Entropy and the Laws of Thermodynamics: Chemistry and Engineering Applications

Entropy and the Second Law of Thermodynamics

Introduction to Entropy and Thermodynamics

Thermodynamics is a branch of physics and chemistry that studies energy transformations in physical and chemical processes. Entropy is a central concept, quantifying the degree of disorder or randomness in a system. The Second Law of Thermodynamics governs the direction of spontaneous processes and the increase of entropy in the universe.

• Entropy (S): A measure of the number of possible arrangements (microstates) of particles in a system; higher entropy means greater disorder.

• Spontaneous Process: A process that occurs without continuous external energy input.

• Second Law of Thermodynamics: In any spontaneous process, the total entropy of the universe increases: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0$

• State Function: Entropy is a state function, meaning its value depends only on the current state of the system, not the path taken.

Statistical Basis of Entropy

Entropy can be understood statistically as the number of ways particles can be arranged while maintaining the same energy.

• Microstate: A specific arrangement of particles and energy in a system.

• Boltzmann Equation: Relates entropy to the number of microstates ($Q$): $S = k_B \ln Q$ where $k_B$ is the Boltzmann constant ($1.38064852 \times 10^{-23}$ J/K).

• Probability: The probability of a highly ordered arrangement is extremely low for systems with many particles.

Entropy Changes in Physical and Chemical Processes

Several types of changes result in increased entropy:

• Phase Changes: Melting of a solid to a liquid increases entropy due to increased particle movement.

• Increase in Number of Particles: Reactions that produce more moles of gas increase entropy.

• Temperature Increase: Heating a substance increases the number of possible energy distributions (microstates).

Calculating Entropy Change

Standard molar entropy ($S^\circ$) values are used to calculate the entropy change for a reaction:

• Formula: $\Delta S^\circ = \sum S^\circ_{products} - \sum S^\circ_{reactants}$

• Example: For the reaction $2 \text{C}_2\text{H}_6(g) + \text{H}_2(g) \rightarrow \text{C}_4\text{H}_{10}(g)$, use tabulated $S^\circ$ values to compute $\Delta S^\circ$.

Spontaneity, Enthalpy, and Free Energy

Enthalpy and Spontaneity

Enthalpy (H) is the heat content of a system. While exothermic reactions (negative $\Delta H$) are often spontaneous, enthalpy alone does not determine spontaneity.

• Exothermic ($\Delta H < 0$): Likely to be spontaneous.

• Endothermic ($\Delta H > 0$): Likely to be non-spontaneous unless compensated by entropy increase.

Gibbs Free Energy

The Gibbs free energy (G) combines enthalpy and entropy to predict spontaneity at constant pressure and temperature.

• Definition: $G = H - TS$

• Change in Free Energy: $\Delta G = \Delta H - T\Delta S$

• Spontaneity Criterion: $\Delta G < 0$ → spontaneous process $\Delta G > 0$ → non-spontaneous process $\Delta G = 0$ → equilibrium

Possible Combinations of $\Delta H$ and $\Delta S$

$\Delta H$	$\Delta S$	Spontaneity
-	+	Spontaneous at all temperatures
+	-	Never spontaneous
-	-	Spontaneous only at low temperatures
+	+	Spontaneous only at high temperatures

Temperature Dependence of Spontaneity

• The temperature at which a reaction changes from spontaneous to non-spontaneous can be calculated: $T = \frac{\Delta H}{\Delta S}$

• Enthalpy-driven reactions: Spontaneous at low temperatures ($\Delta H < 0$, $\Delta S < 0$).

• Entropy-driven reactions: Spontaneous at high temperatures ($\Delta H > 0$, $\Delta S > 0$).

Gibbs Free Energy and Maximum Work

• Maximum Useful Work: The change in Gibbs free energy equals the maximum work obtainable from a process: $\Delta G = -w_{max}$

• Actual work is less than $w_{max}$ for irreversible processes.

The Third Law of Thermodynamics

Standard Molar Entropy and Absolute Zero

The Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero is zero.

• Standard Molar Entropy ($S^\circ$): Entropy of one mole of a substance under standard conditions.

• Used to calculate entropy changes in chemical reactions.

Applications: Recycling of Plastics

Manufacture and Recycling of PET Plastics

Poly(ethylene terephthalate) (PET) is commonly used for soft drink bottles. Its manufacture and recycling involve thermodynamic and economic considerations.

• PET is produced from ethylene glycol and dimethyl terephthalate in a two-step process.

• Recycling involves sorting, washing, melting, and reprocessing PET into new products.

• Recycled PET is not used for food containers due to contamination risks and polymer degradation.

• Recycling is less economically favorable for plastics than for metals like aluminum.

Thermodynamics of Polymerization and Recycling

• Polymerization reactions (e.g., formation of PMMA) are often exothermic and may have negative entropy changes.

• Thermolysis (depolymerization) is endothermic and entropy-increasing, becoming spontaneous at high temperatures.

• Example: The melting point of polyethylene can be estimated using $T = \frac{\Delta H}{\Delta S}$.

Economic and Scientific Obstacles to Recycling

• Recycled plastics have shorter polymer chains and different properties than virgin plastics.

• Aluminum recycling is more efficient and produces material with identical properties to virgin aluminum.

• Economic factors (cost of sorting, energy requirements) influence recycling rates.

Summary Table: Thermodynamic Quantities

Quantity	Symbol	Definition
Enthalpy	H	Heat content of a system
Entropy	S	Degree of disorder/randomness
Gibbs Free Energy	G	Energy available to do work

References

• Holme, T.A. Chemistry for Engineering Students.

• Masterton & Hurley. Chemistry Principles and Reactions.

• Silberberg. Chemistry: The Molecular Nature of Matter and Change.

• Brown et al. Chemistry: The Central Science.

• Zumdahl. Principles of General Chemistry.

Additional info: These notes expand on the original slides and text, providing full academic context, definitions, and examples for thermodynamic concepts relevant to both chemistry and engineering students.