One of the most important consequences of the Third Law of Thermodynamics is that a perfectly ordered crystal at absolute zero has zero entropy:
\[ \boxed{ S(0\ \mathrm{K}) = 0 } \]
This provides a natural zero for the entropy scale and makes it possible to calculate the absolute entropy of a substance.
Beginning at absolute zero, the entropy at a temperature \(T\) can be obtained by integrating the heat capacity:
\[ S(T) = \int_0^T \frac{C_p}{T}\,dT \]
If the substance undergoes one or more phase changes, the entropy changes associated with those phase transitions must also be included:
\[ S(T) = \int \frac{C_p}{T}\,dT + \sum \frac{\Delta H_{\mathrm{transition}}} {T_{\mathrm{transition}}} \]
Thus, calculating a Third-Law entropy amounts to moving from \(0\ \mathrm{K}\) to the desired temperature while accounting for all heat capacity contributions and phase changes encountered along the way.
In principle, calculating an absolute entropy is straightforward. In practice, however, measuring heat capacities near absolute zero is extremely difficult.
The problem is that
\[ S(T) = \int_0^T \frac{C_p}{T}\,dT \]
requires information all the way down to \(0\ \mathrm{K}\), precisely where experimental measurements become most challenging.
To overcome this difficulty, a model is needed for the behavior of the heat capacity at very low temperatures.
For many solids, the heat capacity becomes proportional to the cube of the temperature at sufficiently low temperatures:
\[ \boxed{ C_p = aT^3 } \]
where \(a\) is a constant determined from experimental measurements.
This behavior is known as the Debye extrapolation and is a consequence of the quantized vibrational motions of atoms within a crystal.
If the heat capacity is measured at a low temperature \(T_m\), then the constant \(a\) can be determined from
\[ a = \frac{C_p(T_m)} {T_m^3} \]
Once \(a\) is known, the heat capacity can be extrapolated from \(0\ \mathrm{K}\) to the lowest experimentally accessible temperature.
Substituting
\[ C_p = aT^3 \]
into
\[ S(T) = \int \frac{C_p}{T}\,dT \]
gives
\[ S(T) = \int aT^2\,dT \]
which integrates to
\[ S(T) = \frac{a}{3}T^3 \]
Thus, the entropy contribution from the low-temperature Debye region can be calculated analytically and used to bridge the gap between absolute zero and the lowest measured temperature.
The remaining entropy contributions can then be obtained from measured heat capacities and any phase transitions encountered at higher temperatures.
Big picture: Third-Law entropies are obtained by integrating \(C_p/T\) from absolute zero to the temperature of interest. Because heat capacities are difficult to measure near \(0\ \mathrm{K}\), the Debye extrapolation provides a physically reasonable model, \(C_p \propto T^3\), that allows the missing low-temperature contribution to be estimated.
A crystalline solid is found to have a molar heat capacity of
\[ C_p = 12.0\ \mathrm{J\,mol^{-1}\,K^{-1}} \]
at \(9.0\ \mathrm{K}\).
Assuming the Debye model
\[ C_p = aT^3 \]
holds over this temperature range, calculate the molar entropy of the substance at \(15.0\ \mathrm{K}\).
From the heat-capacity measurement,
\[ a = \frac{C_p}{T^3} \]
\[ a = \frac{12.0\ \mathrm{J\,mol^{-1}\,K^{-1}}} {(9.0\ \mathrm{K})^3} \]
\[ a = 1.65\times10^{-2} \ \mathrm{J\,mol^{-1}\,K^{-4}} \]
According to the Third Law,
\[ S(15\ \mathrm{K}) = \int_0^{15} \frac{C_p}{T}\,dT \]
Substituting the Debye expression,
\[ S(15\ \mathrm{K}) = \int_0^{15} \frac{aT^3}{T}\,dT \]
\[ S(15\ \mathrm{K}) = \int_0^{15} aT^2\,dT \]
\[ S(15\ \mathrm{K}) = \frac{a}{3}T^3 \Bigg|_0^{15} \]
\[ S(15\ \mathrm{K}) = \frac{(1.65\times10^{-2})}{3} (15)^3 \]
\[ S(15\ \mathrm{K}) = 18.5 \ \mathrm{J\,mol^{-1}\,K^{-1}} \]
\[ \boxed{ S(15\ \mathrm{K}) = 18.5 \ \mathrm{J\,mol^{-1}\,K^{-1}} } \]
Test of reasonableness: The entropy is positive, as required by the Third Law. Because the temperature is still quite low, the entropy remains much smaller than typical room-temperature molar entropies. As the temperature increases, additional vibrational states become accessible, causing both the heat capacity and the entropy to increase.
A low-temperature heat capacity measurement will be generated at random. Assume the Debye model \[ C_p = aT^3 \] and calculate the molar entropy at the requested temperature.
Big picture: Third-Law entropies are obtained by integrating \(C_p/T\) from absolute zero to the temperature of interest. Because heat capacities are difficult to measure near \(0\ \mathrm{K}\), the Debye extrapolation provides a physically motivated model for estimating the missing low-temperature contribution. Together, the Third Law and the Debye model make it possible to calculate absolute entropies for real substances.