The concept of entropy describes the degree of disorder or randomness within a system, reflecting how energy and matter are dispersed. Entropy is a thermodynamic property that helps determine the direction of spontaneous processes in nature. Calculating the change in entropy (\(\Delta S\)) quantifies this change in disorder between an initial and a final state. Specific formulas allow for the precise determination of \(\Delta S\) for various physical and chemical processes. This calculation is foundational to understanding energy transformations and the feasibility of reactions across many scientific disciplines.
The Fundamental Formula for Change in Entropy
The formal thermodynamic definition of a change in entropy is established through the concept of heat transfer. The fundamental formula for the entropy change of a system is \(\Delta S = q_{rev}/T\). This equation links the change in disorder (\(\Delta S\)) to the heat transferred (\(q_{rev}\)) and the absolute temperature (\(T\)) at which the transfer occurs.
The term \(q_{rev}\) represents the heat transferred along a hypothetical reversible path between the initial and final states. Since entropy is a state function, depending only on the initial and final states, calculating it along this idealized path yields the correct value even for real, irreversible processes.
The absolute temperature \(T\) in the denominator shows that the same amount of heat transfer has a greater impact on entropy at lower temperatures. At low temperatures, the system is more ordered, so adding energy significantly increases disorder. Conversely, at high temperatures, the system is already highly disordered, resulting in a smaller change in entropy.
Calculating Entropy Changes During Phase Transitions
Phase transitions, such as melting or boiling, occur at a constant temperature and pressure. These transitions are considered reversible when they happen at their equilibrium temperature. Since the temperature remains constant, the fundamental formula simplifies for these specific conditions.
During a phase change, the heat transferred (\(q_{rev}\)) at constant pressure equals the change in enthalpy (\(\Delta H\)) for that transition. The formula becomes \(\Delta S = \Delta H_{trans}/T_{trans}\), where \(\Delta H_{trans}\) is the enthalpy of the transition (e.g., vaporization or fusion).
Entropy increases when moving from a more ordered state (liquid) to a less ordered state (gas). Molecular motion increases significantly during vaporization, leading to a positive \(\Delta S\) value. Conversely, freezing or condensation results in a decrease in disorder and a negative \(\Delta S\).
Calculating Entropy Changes with Temperature Variation
When a substance is heated or cooled between two temperatures, \(T_1\) and \(T_2\), without undergoing a phase change, the temperature is not constant. Therefore, the simple \(\Delta S = q_{rev}/T\) formula cannot be used directly. Instead, the calculation requires integrating the change in heat divided by the continuously changing temperature over the desired range.
For a process occurring at constant pressure, the heat transferred is related to the constant pressure heat capacity (\(C_p\)) and the temperature change (\(dT\)) by \(dq_{rev} = n C_p dT\), where \(n\) is the number of moles. Substituting this into the integral form of the entropy equation yields the practical formula: \(\Delta S = n C_p \ln(T_2/T_1)\).
This method applies to heating or cooling a single physical phase (solid, liquid, or gas) as long as no phase change occurs within the temperature range. \(C_p\) represents the heat energy required to raise the temperature of the substance by one degree. Since \(C_p\) is positive, this formula confirms that entropy always increases as temperature increases.
Calculating Entropy Changes for Chemical Reactions
The most practical method to determine the change in entropy for a chemical transformation involves using tabulated standard molar entropy values. This approach calculates the standard entropy change (\(\Delta S^\circ\)) for a reaction, which is the entropy change when the reaction occurs under standard conditions, typically 298 K (25°C) and 1 bar pressure.
The standard molar entropy (\(S^\circ\)) for a substance is the absolute entropy of one mole of that substance at standard conditions, measured relative to a perfect crystal at absolute zero. Unlike enthalpy of formation, the standard molar entropies of elements in their standard state are not zero. These values are experimentally determined and compiled in thermodynamic tables.
The standard entropy change for the overall chemical reaction is calculated by summing the standard molar entropies of the products and subtracting the sum of the reactants. This is expressed by the summation formula: \(\Delta S^\circ = \sum S^\circ (\text{products}) – \sum S^\circ (\text{reactants})\). Stoichiometric coefficients from the balanced chemical equation must be used as multipliers for each substance’s \(S^\circ\) value.