Review

1. Checklist of Key Concepts

1. Entropy as a State Function

   • Definition: For a reversible process, the change in entropy is

   \[dS \;=\;\frac{\delta q_\text{rev}}{T}\]

   • Key Property: \(S\) depends only on the initial and final states—not on the path taken.

   • Physical Meaning: Measures dispersal of energy; quantifies how heat “spreads out” in a system.

2. Spontaneity and the Second Law

   • Clausius Statement: Heat does not spontaneously flow from cold to hot.

   • Entropy Criterion:

     • For an isolated system, \(\Delta S_\text{tot} \ge 0\).

     • Equality holds only for reversible processes; strict inequality for irreversible ones.

   • Macroscopic Implication: Determines the “arrow of time”—which processes can occur spontaneously.

3. Carnot Cycle & Maximum Efficiency

   • Reversible Engine: Comprised of two isotherms \(\left( T_h, T_c \right)\) and two adiabats.

   • Efficiency:

   \[\eta_\text{Carnot} \;=\; 1 - \frac{T_c}{T_h} \quad\Longrightarrow\quad \text{This is the upper limit for any heat engine.}\]

   • Significance: Links thermodynamic reversibility to practical limits on converting heat to work.

4. Microscopic Origin of Entropy

   • Boltzmann’s Formula:

   \[S \;=\; k_B \ln \Omega\]

   where \(\Omega\) is the number of accessible microstates at given \(U, V, N\).

   • Statistical Interpretation: Entropy quantifies uncertainty about which microstate the system occupies.

   • Approach to Equilibrium: Systems evolve towards macrostates with higher \(\Omega\); thus \(S\) increases until equilibrium.

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2. Checklist of Most Important Equations

Equation	Variable Definitions	Meaning & Use Cases	Caveats/Approximations
\(dS = \frac{\delta q_\text{rev}}{T}\)	\(\delta q_\text{rev}\): infinitesimal heat in a reversible process \(T\): absolute temperature	Fundamental relation defining entropy. Use to compute \(\Delta S\) for isothermal, isobaric, mixing, etc. processes.	Valid only for reversible paths; for irreversible processes, \(\Delta S > \int \delta q/T\).
\(dU = T\,dS - P\,dV\)	\(U\): internal energy \(P\): pressure \(V\): volume	First law written in natural variables \((S, V)\). Use to derive Maxwell relations and fundamental thermodynamic identities.	Assumes \(U=U(S,V)\) is a well-defined state function; neglects non-PV work.
\(\eta_\text{Carnot} = 1 - \dfrac{T_c}{T_h}\)	\(T_h\), \(T_c\): temperatures of hot and cold reservoirs	Sets the maximum efficiency for any heat engine operating between \(T_h\) and \(T_c\). Use in engine design and in defining thermodynamic reversibility.	Requires fully reversible (Carnot) cycle; real engines incur friction, heat leaks, and irreversibilities.
\(\Delta S_\text{tot} \ge 0\)	\(\Delta S_\text{tot}\): entropy change of system + surroundings	General statement of the Second Law for isolated systems. Predicts direction of spontaneous processes.	Equality only for reversible processes; in open or non-isolated systems, must include entropy change of environment.
\(S = k_B \ln \Omega\)	\(k_B\): Boltzmann constant \(\Omega\): number of microstates	Bridges macroscopic entropy with molecular statistics. Use to calculate \(S\) from combinatorial counts or partition functions.	Applies to isolated systems with well-defined \(\Omega\). Counting microstates can be intractable for large systems.
\(p_i = \dfrac{e^{-\beta E_i}}{Q}\); \(\beta = 1/(k_B T)\)	\(E_i\): energy of microstate \(i\) \(Q\): canonical partition function	Probability of occupying microstate \(i\) in the canonical ensemble. Use in statistical mechanics to compute ensemble averages.	Assumes equilibrium with a heat bath at fixed \(T\); neglects quantum degeneracy unless included in \(E_i\).
\(S = \frac{U}{T} + k_B \ln Q\)	\(Q = \sum_i e^{-\beta E_i}\): partition function	Expresses entropy in terms of the partition function and internal energy. Use to derive free energies, heat capacities, and response functions.	Derived under canonical ensemble assumptions. Valid when \(U\) and \(Q\) are known or can be approximated.