Reference: Daniel V. Schroeder, An Introduction to Thermal Physics, (Addison-Wesley, 2000) – Problem 1.43.
The specific heat capacity of water is , so how does this translate into the heat capacity per water molecule? The molar weight of water is so in 1gram of water there are
The heat capacity of a single molecule is therefore
where is Boltzmann’s constant.
Thus to raise the temperature of a sample of water by , we need to raise the energy of each molecule by around . If we assumed that all the thermal energy of water is due to quadratic degrees of freedom, then because each degree of freedom contributes to the energy, we would need 18 degrees of freedom in a water molecule. As we’ve seen, water has around 10 such degrees of freedom, so the thermal energy must also be stored elsewhere.