Thermodynamic Equations

Why This Matters

Thermodynamics isn't just about memorizing formulas—it's the framework that explains why reactions happen, how energy flows, and whether a process will occur spontaneously. In General Chemistry II, you're being tested on your ability to connect enthalpy, entropy, and Gibbs free energy into a coherent picture of chemical change. These equations show up everywhere: predicting reaction spontaneity, calculating heat transfer, understanding equilibrium shifts, and analyzing phase changes.

Here's the key insight: every thermodynamic equation answers a specific question about energy or disorder. The First Law asks "where did the energy go?" Gibbs free energy asks "will this happen on its own?" The Van 't Hoff equation asks "how does temperature shift equilibrium?" Don't just memorize the math—know what question each equation answers and when to apply it. That's what separates a 3 from a 5 on exam day.

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The Fundamental Laws: Energy Conservation and Entropy

These are the bedrock principles that govern all thermodynamic processes. Every other equation in thermodynamics derives from or connects back to these laws.

First Law of Thermodynamics

• Energy is conserved—it transforms between forms (heat, work, internal energy) but is never created or destroyed
• $\Delta U = Q - W$ relates internal energy change to heat added ($Q$) minus work done by the system ($W$)
• Sign conventions matter: positive $Q$ means heat flows into the system; positive $W$ means work done by the system

Second Law of Thermodynamics

• Entropy of an isolated system never decreases—this defines the direction of spontaneous change
• Irreversibility is built into nature; real processes always increase total entropy of the universe
• Spontaneity criterion: for any spontaneous process, $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} > 0$

Third Law of Thermodynamics

• Entropy of a perfect crystal at 0 K equals zero—this provides the absolute reference point for entropy measurements
• Absolute entropies can be calculated because we have this defined starting point, unlike enthalpy which only measures changes
• Practical implication: all substances have positive entropy values at temperatures above absolute zero

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Enthalpy: Tracking Heat Flow

Enthalpy equations quantify heat transfer at constant pressure, which is how most reactions occur in the lab. The key insight is that enthalpy is a state function—only initial and final states matter, not the path.

Enthalpy Change Equation

• $\Delta H = H_{products} - H_{reactants}$ measures heat absorbed or released at constant pressure
• Negative $\Delta H$ means exothermic (heat released); positive $\Delta H$ means endothermic (heat absorbed)
• State function property: $\Delta H$ depends only on initial and final states, enabling indirect calculations

Hess's Law

• Enthalpy changes are additive—total $\Delta H$ equals the sum of $\Delta H$ values for individual steps, regardless of pathway
• Calculation strategy: manipulate known reactions (reverse them, multiply coefficients) to match your target reaction
• Standard enthalpies of formation ($\Delta H_f°$) provide the data you need: $\Delta H_{rxn}° = \Sigma \Delta H_f°(products) - \Sigma \Delta H_f°(reactants)$

Born-Haber Cycle

• Lattice energy calculation—breaks ionic compound formation into measurable steps using Hess's Law
• Component steps include sublimation, ionization energy, electron affinity, and bond dissociation, each with known enthalpy values
• Application: determines lattice energy indirectly since it cannot be measured directly in the lab

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Entropy: Quantifying Disorder

Entropy measures the dispersal of energy and matter. These equations let you calculate entropy changes for specific processes.

Entropy Change Equation

• $\Delta S = \frac{Q_{rev}}{T}$ calculates entropy change from reversible heat transfer at temperature $T$ (in Kelvin)
• Phase transitions have characteristic entropy changes: $\Delta S_{fus} = \frac{\Delta H_{fus}}{T_m}$ and $\Delta S_{vap} = \frac{\Delta H_{vap}}{T_b}$
• Sign interpretation: positive $\Delta S$ means increased disorder (more microstates); negative $\Delta S$ means decreased disorder

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Spontaneity and Equilibrium: The Big Picture Equations

These equations integrate enthalpy and entropy to predict whether reactions occur and how equilibrium responds to conditions. This is where thermodynamics becomes predictive.

Gibbs Free Energy Equation

• $\Delta G = \Delta H - T\Delta S$ combines enthalpy and entropy to determine spontaneity at constant $T$ and $P$
• Spontaneity criteria: $\Delta G < 0$ (spontaneous), $\Delta G > 0$ (non-spontaneous), $\Delta G = 0$ (at equilibrium)
• Temperature dependence: when $\Delta H$ and $\Delta S$ have the same sign, temperature determines spontaneity

Van 't Hoff Equation

• $\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$ predicts how equilibrium constant changes with temperature
• Endothermic reactions ($\Delta H > 0$): $K$ increases with increasing temperature
• Connects to Le Chatelier's principle quantitatively—this equation is the mathematical proof of temperature effects on equilibrium

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Heat Transfer Calculations

These equations handle the practical matter of calculating how much heat is involved in temperature changes. Essential for calorimetry problems.

Heat Capacity Equations

• $q = mc\Delta T$ calculates heat from mass ($m$), specific heat capacity ($c$), and temperature change ($\Delta T$)
• $C_p > C_v$ for gases because constant-pressure heating requires additional work to expand against external pressure
• Calorimetry applications: use $q_{released} = -q_{absorbed}$ to find unknown specific heats or final temperatures

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Quick Reference Table

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Self-Check Questions

1. Which two equations both rely on enthalpy being a state function, and how do their applications differ?

2. A reaction has $\Delta H < 0$ and $\Delta S < 0$. Using the Gibbs equation, at what temperature conditions will this reaction be spontaneous?

3. Compare and contrast how the Second Law of Thermodynamics and the Gibbs Free Energy equation each predict spontaneity—when would you use one versus the other?

4. If an FRQ gives you equilibrium constants at two temperatures and asks for $\Delta H$, which equation do you use and what algebraic rearrangement is required?

5. Why does the Third Law of Thermodynamics allow us to calculate absolute entropies but not absolute enthalpies? What reference point does each quantity use?