CHEM 156: Basic Physical Chemistry II - Second and Third Laws of Thermodynamics

Overview of Spontaneous and Nonspontaneous Processes

• Spontaneous Process:     * Definition: A spontaneous process is one that proceeds of its own accord without any external assistance.     * Spontaneity: Generally, the natural tendency of a process to occur is referred to as its spontaneity.     * Examples provided:         * Heat flow (from hot to cold).         * Gas flow.         * Melting of ice.         * Rusting of iron.

• Nonspontaneous Process:     * Definition: A process that does not proceed on its own and requires external intervention or energy input.     * Examples provided:         * Charging of a battery.         * Photosynthesis.         * Electrolysis of water.

Characteristics of a Spontaneous Process

• 1. Unidirectional Process:     * A spontaneous change is a one-way process that occurs naturally in a specific direction.     * To reverse a spontaneous process, external work must be performed, making the reverse direction nonspontaneous.

• 2. Time Independence:     * Spontaneity is distinct from the rate of reaction; being spontaneous does not necessarily mean a process is fast.     * Spontaneous processes can occur at varying speeds: combustion is very quick, while the rusting of iron is very slow.

• 3. Driven by Instability (Away from Equilibrium):     * If a system is not currently at equilibrium, it will undergo a spontaneous change toward the state of equilibrium.     * This movement toward equilibrium is inevitable unless there is external interference.

• 4. Equilibrium as the Final State:     * Once a system reaches equilibrium, no further spontaneous changes will occur if the system remains undisturbed.     * Disrupting this equilibrium requires the application of external energy or work.

• 5. Relation to Free Energy ($\Delta G$) and Enthalpy ($\Delta H$):     * Most spontaneous changes are accompanied by a decrease in the free energy of the system (\Delta G < 0).     * A decrease in enthalpy ($\Delta H$) often accompanies spontaneous processes, but it is not a requirement.     * Processes can be spontaneous even with an increase in enthalpy if the increase in entropy ($\Delta S$) is sufficiently large to ensure that the overall Gibbs free energy remains negative.

Concept of Entropy ($S$)

• Definition:     * Entropy is a thermodynamic state quantity that measures the randomness or disorder of the molecules within a system.     * It can also be defined as the measure of the extent to which energy is dispersed throughout a system.     * Changes or processes that increase randomness or disorder are more likely to occur spontaneously than those that bring about order.

• Properties of Entropy:     * Entropy is a state function, meaning its value depends only on the initial state ($i$) and the final state ($f$) of the system.     * The change in entropy is denoted as $\Delta S$:         * $\Delta S = S_f - S_i$     * A process accompanied by an increase in entropy tends to be spontaneous.     * For spontaneous processes: S_f > S_i, therefore $\Delta S$ is positive (\Delta S > 0).

• Discussion Point:     * Is entropy an intensive or extensive property of a system? (Extensive properties depend on the amount of matter in a sample; intensive properties do not).

The Second Law of Thermodynamics

• Core Statement:     * The second law states that whenever a spontaneous process occurs, it is accompanied by an increase in the entropy of the universe.     * The "universe" is defined as the combination of the system and its surroundings.     * $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$

• Principles of Entropy Change:     * Irreversible Spontaneous Processes: When these occur, the total entropy of the system and the surroundings increases (\Delta S_{univ} > 0).     * Reversible Processes: When a reversible process occurs, the total entropy of the system remains constant ($\Delta S_{univ} = 0$).     * Concise Statement: The entropy of the universe is constantly increasing.

Consequences of the Second Law of Thermodynamics

• 1. Irreversibility of Natural Processes:     * Natural processes are one-way (e.g., heat flows from hot to cold spontaneously).     * Entropy prevents the spontaneous "un-mixing" of substances, such as milk from coffee, or the "un-burning" of wood.

• 2. Limits on Heat-to-Work Conversion:     * Heat cannot be fully converted into useful work. No heat engine can achieve $100\%$ efficiency; some energy is always lost as waste heat.     * This principle limits the efficiency of power plants, turbines, and engines.

• 3. Tendency Toward Greater Disorder:     * Systems naturally move toward high entropy states. Examples include gases spontaneously expanding to fill a volume and ice melting at room temperature.

• 4. Direction of Time ("Arrow of Time"):     * The Second Law assigns a direction to time, moving from low entropy (order) to high entropy (disorder). In a thermodynamic sense, time cannot be reversed.

• 5. Equilibrium as Maximum Entropy:     * Systems evolve toward thermodynamic equilibrium, the state where entropy is maximized and spontaneous change ceases.

• 6. Work Requirement for Refrigeration:     * Heat does not flow spontaneously from cold to hot. To transfer heat "uphill" (as in air conditioners or refrigerators), external work must be performed.

• 7. Maintenance of Living Systems:     * Living systems survive by importing energy from their environment to avoid internal disorder/entropy increase.     * Plants import energy via sunlight (photosynthesis), and animals via food.     * This energy is used to maintain order (growth, repair, reproduction, movement).     * Living systems release waste energy (heat) and products ($CO_2$, etc.) back into the environment, increasing the environment's entropy.

• 8. Heat Death:     * A theoretical ultimate consequence where the universe reaches maximum entropy and no more work can be performed.

The Third Law of Thermodynamics

• Temperature Dependence:     * The entropy of a substance varies directly with its temperature.     * As temperature decreases, entropy decreases.

• Core Statement:     * At absolute zero temperature ($0\,K$), the entropy of a pure crystal is zero.     * Mathematically: $S = 0$ at $T = 0\,K$.

Numerical and Statistical Definitions of Entropy

• Clausius’ Thermodynamic Formulation (1850-1865):     * In a system not undergoing physical/chemical changes, entropy is constant when there is no heat transfer.     * If heat ($q$) flows into a system, entropy increases by $\frac{q}{T}$. If heat flows out, entropy decreases.     * For a reversible change at a fixed temperature ($T$):         * $\Delta S = \frac{q_{rev}}{T}$     * Signs and Interpretation:         * If heat is absorbed ($+q_{rev}$), $\Delta S$ is positive (increase in entropy).         * If heat is evolved ($-q_{rev}$), $\Delta S$ is negative (decrease in entropy).     * Units of Entropy:         * Entropy units ('eu'): $cal\,K^{-1}\,mol^{-1}$.         * Standard SI units: $J\,K^{-1}\,mol^{-1}$.         * Conversion: $1\,eu = 1\,cal\,K^{-1}\,mol^{-1} = 4.184\,J\,K^{-1}\,mol^{-1}$.

• Boltzmann’s Statistical Formulation (1877):     * Entropy is a measure of the statistical disorder related to the number of microstates compatible with a system's energy.     * Formula:         * $S = k \ln(W)$         * $W$: The number of microstates.         * $k$: Boltzmann’s constant ($k = 1.38 \times 10^{-23}\,J/K$).     * Microstates:         * A microstate is a specific configuration of the locations and energies of all atoms or molecules in a system.         * A system with a higher number of microstates is more disordered (higher entropy) than one with fewer microstates.

Applied Discussions and Calculations

• Microstate and Entropy Scenarios:     * Valve Opening (Gas Expansion): Upon opening a valve between two containers, the number of available microstates ($W$) increases significantly as particles occupy more volume, leading to \Delta S > 0.     * Melting of Ice: Transitioning from solid to liquid increases molecular mobility and randomness ($W$ increases), therefore \Delta S > 0.     * Freezing of Meat: Transitioning from liquid/flexible state to a solid crystalline structure decreases randomness, therefore \Delta S < 0.     * Hot Tea (Cooling): As heat leaves the system into the surroundings, the entropy of the tea system decreases (\Delta S < 0), though the entropy of the surroundings increases by a larger amount.

• Standard Entropy ($S^{\circ}$):     * Absolute Entropy: The actual amount of entropy a substance has at any temperature above $0\,K$.     * Standard Entropy ($S^{\circ}$): Absolute entropy at $25^{\circ}C$ ($298\,K$) and $1\,atm$ pressure.     * The entropy of any substance above $0\,K$ is always a positive value.

• Chemical Reaction Calculations:     * Standard Entropy Change ($\Delta S^{\circ}$): Calculated as the sum of standard entropies of products minus the sum of standard entropies of reactants:         * $\Delta S^{\circ} = \sum S^{\circ}(\text{products}) - \sum S^{\circ}(\text{reactants})$     * Standard Entropy of Formation ($\Delta S^{\circ}_f$): The entropy change for the formation of one mole of a compound from its constituent elements under standard conditions:         * $\Delta S^{\circ}_f = \sum S^{\circ}(\text{compound}) - \sum S^{\circ}(\text{elements})$