Key Thermodynamic State Functions

Why This Matters

State functions are the backbone of everything you'll encounter in Physical Chemistry II—from predicting whether a reaction will happen spontaneously to understanding why ice melts at 0°C. These quantities depend only on the current state of your system, not on how it got there. That path independence is what makes them so powerful for calculations involving equilibrium, spontaneity, phase transitions, and energy transfer.

You're being tested on more than definitions here. Exam questions will ask you to choose the right state function for a given set of conditions, derive relationships between them using Maxwell relations, and predict process spontaneity. Don't just memorize formulas—know when each function applies (constant $T$? constant $P$? isolated system?) and what physical insight each one provides.

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Energy Content Functions

These state functions quantify the total energy stored in a system. Internal energy captures everything at the molecular level, while enthalpy adds a correction for systems that must "make room" against external pressure.

Internal Energy (U)

• Total microscopic energy—includes all kinetic energy of molecular motion plus potential energy from intermolecular forces
• First law foundation: $dU = \delta q + \delta w$, making $U$ the central quantity for energy conservation in closed systems
• Natural variables are $S$ and $V$—this means $U$ is most useful for processes at constant entropy or constant volume (isochoric processes)

Enthalpy (H)

• Defined as $H = U + PV$—the "pressure-volume work" term accounts for energy needed to displace the surroundings
• Constant-pressure heat flow: $\Delta H = q_P$, which is why calorimetry experiments at atmospheric pressure directly measure enthalpy changes
• Reaction energetics: bond enthalpies, Hess's law, and standard formation values ($\Delta H_f^\circ$) all rely on this function

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Disorder and Directionality

Entropy captures the statistical nature of thermodynamics—why systems evolve toward more probable configurations. It's the key to understanding irreversibility and the arrow of time.

Entropy (S)

• Measures multiplicity of microstates: $S = k_B \ln W$, connecting macroscopic thermodynamics to molecular statistics
• Second law criterion: for an isolated system, $\Delta S_{total} \geq 0$; equality holds only for reversible processes
• Temperature dependence: $dS = \frac{\delta q_{rev}}{T}$, meaning the same heat transfer produces larger entropy changes at lower temperatures

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Free Energy Functions

Free energies combine energy and entropy to predict spontaneity under specific constraints. They represent the maximum useful work extractable from a process—the quantity nature "wants" to minimize.

Gibbs Free Energy (G)

• Defined as $G = H - TS$—subtracts the "unavailable" energy (entropy term) from enthalpy
• Spontaneity at constant $T$ and $P$: $\Delta G < 0$ means spontaneous; this is the condition for most laboratory and biological processes
• Equilibrium connection: $\Delta G^\circ = -RT \ln K$, directly linking free energy to equilibrium constants

Helmholtz Free Energy (A)

• Defined as $A = U - TS$—the constant-volume analog of Gibbs free energy
• Spontaneity at constant $T$ and $V$: $\Delta A < 0$ indicates a spontaneous process; common in rigid containers and statistical mechanics
• Maximum work: $-\Delta A$ equals the maximum work (excluding $PV$ work) obtainable from a constant-$T$, constant-$V$ process

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Fundamental Measurable Properties

These intensive and extensive properties define the thermodynamic state and appear in every equation of state. They're the variables you control experimentally.

Pressure (P)

• Molecular origin: force per unit area from particle collisions with container walls; intensive property independent of system size
• Equation of state variable: appears in ideal gas law ($PV = nRT$) and more sophisticated equations like van der Waals
• Conjugate to volume: the product $P \cdot dV$ represents pressure-volume work in expansion/compression

Volume (V)

• Extensive property—doubles when you double the amount of substance; defines the spatial extent of your system
• Work calculations: $w = -\int P \, dV$ for reversible expansion; sign convention means expansion does negative work on surroundings
• Partial molar quantities: molar volume and its derivatives connect to compressibility and thermal expansion coefficients

Temperature (T)

• Average kinetic energy: $\langle KE \rangle = \frac{3}{2}k_B T$ for ideal gases; determines the "thermal intensity" of a system
• Spontaneity arbiter: the $TS$ term in free energy equations means temperature determines whether entropy or enthalpy dominates
• Phase behavior: controls which phase is stable; appears in Clausius-Clapeyron equation for phase boundaries

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Quick Reference Table

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Self-Check Questions

1. Which two state functions both predict spontaneity, and what experimental conditions determine which one to use?

2. If a reaction has $\Delta H > 0$ and $\Delta S > 0$, at what temperatures will it be spontaneous? Which state function tells you this?

3. Compare internal energy and enthalpy: for an ideal gas at constant temperature, how do $\Delta U$ and $\Delta H$ differ during expansion?

4. A bomb calorimeter operates at constant volume. Which free energy function determines spontaneity inside it, and why isn't Gibbs free energy appropriate?

5. Using the definitions $G = H - TS$ and $H = U + PV$, derive an expression for $G$ in terms of $U$, $P$, $V$, $T$, and $S$. What does this tell you about the relationship between Gibbs and Helmholtz free energy?