# Why is the heat capacity of water $9R$ and not $6R$?

From the equipartition theorem, the relationship between energy and temperature in a substance is $$U=\frac{NRT}{2}$$ for $$N$$ quadratic degrees of freedom associated with a particle of that substance. For water, which is triatomic with no rotational symmetry, there are (assuming high enough temperature for each mode to be occupied) $$3$$ translational degrees of freedom, $$3$$ rotational degrees of freedom, $$3$$ kinetic vibrational degrees of freedom, and $$3$$ potential vibrational degrees of freedom. This results in $$12$$ total quadratic degrees of freedom or $$U = 6RT$$. From thermodynamics, another definition for the temperature dependence of internal energy is $$U = C_VT$$. Comparing the experimental heat capacity of water to the gas constant, it looks like (at room temperature) $$C_V = 9R$$.

I understand that the equipartition theorem is a classical result and we now know that it is not entirely accurate, but for other materials at room temperature it appears to hold very well. For example, monoatomic substances like He, Ne, Ar, Xe, Kr, $$C_V$$ is almost exactly $$\frac{3}{2}R$$. For diatomic substances like Oxygen and Nitrogen, $$C_V$$ is almost exactly $$\frac{5}{2}R$$. The fact that the heat capacity of water is almost exactly an integer multiple of $$\frac{R}{2}$$ seems to indicate that the equipartition theorem also holds for water, but that there are degrees of freedom unaccounted for.

Where do these "extra" $$6$$ quadratic degrees of freedom in water come from?

• Water at room temperature is a liquid. The counting you have done assumes that you have an ideal gas. Liquid water is not an ideal gas. Commented May 8, 2023 at 21:26

Water is a complicated beast, therefore let’s first recapitulate the expectations for an ideal “normal” substance:

• In a crystal, molecules can only vibrate around their position, leading to three vibrational degrees of freedom. As these count twice, you have a molar heat capacity: $$C_\text{mol} = 3R$$. This is the (Dulong–Petit law). The distinction between isochoric and isobaric matters little as solids have little compressibility.

• In a gas, you have three translational and three rotational degrees of freedom, resulting in a molar isochoric heat capacity $$c_{V, \text{mol}} = 3R$$ and a molar isobaric heat capacity of $$c_{p, \text{mol}} = 4R$$ (assuming an ideal gas). For monoatomic or diatomic gases, the rotational degrees of freedom reduce, but this does not matter here. Internal vibrational degrees of freedom (from the molecule contracting and similar) are usually frozen at room temperature.

• In a liquid, you would expect the best of both worlds: Molecules can freely move, but also interact with each other, resulting in the degrees of freedom from the solid and gaseous phase to add up.

Now, for water around 0°C:

• For ice, we roughly have $$c_{p, \text{mol}} = 4.4 R$$ (source).
• For water vapour, we have $$c_{p, \text{mol}} = 4 R$$ (source).
• Finally, for liquid water, we have $$c_{p, \text{mol}} = 9.1 R$$ (source).

As you can see, the heat capacity for ice already exceeds our idealistic expectations, and the one for liquid water even more so. In my understanding, the established explanation for this are the complex molecular structure of ice and hydrogen bonds, although I could not find a robust source for this. Also see this question on Chemistry SE.