The change in entropy (\(\Delta S\)) is a fundamental concept in chemistry and physics that quantifies the dispersal of energy or matter within a system. Entropy measures how spread out energy is at a specific temperature or how many different microscopic arrangements a system can have. Calculating \(\Delta S\) is important because it provides a direct way to predict the direction of a process, known as spontaneity, under certain conditions. The calculation depends on the nature of the process being analyzed, whether it involves a chemical transformation, a physical change of state, or a comparison to the entire universe.
Determining Entropy Change from Tabulated Data
The most common method for determining the entropy change for a chemical reaction involves using published tables of standard molar entropies (\(S^\circ\)). \(S^\circ\) represents the absolute entropy content of one mole of a substance under standard conditions (1 bar pressure and 298.15 Kelvin). Unlike the standard enthalpy of formation, \(S^\circ\) for an element in its standard state is a positive, non-zero value.
The calculation relies on the “products minus reactants” rule because entropy is a state function. The change in standard entropy for a reaction (\(\Delta S^\circ_{rxn}\)) is found by summing the standard molar entropies of all products and subtracting the sum of the standard molar entropies of all reactants. The balanced chemical equation’s stoichiometric coefficients must be applied to each substance’s \(S^\circ\) value. For example, in the reaction \(aA + bB \rightarrow cC + dD\), the formula is \(\Delta S^\circ_{rxn} = [c \cdot S^\circ(C) + d \cdot S^\circ(D)] – [a \cdot S^\circ(A) + b \cdot S^\circ(B)]\).
The resulting \(\Delta S^\circ_{rxn}\) value (usually expressed in J/K·mol) indicates the change in disorder of the reacting system. A positive \(\Delta S^\circ_{rxn}\) suggests the products have a greater dispersal of energy than the reactants, often correlating with an increase in the number of gaseous molecules. Conversely, a negative value indicates a decrease in the system’s disorder.
Determining Entropy Change During Phase Transitions
A distinct approach is necessary when calculating the entropy change for a physical phase transition, such as the melting of ice or the boiling of water. These processes, including fusion, vaporization, and sublimation, occur at a constant temperature and pressure. Because a phase transition at its equilibrium point is considered a reversible process, the entropy change (\(\Delta S\)) is calculated using the fundamental thermodynamic definition: \(\Delta S = q_{rev}/T\).
For a phase change taking place at constant pressure, the heat transferred (\(q_{rev}\)) is equal to the enthalpy change (\(\Delta H\)) for that process. Therefore, the formula simplifies to \(\Delta S = \Delta H / T\). Here, \(\Delta H\) is the molar enthalpy of fusion (\(\Delta H_{fus}\)) or the molar enthalpy of vaporization (\(\Delta H_{vap}\)). This equation is only valid precisely at the temperature where the two phases coexist in equilibrium.
For instance, the entropy of fusion (\(\Delta S_{fus}\)) is found by dividing the molar enthalpy of fusion (\(\Delta H_{fus}\)) by the melting temperature (\(T_m\)) in Kelvin. Since phase transitions involve moving to a more disordered state, the calculated \(\Delta S\) for these processes is always a positive value. This reflects the increased dispersal of energy that occurs when a solid turns into a liquid or a liquid turns into a gas.
Relating System Entropy to the Universe
While calculating the entropy change of the system (\(\Delta S_{sys}\)) provides valuable information, the Second Law of Thermodynamics dictates that the ultimate predictor of spontaneity is the total entropy change (\(\Delta S_{total}\)). \(\Delta S_{total}\) is the sum of the entropy change of the system and the entropy change of the surroundings. A process is considered spontaneous only if \(\Delta S_{total}\) is positive (\(\Delta S_{total} > 0\)).
The entropy change of the surroundings (\(\Delta S_{surroundings}\)) is determined by the heat flow between the system and its environment. Since the surroundings are considered very large, any heat transfer is treated as a reversible process occurring at a constant temperature (\(T\)). The heat gained or lost by the surroundings is equal in magnitude but opposite in sign to the enthalpy change of the system (\(\Delta H_{sys}\)). This relationship allows for the calculation: \(\Delta S_{surroundings} = -\Delta H_{sys} / T\).
By combining the calculated system entropy change (\(\Delta S_{sys}\)) with the surroundings’ entropy change, the total entropy change is found: \(\Delta S_{total} = \Delta S_{sys} + (-\Delta H_{sys} / T)\). Determining this total value is the final step in using entropy to predict whether a chemical reaction or a physical process will occur on its own without a continuous external energy input. A positive result for \(\Delta S_{total}\) confirms that the process is spontaneous, reflecting the natural tendency of the universe towards greater energy dispersal.