What does it mean that the standard molar entropy value is the amount of energy that a substance must have to exist at a certain temperature?

Question

What does it mean that the standard molar entropy value is the amount of energy that a substance must have to exist at a certain temperature?

According to this reference: http://khimiya.org/volume15/Entropia.pdf the meaning of standard molar entropy is the relative amount of energy that the substance must have to exist at for example 298 Kelvin.

An example is given of diamond and graphite, which has 2.4 J/K and 5.7 J/K respectively. I see that graphite has less strong bonds than diamond, i.e. diamond has higher potential energy stored inside its bonds. But I don't understand how that translates into a lower amount of energy needed to exist at a certain temperature? Is this the energy that needs to be supplied to the diamoned to raise its temperature to 298 kelvin? But isn't it than just a kind of heat capacity? What happens if you did not transfer enough energy to make it exist stably at that temperature?

1. The meaning of standard molar entropy values When seen in their relation to the energy content of a substance, standard molar entropy values, $S^0_{298}$, give useful insight that is absent when - as is usual - those J/K values are treated as merely abstract numbers to be added or subtracted in determining a $dS_0$ reaction . A $S^0_{298}$ value for a substance is the number of joules of energy/T transferred incrementally (reversibly, from the surroundings at each T) to a mole of substance from 0 K to 298 K. Thus, this number is a rough indicator or approximate index (not the joules dispersed at 298 K, nor the total joules dispersed from 0 K to 298 K!) of the relative amount of energy that the substance must have to be exist stably at 298 K. This is why $S^0_{298}$ values illuminate e.g., the difference in rigidity of bonding: the more rigid bonds of diamond (2.4 J/K) vs. the looser interatomic bonds in graphite (5.7 J/K) ,

Answer 1

The wording of the cited paragraph is confusing, but it does not say that "the standard molar entropy value is the amount of energy that a substance must have to exist at a certain temperature" (as claimed in the question). In fact the author says specifically that this is not the case. Rather, it is "is a rough indicator or approximate index" of the amount of energy.

Mathematically, the change in internal energy of a substance as it is heated (at constant volume to avoid PV work) from absolute zero to a temperature T would be $$\Delta U = \int_0^T C_V dT,$$ where $C_V$ is the heat capacity at constant volume and is a function of T. For one mole of substance, we would use the molar heat capacity $\overline{C}_V$.

This is similar to but not the same as the expression for absolute entropy, which is $$S^\circ_T = \int_0^T \dfrac{C_P}{T} dT,$$ with the crucial difference being the T in the denominator. (The difference between $C_P$ and $C_V$ is of secondary importance.)

As long as the change in the heat capacity as a function of temperature is similar between two substances, then the relative ranking based on absolute entropy will be similar to a ranking of the internal energy, which is what the author means when he says that the former can be used as "a rough indicator or approximate index" of the latter. But clearly they are by no means the same and I would argue that it's somewhat misleading to suggest that the entropy should be used in this way.