Thermodynamics Equations

Why This Matters

Thermodynamics is the backbone of predicting whether reactions happen and how much energy they involve. On the AP Chemistry exam, you're being tested on your ability to connect energy conservation, heat transfer, and spontaneity—not just plug numbers into formulas. These equations show up everywhere: in calorimetry calculations, Hess's Law manipulations, and predicting reaction direction using Gibbs free energy. Master the relationships between enthalpy, entropy, and free energy, and you'll unlock the logic behind Unit 6 and Unit 9.

Don't just memorize these equations—understand what each one tells you about a system. Can you explain why $\Delta G$ combines both $\Delta H$ and $\Delta S$? Do you know when to use $q = mc\Delta T$ versus $q = C\Delta T$? The exam rewards students who see thermodynamics as a connected framework, not a list of disconnected formulas. Each equation below represents a specific concept: energy conservation, heat measurement, enthalpy pathways, or spontaneity criteria.

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Energy Conservation and the First Law

The first law of thermodynamics is fundamentally about bookkeeping—energy entering a system must equal energy leaving plus any change stored internally. This principle underlies every calorimetry problem and connects heat and work to internal energy changes.

First Law of Thermodynamics: $\Delta U = q + w$

• Internal energy change ($\Delta U$)—equals heat transferred to the system ($q$) plus work done on the system ($w$)
• Energy conservation means energy transforms between forms but the total remains constant in any process
• Sign conventions matter: $q > 0$ when heat is absorbed (endothermic), $w > 0$ when work is done on the system (compression)

Enthalpy-Internal Energy Relationship: $\Delta H = \Delta U + P\Delta V$

• Enthalpy ($\Delta H$)—the heat change at constant pressure, which is what most lab reactions experience
• $P\Delta V$ term accounts for expansion/compression work; for reactions with no gas volume change, $\Delta H \approx \Delta U$
• Constant-pressure calorimetry (coffee-cup) directly measures $\Delta H$, while bomb calorimeters measure $\Delta U$

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Heat Transfer and Calorimetry

Calorimetry problems require you to track heat flow between system and surroundings. The key principle: heat lost by one substance equals heat gained by another ($q_{system} + q_{surroundings} = 0$).

Heat Capacity Equation: $q = mc\Delta T$

• Specific heat capacity ($c$)—the energy required to raise 1 gram of a substance by 1°C, expressed in J/g·°C
• Water's specific heat is 4.184 J/g·°C, a value you should memorize for calorimetry calculations
• $\Delta T = T_{final} - T_{initial}$; a positive $q$ means heat absorbed, negative means heat released

Calorimeter Heat Capacity: $q = C\Delta T$

• Heat capacity ($C$)—total energy to raise an entire object by 1°C, used for calorimeters themselves
• Calorimeter constant ($C_{cal}$) must be determined experimentally before accurate measurements
• Molar heat capacity ($C_m$) uses J/mol·K and appears when working with mole-based calculations

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Enthalpy Pathways and Hess's Law

Because enthalpy is a state function, the pathway doesn't matter—only the starting and ending states. This allows you to calculate $\Delta H$ for reactions that can't be measured directly by combining known reactions.

Hess's Law: $\Delta H_{total} = \Delta H_1 + \Delta H_2 + \Delta H_3 + ...$

• Additivity of enthalpy—sum the $\Delta H$ values of individual steps to find the overall reaction enthalpy
• Manipulating equations: reversing a reaction flips the sign of $\Delta H$; multiplying coefficients multiplies $\Delta H$ by the same factor
• Intermediate species cancel when you add equations, leaving only reactants and products of the target reaction

Standard Enthalpy of Formation: $\Delta H°_{rxn} = \Sigma \Delta H°_f (products) - \Sigma \Delta H°_f (reactants)$

• $\Delta H°_f$—enthalpy change when 1 mole of compound forms from elements in their standard states
• Elements in standard states have $\Delta H°_f = 0$ by definition (e.g., $O_2(g)$, $C(graphite)$)
• Table lookup method lets you calculate any reaction's $\Delta H°$ without experimental data

Born-Haber Cycle

• Lattice energy determination—breaks ionic compound formation into measurable steps: sublimation, ionization, electron affinity, and bond dissociation
• Applies Hess's Law to ionic solids where direct lattice energy measurement is impossible
• Stability analysis: larger lattice energies indicate more stable ionic compounds with higher melting points

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Spontaneity and Free Energy

Spontaneity isn't just about energy release—it's about the balance between enthalpy and entropy. The Gibbs free energy equation captures this relationship and tells you whether a process will occur naturally.

Gibbs Free Energy: $\Delta G = \Delta H - T\Delta S$

• Spontaneity criterion: $\Delta G < 0$ means spontaneous, $\Delta G > 0$ means non-spontaneous, $\Delta G = 0$ means equilibrium
• Temperature dependence: at high $T$, the $T\Delta S$ term dominates; at low $T$, $\Delta H$ dominates
• Four scenarios to memorize: exothermic + entropy increase (always spontaneous), endothermic + entropy decrease (never spontaneous), and two temperature-dependent cases

Entropy Change: $\Delta S = \frac{q_{rev}}{T}$

• Entropy ($\Delta S$)—measures the dispersal of energy or randomness in a system, in units of J/mol·K
• Reversible heat transfer at temperature $T$ defines the entropy change for a process
• Entropy increases when gases form, solids dissolve, or temperature rises; decreases for the reverse

Second Law: $\Delta S_{universe} > 0$ for Spontaneous Processes

• Universe entropy always increases for any real (spontaneous) process—this is nature's direction
• $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$; both must be considered together
• Connects to $\Delta G$: the Gibbs equation is derived from requiring $\Delta S_{universe} > 0$

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Temperature Dependence Equations

These equations connect thermodynamic quantities to temperature changes, essential for understanding phase transitions and equilibrium shifts.

Clausius-Clapeyron Equation: $\ln\left(\frac{P_2}{P_1}\right) = \frac{\Delta H_{vap}}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)$

• Vapor pressure-temperature relationship—predicts how vapor pressure changes with temperature using $\Delta H_{vap}$
• $R = 8.314$ J/mol·K when $\Delta H_{vap}$ is in J/mol; temperatures must be in Kelvin
• Linear form: plotting $\ln P$ vs. $1/T$ gives a straight line with slope $-\Delta H_{vap}/R$

Van 't Hoff Equation: $\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H°}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

• Equilibrium constant temperature dependence—shows how $K$ changes when temperature changes
• Endothermic reactions ($\Delta H° > 0$): $K$ increases with temperature; exothermic: $K$ decreases
• Connects to Le Chatelier's principle quantitatively—predicts how much $K$ shifts, not just direction

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Quick Reference Table

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Self-Check Questions

1. Which two equations share the same mathematical form relating a thermodynamic quantity to temperature, and what distinguishes when you use each one?

2. A reaction has $\Delta H > 0$ and $\Delta S > 0$. Under what temperature conditions will this reaction be spontaneous, and which term in the Gibbs equation explains why?

3. Compare the information you get from a coffee-cup calorimeter versus a bomb calorimeter. Which thermodynamic quantity does each measure directly, and why do they differ?

4. If an FRQ gives you $\Delta H°_f$ values for all species and asks for $\Delta H°_{rxn}$, what equation do you use? How would your approach change if you were given a series of reactions with their $\Delta H$ values instead?

5. Explain why $\Delta H < 0$ alone is insufficient to determine spontaneity. What additional information do you need, and how does the Gibbs equation incorporate it?