At very low temperature, the heat capacity of crystals is equal to \(C=a T^{3}\), where \(a\) is a constant. Find the entropy of a crystal as a function of temperature in this temperature interval. (a) \(S=\frac{a \cdot T^{3}}{3}\) (b) \(S=a T^{3}\) (c) \(\frac{a \cdot T^{2}}{2}\) (d) \(\frac{a \cdot T}{3}\)

Understanding the heat capacity of materials, particularly crystals, at low temperatures offers valuable insights into their thermodynamic behavior. Heat capacity, denoted as 'C,' is a measure of the amount of heat energy needed to raise the temperature of a substance.

At very low temperatures, we see a notable deviation from the classical constant behavior predicted by the Dulong-Petit law. Instead, the heat capacity of crystals follows a temperature dependency expressed as \( C = aT^3 \), where 'a' is a material-specific constant, and 'T' represents temperature. This proportionality indicates that as the temperature approaches absolute zero, the heat capacity decreases rapidly, eventually tending towards zero.

This relationship is of interest because it helps us understand how entropy behaves as well, which is directly linked to the third law of thermodynamics.

The third law of thermodynamics is a key principle within the field of physics that has significant implications for the behavior of systems at low temperatures. It essentially states that as a system approaches absolute zero, the entropy, or the measure of disorder, of a perfect crystalline substance approaches zero as well.

This implies that at absolute zero, a perfectly organized crystal has only one state with minimum energy: its ground state. Therefore, there is no randomness or positional disorder, resulting in an entropy of zero. The third law provides the foundation for calculating changes in entropy, as it sets a reference point – the zero of entropy – at this lowest attainable temperature.

Applying the third law to our exercise, it tells us that the constant of integration when calculating the entropy function for a crystal at low temperatures is considered to be zero. This is why, when we calculate the entropy by integrating the heat capacity over temperature, the integration constant is negated.

A fundamental thermodynamic relation involved in calculating entropy is the relationship \( dS = \frac{dQ}{T} \). This formula states that the change in entropy, 'dS', can be found by dividing the infinitesimal amount of heat added to the system, 'dQ', by the absolute temperature 'T' at which the heat is added.

In practical situations, where a system follows a specific heat capacity behavior, the change in heat 'dQ' can be expressed in terms of heat capacity and temperature change as 'CdT'. Thus, the formula becomes \( dS = \frac{CdT}{T} \). This equation leads us to the understanding that integrating the heat capacity over temperature can yield the entropy of a system.

When applied to a crystal with a heat capacity of \( C = aT^3 \) at low temperatures, as given in our exercise, this relation underscores the process for determining entropy. By substituting the expression for heat capacity and integrating, we find an explicit formula for the entropy in terms of temperature, which is a critical step in understanding the thermodynamic behavior of crystals near absolute zero.