6.1 Third Law and absolute entropy

Absolute Entropy and the Third Law

Definition and Relationship

The Third Law of Thermodynamics establishes something the first two laws don't: a fixed reference point for entropy. It states that the entropy of a perfect crystal at absolute zero (0 K) is exactly zero. This matters because without it, you could only ever talk about changes in entropy, never the total entropy a substance actually has.

Absolute entropy is the total entropy of a substance at a given temperature, measured from that zero-entropy reference point at 0 K. As temperature increases, absolute entropy increases too, because molecules gain access to more microstates through vibrations, rotations, and eventually translations.

Why must it be a perfect crystal? In a perfect crystal at 0 K, every atom sits in exactly one arrangement. There's only one microstate ($W = 1$), so from Boltzmann's equation $S = k_B \ln W$, the entropy is $S = k_B \ln 1 = 0$. Any residual disorder, like a frozen-in orientational randomness in CO or $\text{N}_2\text{O}$, leaves a small residual entropy even at 0 K.

Temperature Dependence and Calculation

Because the Third Law gives you a definite starting point ($S = 0$ at $T = 0$ K), you can calculate the absolute entropy at any temperature $T$ by integrating the heat capacity:

$S(T) = \int_{0}^{T} \frac{C_p}{T}\, dT$

• $S(T)$ is the absolute entropy at temperature $T$
• $C_p$ is the constant-pressure heat capacity (in J/(mol·K) for molar entropy)
• The factor of $1/T$ means that a given amount of heat added at low temperature contributes more to entropy than the same heat added at high temperature

If the substance undergoes any phase transitions between 0 K and $T$, you must include the entropy of each transition:

$S(T) = \int_{0}^{T_{\text{fus}}} \frac{C_p(\text{s})}{T}\, dT + \frac{\Delta H_{\text{fus}}}{T_{\text{fus}}} + \int_{T_{\text{fus}}}^{T_{\text{vap}}} \frac{C_p(\text{l})}{T}\, dT + \frac{\Delta H_{\text{vap}}}{T_{\text{vap}}} + \int_{T_{\text{vap}}}^{T} \frac{C_p(\text{g})}{T}\, dT$

Each $\Delta H / T$ term accounts for the discontinuous jump in entropy at a phase change.

Implications of the Third Law at Absolute Zero

Molecular Behavior and Energy

At 0 K in a perfect crystal, the system occupies its single lowest-energy quantum state. There is no thermal motion beyond the quantum mechanical zero-point energy (which does not contribute to entropy because it doesn't create additional accessible microstates).

A key consequence: it is impossible to reach absolute zero in a finite number of steps. Each successive cooling step removes less and less entropy, and you'd need infinitely many steps to extract it all. This is sometimes called the unattainability principle and is an equivalent statement of the Third Law.

As $T \to 0$, the heat capacity $C_p$ also approaches zero. This isn't just an empirical observation; it's a thermodynamic requirement. If $C_p$ stayed finite as $T \to 0$, the integral $\int_0^T C_p / T\, dT$ would diverge, giving infinite entropy, which contradicts the Third Law. The Debye model captures this behavior for crystalline solids, predicting $C_p \propto T^3$ at very low temperatures.

Quantum Mechanical Effects

Near absolute zero, quantum effects dominate macroscopic behavior:

• Superconductivity: Certain materials (e.g., lead below 7.2 K, mercury below 4.2 K) lose all electrical resistance. Cooper pairs of electrons form a coherent quantum ground state, and the entropy of the superconducting state drops below that of the normal state.
• Superfluidity: Helium-4 below about 2.17 K (the lambda point) flows with zero viscosity. This reflects a Bose-Einstein condensation into the ground state, consistent with the Third Law's picture of systems settling into highly ordered, low-entropy configurations.

These phenomena aren't just curiosities; they confirm the Third Law's prediction that matter approaches a highly ordered ground state as $T \to 0$.

Calculating Absolute Entropy

Experimental Techniques and Data

To build the $C_p(T)$ curve needed for the entropy integral, you need heat capacity data from near 0 K up to your target temperature. Here's how it's typically done:

1. Low-temperature calorimetry (adiabatic calorimetry down to ~1 K, or $^3$He cryostats for even lower) measures $C_p$ in the cryogenic range.
2. Debye extrapolation fills in the gap from 0 K to the lowest measured temperature. You fit the low-$T$ data to $C_p = aT^3$ and integrate analytically: $S(T_{\text{low}}) = \frac{aT_{\text{low}}^3}{3}$.
3. Differential scanning calorimetry (DSC) or drop calorimetry covers higher temperature ranges, including through phase transitions.
4. Enthalpies of transition ($\Delta H_{\text{fus}}$, $\Delta H_{\text{vap}}$) are measured separately and added as $\Delta H / T_{\text{tr}}$ at each transition temperature.

Accurate data across the full temperature range is essential. Gaps or errors at low temperature are especially damaging because the $1/T$ weighting amplifies their effect on the final entropy value.

Integration Methods

Since $C_p(T)$ is typically a set of discrete experimental data points rather than a neat analytical function, you evaluate the integral numerically:

1. Plot $C_p / T$ versus $T$.
2. Use numerical integration (trapezoidal rule, Simpson's rule, or spline fitting) to find the area under the curve for each single-phase region.
3. Add the $\Delta H / T$ contributions at each phase transition.
4. Sum everything to get $S(T)$.

The values you get this way are the standard molar entropies ($S^\circ$) tabulated in thermodynamic data tables, typically reported at 298.15 K and 1 bar. For example, $S^\circ$ for $\text{H}_2\text{O(l)}$ at 298 K is about 69.9 J/(mol·K), while for $\text{H}_2\text{O(g)}$ it's about 188.8 J/(mol·K), reflecting the much greater disorder in the gas phase.

Third Law vs. Other Thermodynamic Laws

Focus and Scope

Each law of thermodynamics addresses a different fundamental question:

• Zeroth Law: Defines thermal equilibrium and makes temperature a meaningful, measurable quantity.
• First Law: Energy is conserved. Heat and work are interconvertible, but the total energy of an isolated system doesn't change.
• Second Law: The total entropy of an isolated system never decreases. This determines the direction of spontaneous processes.
• Third Law: Entropy has an absolute zero point. A perfect crystal at 0 K has $S = 0$.

Specific Comparisons

The First and Second Laws let you calculate changes in entropy ($\Delta S$) for processes, but they can't tell you the absolute value of $S$ for any substance. The Third Law fills that gap by anchoring the entropy scale.

The Second Law says entropy of an isolated system always increases (or stays the same). The Third Law provides the lower bound: entropy can never go below zero for a perfect crystal, and it equals zero only at 0 K.

The Zeroth Law's concept of temperature is essential for the Third Law to even make sense. Without a well-defined temperature scale, you couldn't talk about "absolute zero" or integrate $C_p/T$ over temperature.