Why the specific heat capacity of a substance is different at different temperatures?

The following question was asked by Ruchit Panchal :

“While solving problems of Thermodynamics in class, we encountered an example where we were provided with the specific heat of Water at -13 degrees C(ice) as 0.5 and in liquid state as 1.0. We were wondering why the data said two different values for same substance at different temperatures? According to the definition, its dQ/dT. So I'm wondering why water takes more heat for a degree rise in temperature at higher temperature (liquid) than at lower temperatures (ice)?”

By definition, heat capacity of a material is the amount of heat that must be supplied to the unit mass of that material to raise its temperature by one kelvin or degree Celsius. In order to understand why this amount of heat should be different at different temperatures for the same material, we must first understand what happens to the material when we supply it with a definite amount of heat. When we supply a certain amount of heat, say Q, to a material we observe an increase in its temperature. This increase is directly related to the vibrations or random thermal motion of the atoms/molecules/ions or the collective oscillations of the whole lattice (known as phonons) in the material. The more the vibrations the more will be the temperature of the material. Thus, suppose you have two kinds of  materials, say A and B, and if you supply Q amount of heat to both of them, if the molecules of A tend to vibrate easily and more vigorously compared to that of B, then A will have greater increase in its temperature compared to B. This means to increase the temperature by the same amount, we would need less heat for A compared to B, hence, specific heat of A will be smaller than B.

The vibrations or thermal agitation within a material depend on how the material is held together, i.e. on the bond strength, types of bonds in the material, crystal structure and in addition, it also depends on external factors which affect the vibrations such as pressure. So, in conclusion, heat capacity of a material depend on the behaviour of the bonds in a material (in solids and liquids) and on external factors such as pressure (in gases, pressure have negligible effect of heat capacities of solids/liquids).

Now it is well known fact that the behaviour of bonds is different at different temperatures. A material is more loosely held at higher temperatures than at lower temperatures. Since, the behaviour of the bonds are different at different temperatures, the heat capacity of a material must also be different at different temperatures as vibrations are related directly to the behaviour of the bonds.

Therefore, variation of heat capacity of a substance with temperature boils down to analyzing the variations of vibrations/thermal motion within the substance with temperature. In fact there is a definite relationship between these vibrations and the heat capacity given by Peter Debye in 1912 and is known as the Debye model of specific heat. This model relates the heat capacity of a substance (solid) and the vibrations of the lattice and explicitly derives that the heat capacity of a substance increases as we increase the temperature and is true for every (solid) material.

This is because as we increase the temperature of a material, molecules within the material start vibrating vigorously. The more vigorously the molecules are vibrating (at high temperatures) the more difficult it gets to further increase these vibrations (and hence the temperature) by a small supply of heat. For example, consider a material at 10 kelvin and we want to change its temperature to 11 kelvin. Now consider the same material at 500 kelvin and we want to increase its temperature to 501 kelvin. Obviously the heat required to change temperature from 500 to 501 will be much more because the molecules are already in a state of vigorous vibrations and supplying small amount heat won’t affect the vibrations much. In order to cause any observable change in its vibrations (hence the temperature) the amount of heat we need to supply should be comparable to its heat content at that temperature. If at 500K the material has 2000J of heat content then suppying 1J would not make much of difference as 1J is negligible compared to 2000J. On the other hand, if at 10K material has 20J of heat content then supplying 1J of heat will certainly produce observable change as 1J is not negligible compared to 20J. Thus more heat will be required to raise the temperature when a material is at a higher temperature compared to when it is at a lower temperature. Hence, heat capacity of the material will be more at 500 K compared to 10 K and this is, in general, true for all the (solid) materials in temperature ranges in which no (second order) phase transitions are involved.

Bottom line is, heat capacity of a material depends on the bonds and structure of the material which varies with temperature. From a graph of C (heat capacity) vs. T (Temperature) of a material you can make some rough statements about the behaviour of the bonds within that material at different temperatures.

As an exercise, I would recommend you to make a plot of specific heat of water in the temperature range 5K to 200K. The data should be readily available on internet. You’ll find some interesting things about the behaviour of water between 5K to 45K.

Note : There is a limit to the extent to which the molecules in a material can vibrate beyond which the material undergoes phase transitions.