagonalize the Hamiltonian to find new normal modes, and with strong interactions among a lot of atoms you'll find plenty of low-lying collective modes. Consider a square of 100 x 100 particles, each connected with a very stiff spring, which, if it were by itself, would vibrate at 3600 cm^-1. What will be the lowest collective mode? Way lower than 3600 cm^-1. That's what I'm saying.

Having done experimental Raman and IR spectroscopy, I can assure you that the internal modes of molecules are not greatly* affected by moving from crystal to solution to gas phase. The reason for this is that the bonds between molecules have considerably smaller force constants than the intramolecular bonds do, so the internal modes are only weakly coupled between molecules. This circumstance is very different than the situation in atomic crystals, in which all the bonds are about the same.

Further, most vibrations are localized. Thus, if you take a protein and decompose it by hydrolysis into its component amino acids, most but not all of the vibrational modes are at very much the same frequency as they were before. The one major change is the amide vibrations, the vibrations of the - (C=O)- NH - group that is removed when you hydrolyze the protein. This issue caused a mild scientific foofaraw, scandal being too strong a word, when the FDA claimed 40 year or so ago that an alleged cure for cancer was not a protein but was instead a particular amino acid. There was then a response in iirc Science, signed by most of the people in the field, explaining that the particular vibrational mode was seen regularly in proteins containing the amino acid, at about the same frequency as the mode was found in the free amino acid, so the FDA demonstration in question was invalid.

Readers interested in how this vibration issue works may consider a linear diatomic with two atoms mass m linked by a spring of force constant k. Compute the normal mode frequency. Now (resort to Mathematica will save pain) line a series of these molecules up, and link them by bonds of strength k/100. There are indeed normal modes, but the high-frequency modes will still be there, not greatly perturbed by the weak intermolecular bonds. This outcome is extremely different from the classic freshman or so physics problem in which a line of masses are all coupled by bonds having the same strength, and you get modes having very different frequencies.

What is the difference between the spring constants of hydrogen bonds and covalent O-H?

Excellent point that I should have considered more carefully. As an example of the point I was making, note http://www.hindawi.com/journals/dpis/2013/329406/fig1/

and compare the liquid and gas cell spectra. The OH vibrations are the band near 3300-3600 cm^-1. There is only a modest change between gas and liquid. I should however have allowed that water is a special case. Note, however that there is also in water a mode

O-H :O O: H-O- in which the proton H wanders between molecules

So, in summary, hydrogen bonds ARE responsible for the unusually high heat capacity of water?

George, you've forgotten that there are *always* high frequency collective modes when you rediagonalize the Hamiltonian, and they do indeed look essentially just like local modes (and there are always a lot of them).  None of that says boo about how many low lying collective modes you have.  And the answer is: a lot.  Look at it this way, all of the hydrodynamic behaviour of water is a result of those long-wavelength low-frequency modes, right?  There's plenty of them, and their characteristics are important for all the hydrodynamic behaviour of water -- and the heat capacity can reasonably be regarded as one of those, in sort of an infinite time limit.  You're coming at this from a high-frequency spectroscopy point of view, which is not the way to see it.  The collective behaviour of water isn't going to touch anything in IR spectroscopy.  Think like a liquids theorist. 

i hope you dont mind me posting here...does anyone know how to increase the thermal capacity of water? I knowif you add solutes to the water you would make a more ionic soulution that cools faster than normal water; so i want to know if there is a way to get water to loose or absorb heat even more slower than it currentlydoes (4.187KJ/KgoC)

Your question is far from trivial.

One could say: ok, specific heat capacity depends on the number of molecules per gram and the number of bonds to break - water is a small molecule with high ability to form H-bonds - this explains its high specific C.

Yet, H2O2 is a small-molecule H-bonded material that does not share any of water's peculiarities.

The trick could reside in water's local tetrahedral arrangement and high cooperativity - indeed, the Stillinger-Weber family of liquids (Si, Ge, etc.) follow a water-like hierarchy of anomalies, with an increase in  heat capacity on isobaric cooling at high tetrahedralities.

How exactly the mechanism works in water has been matter of a lo(ooooo)ng debate, which is still in progress. Take a look at water's model(s) and hypothesized thermodynamics in the no man's land of the phase diagram (for example,  https://arxiv.org/pdf/1404.4031.pdf)