Thermodynamic Equations

Why This Matters

Thermodynamics is the framework that explains why reactions happen, how energy flows, and whether a process will occur spontaneously. In General Chemistry II, you need 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.

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.

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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.

Be careful with sign conventions. Some textbooks define the First Law as $\Delta U = Q + W$, where $W$ represents work done on the system. The physics is identical, but the sign of $W$ flips. Check which convention your course uses before plugging numbers in.

Second Law of Thermodynamics

• The entropy of an isolated system never decreases. This defines the direction of spontaneous change.
• Real processes are irreversible and always increase the 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

• The entropy of a perfect crystal at 0 K equals zero. This provides the absolute reference point for entropy measurements.
• Because of this defined starting point, absolute entropies (often written $S°$) can be looked up and used directly. Enthalpy has no such absolute reference, which is why we only ever measure changes in enthalpy ($\Delta H$).
• 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. Enthalpy is a state function, so only initial and final states matter, not the path taken.

Enthalpy Change Equation

• $\Delta H = H_{products} - H_{reactants}$ measures heat absorbed or released at constant pressure.
• Negative $\Delta H$ = exothermic (heat released to surroundings). Positive $\Delta H$ = endothermic (heat absorbed from surroundings).
• Because $\Delta H$ is a state function, you can calculate it indirectly through any convenient pathway.

Hess's Law

Hess's Law says enthalpy changes are additive: the total $\Delta H$ for a reaction equals the sum of $\Delta H$ values for individual steps, regardless of the pathway taken.

To use Hess's Law in practice:

1. Write out the target reaction you need $\Delta H$ for.
2. Identify the given reactions whose enthalpies you know.
3. Manipulate each given reaction as needed: reverse it (which flips the sign of $\Delta H$) or multiply all coefficients by a factor (which multiplies $\Delta H$ by the same factor).
4. Add the manipulated reactions so that intermediate species cancel, leaving only the target reaction.
5. Sum the adjusted $\Delta H$ values to get $\Delta H$ for the target reaction.

When you have standard enthalpies of formation ($\Delta H_f°$), there's a direct shortcut:

$\Delta H_{rxn}° = \Sigma \, n \, \Delta H_f°(\text{products}) - \Sigma \, n \, \Delta H_f°(\text{reactants})$

where $n$ represents the stoichiometric coefficients. Remember that $\Delta H_f°$ for any element in its standard state is zero by definition.

Born-Haber Cycle

The Born-Haber cycle is a specific application of Hess's Law used to find lattice energy for ionic compounds. Lattice energy can't be measured directly in the lab, so you break ionic compound formation into measurable steps:

1. Sublimation of the metal (solid → gas)
2. Ionization energy (removing electrons from the gaseous metal)
3. Bond dissociation of the nonmetal (e.g., splitting $Cl_2$ into individual atoms)
4. Electron affinity (adding electrons to the gaseous nonmetal atoms)
5. Formation of the ionic solid from gaseous ions (this is the lattice energy step)

Since you know $\Delta H_f°$ for the compound and the enthalpy of each other step, you solve for lattice energy algebraically. The key is setting up the cycle so that all steps sum to the overall formation reaction, then isolating the unknown.

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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 occur at constant temperature and pressure, so they have clean entropy expressions:
  • $\Delta S_{fus} = \frac{\Delta H_{fus}}{T_m}$ (at the melting point)
  • $\Delta S_{vap} = \frac{\Delta H_{vap}}{T_b}$ (at the boiling point)
• Sign interpretation: positive $\Delta S$ means increased disorder (more microstates accessible); negative $\Delta S$ means decreased disorder.

For standard reaction entropy changes, use a formula parallel to the enthalpy one:

$\Delta S_{rxn}° = \Sigma \, n \, S°(\text{products}) - \Sigma \, n \, S°(\text{reactants})$

Note a critical difference from the enthalpy version: unlike $\Delta H_f°$, the standard entropy $S°$ of elements is not zero. Every substance above 0 K has a positive absolute entropy (Third Law). You use the actual tabulated $S°$ values for every species, elements included.

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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

$\Delta G = \Delta H - T\Delta S$

This combines enthalpy and entropy to determine spontaneity at constant temperature and pressure.

Spontaneity criteria:

• $\Delta G < 0$: spontaneous (thermodynamically favorable)
• $\Delta G > 0$: non-spontaneous
• $\Delta G = 0$: system is at equilibrium

Temperature dependence is the most commonly tested aspect. When $\Delta H$ and $\Delta S$ have the same sign, temperature determines which term dominates:

For the last two cases, the crossover temperature where $\Delta G = 0$ is:

$T = \frac{\Delta H}{\Delta S}$

Make sure $\Delta H$ and $\Delta S$ are in compatible units before dividing (both in J or both in kJ). A very common mistake is using $\Delta H$ in kJ and $\Delta S$ in J/K, which gives a crossover temperature off by a factor of 1000.

Two other relationships connect $\Delta G$ to equilibrium and non-standard conditions:

• $\Delta G° = -RT\ln K$ links the standard free energy change to the equilibrium constant. A large positive $K$ (products favored) corresponds to a negative $\Delta G°$, and vice versa.
• $\Delta G = \Delta G° + RT\ln Q$ adjusts for non-standard conditions using the reaction quotient $Q$. When $Q = K$, the $RT\ln Q$ term exactly cancels $\Delta G°$, giving $\Delta G = 0$ (equilibrium).

Van 't Hoff Equation

$\ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H°}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$

This predicts how the equilibrium constant changes with temperature.

• Endothermic reactions ($\Delta H° > 0$): $K$ increases as temperature increases.
• Exothermic reactions ($\Delta H° < 0$): $K$ decreases as temperature increases.

This equation is the quantitative version of Le Chatelier's principle for temperature changes. If a problem gives you $K$ values at two temperatures and asks for $\Delta H°$, rearrange to solve:

$\Delta H° = -R \cdot \frac{\ln(K_2/K_1)}{(1/T_2 - 1/T_1)}$

Here, $R = 8.314 \text{ J/(mol·K)}$ and temperatures must be in Kelvin. The equation assumes $\Delta H°$ is roughly constant over the temperature range, which is a good approximation for moderate temperature changes.

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Heat Transfer Calculations

These equations handle the practical matter of calculating how much heat is involved in temperature changes and phase changes. They're essential for calorimetry problems.

Heat Capacity Equations

• $q = mc\Delta T$ calculates heat from mass ($m$), specific heat capacity ($c$), and temperature change ($\Delta T$). You can also write this as $q = nC\Delta T$ using moles ($n$) and molar heat capacity ($C$).
• This equation only applies when there's a temperature change with no phase change. During a phase change (melting, boiling), temperature stays constant and you use $q = n\Delta H_{fus}$ or $q = n\Delta H_{vap}$ instead.
• $C_p > C_v$ for gases because heating at constant pressure requires additional energy to do expansion work against the surroundings. For an ideal gas, the relationship is $C_p - C_v = R$.
• Calorimetry principle: $q_{released} = -q_{absorbed}$. The heat lost by one substance equals the heat gained by another (assuming no heat escapes the calorimeter).

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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 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 a problem 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?