The Second Law of Thermodynamics, Entropy, and the Carnot Cycle

Chapter 5A: The Second Law of Thermodynamics

Spontaneous and Nonspontaneous Processes

The Second Law of Thermodynamics helps us understand the direction of spontaneous change and its importance to chemical systems and processes.

• Spontaneous Process: A process that occurs naturally under given conditions and does not require work to be done to bring about the change.

• Nonspontaneous Process: A process that requires work to be done to bring about the change.

Spontaneity is a key concept in predicting whether a chemical or physical process will occur without external intervention.

State Functions and Entropy

We introduce a new state function, entropy (S), which is a measure of disorder or randomness in a system. Entropy is defined by modifying the heat transfer (q) with an integrating factor (1/T):

• For a reversible process: $dS = \frac{dq_{rev}}{T}$

The First Law of Thermodynamics states that energy is conserved in isolated systems, but it does not predict the direction of processes. The Second Law of Thermodynamics states that the total entropy of the universe must increase for a process to occur spontaneously and irreversibly.

Definitions of the Second Law

• Reversible Case: The total entropy change of the universe is zero ($\Delta S_{univ} = 0$).

• Irreversible Case: The total entropy change of the universe is greater than zero ($\Delta S_{univ} > 0$).

Calculating Entropy Change ($\Delta S$)

There are specific guidelines for calculating entropy changes in different scenarios:

• For phase changes: Use $\Delta S = \frac{q_{rev}}{T}$, where $q_{rev}$ is the heat absorbed or released reversibly at constant temperature.

• For temperature changes: Use $\Delta S = nC \ln\left(\frac{T_2}{T_1}\right)$, where $C$ is the heat capacity, $n$ is the number of moles, and $T_1$ and $T_2$ are the initial and final temperatures.

It is important to know how to modify these equations for ideal gases and other conditions (isothermal, isochoric, isobaric, adiabatic).

Visualizing and Interpreting Entropy

• Entropy quantifies disorder, which arises from the distribution of energy among microstates.

• The macroscopic state we observe is made up of many microscopic states (microstates).

• The number of microstates ($W$) is related to entropy by Boltzmann's equation: $S = k_B \ln W$, where $k_B$ is Boltzmann's constant.

Processes where entropy increases or decreases can be predicted by considering the change in the number of microstates.

Chapter 5B: Entropy and the Third Law of Thermodynamics

Clausius Inequality and Entropy Change

The Clausius Inequality states that the amount of heat exchanged between the system and surroundings for a reversible process is always greater than for an irreversible process:

• For a reversible process: $\Delta S_{univ} = 0$

• For an irreversible process: $\Delta S_{univ} > 0$

For an isolated system, $\Delta S_{univ}$ has meaning only in terms of entropy, not energy.

Guidelines for Calculating $\Delta S$

• Reversible process (at equilibrium): $\Delta S_{sys} = \frac{q_{rev}}{T}$

• Irreversible process (not at equilibrium): $\Delta S_{sys} \neq \frac{q}{T}$; instead, calculate $\Delta S$ using state functions, not the actual path.

• Total entropy change: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr}$

The temperature of the surroundings often determines whether a process is spontaneous.

The Third Law of Thermodynamics

The Third Law states that the entropy of a perfect crystal at absolute zero is zero. This law allows us to determine absolute entropies for substances.

• Standard entropy of reaction ($\Delta S^\circ$): Calculated from standard entropies of products and reactants:

$\Delta S^\circ = \sum n S^\circ_{products} - \sum n S^\circ_{reactants}$

Residual vs. Absolute Entropy

• Absolute entropy: The total entropy of a substance at a given temperature, measured from absolute zero.

• Residual entropy: The entropy present in a substance at absolute zero due to disorder in the arrangement of molecules (e.g., in crystals with positional disorder).

Heat Engines, Efficiency, and the Carnot Cycle

Heat Engines and the Second Law

Heat engines convert disordered molecular energy (heat, $q$) into ordered macroscopic energy (work, $w$). In a cyclic process, the working substance returns to its initial state, and the net change in state functions over one cycle is zero.

• A system can never convert all absorbed heat into work on the surroundings (Second Law).

• Efficiency: The fraction of work output achieved per heat energy input.

• Heat engines and heat pumps move heat from a hot reservoir to a cold reservoir, rejecting any energy not converted to work.

The Carnot Cycle

The Carnot Cycle is a hypothetical 4-step reversible cyclic process that represents the most efficient heat engine possible between two temperatures, $T_{hot}$ and $T_{cold}$.

• The four steps typically include two isothermal and two adiabatic processes.

• The efficiency of a Carnot engine is given by:

$\text{Efficiency} = 1 - \frac{T_{cold}}{T_{hot}}$

• Net work done by the system over one cycle is the area enclosed by the cycle on a PV diagram.

• Work is done by the system during expansion steps and on the system during compression steps.

Summary Table: Key Thermodynamic Quantities

Quantity	Symbol	Definition/Formula
Entropy	S	$S = k_B \ln W$
Change in Entropy (reversible)	$\Delta S$	$\Delta S = \frac{q_{rev}}{T}$
Standard Entropy Change	$\Delta S^\circ$	$\Delta S^\circ = \sum n S^\circ_{products} - \sum n S^\circ_{reactants}$
Efficiency (Carnot Engine)	—	$1 - \frac{T_{cold}}{T_{hot}}$

Example: Calculating Entropy Change for Melting Ice

• Suppose 1 mol of ice melts at 0°C (273 K). The enthalpy of fusion ($\Delta H_{fus}$) is 6.01 kJ/mol.

• Calculate $\Delta S$ for the process:

$\Delta S = \frac{q_{rev}}{T} = \frac{6.01 \times 10^3\ \text{J}}{273\ \text{K}} = 22.0\ \text{J/K}$

Additional info: The Carnot Cycle is a theoretical construct; no real engine is perfectly reversible, but the Carnot efficiency sets the upper limit for all real engines.