Gibbs Free Energy Equations

Why This Matters

Gibbs free energy tells you whether a reaction will happen on its own or whether you need to put energy in to make it go. In General Chemistry II, you're expected to connect enthalpy, entropy, and temperature into a single predictive framework. Equilibrium problems, electrochemistry questions, and phase change discussions all come back to $\Delta G$.

These equations aren't isolated formulas. They represent different ways of asking the same question: is this process thermodynamically favorable? Whether you're calculating equilibrium constants, predicting cell voltages, or figuring out if a reaction becomes spontaneous at high temperatures, you need to know which equation to reach for and why it fits the situation.

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Predicting Spontaneity: The Core Equation

Gibbs free energy balances two competing thermodynamic drives: the tendency to minimize energy (enthalpy) and the tendency to maximize disorder (entropy).

The Fundamental Gibbs Equation

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

• Negative $\Delta G$ indicates a spontaneous process. This is the single most important criterion for spontaneity at constant T and P.
• $\Delta H$ (enthalpy change) captures heat flow. Exothermic reactions ($\Delta H < 0$) favor spontaneity.
• $T\Delta S$ (entropy term) grows with temperature. Reactions with positive $\Delta S$ become more favorable as T rises.

There are four possible sign combinations of $\Delta H$ and $\Delta S$, and two of them make spontaneity temperature-dependent:

For the temperature-dependent cases, the crossover temperature where $\Delta G = 0$ is:

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

This is the temperature at which the reaction switches between spontaneous and non-spontaneous. Just set $\Delta G = 0$ in the fundamental equation and solve for T.

Gibbs Free Energy and Maximum Work

$\Delta G = w_{max}$ (at constant T and P)

• $\Delta G$ equals the maximum non-expansion work obtainable from a reversible process. Non-expansion work means any work other than pressure-volume work, such as electrical work in a battery.
• Negative $\Delta G$ means the system can perform work on its surroundings; positive $\Delta G$ means work must be done on the system.
• Real processes always yield less work than this maximum due to irreversibilities and friction.

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Connecting to Equilibrium: Standard and Non-Standard Conditions

These equations bridge thermodynamics and equilibrium. $\Delta G$ determines where a reaction "wants" to go, while K tells you where it ends up.

Standard Free Energy and Equilibrium Constant

$\Delta G° = -RT \ln K$

• $\Delta G°$ is the standard free energy change, measured with all species at 1 M (solutions) or 1 atm (gases) at the specified temperature.
• Large K (>1) corresponds to negative $\Delta G°$, meaning products are favored at equilibrium.
• Small K (<1) corresponds to positive $\Delta G°$, meaning reactants are favored.
• The relationship is logarithmic. At 298 K, a change of about 5.7 kJ/mol in $\Delta G°$ shifts K by a factor of 10.

Here, $R = 8.314 \text{ J/(mol·K)}$ and $T$ is in Kelvin. Watch your units: if $\Delta G°$ is given in kJ/mol, convert to J/mol before plugging in. This is one of the most common calculation errors on exams.

Non-Standard Conditions: The Reaction Quotient

$\Delta G = \Delta G° + RT \ln Q$

• Q (reaction quotient) is the current ratio of products to reactants, calculated the same way as K but using current concentrations or pressures, not equilibrium values.
• When Q < K, $\Delta G < 0$ and the reaction proceeds forward spontaneously.
• When Q = K, $\Delta G = 0$ and the system is at equilibrium.
• When Q > K, $\Delta G > 0$ and the reaction proceeds in reverse.

Notice that when all species are at standard conditions, $Q = 1$, so $\ln Q = 0$, and the equation reduces to $\Delta G = \Delta G°$. That's why standard conditions are the "default."

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Electrochemistry Connection

Electrochemical cells convert chemical energy directly to electrical work. The cell potential is just another way of expressing Gibbs free energy.

Gibbs Free Energy and Cell Potential

$\Delta G = -nFE$

• $n$ = moles of electrons transferred in the balanced redox reaction
• $F$ = Faraday's constant = 96,485 C/mol $e^-$
• $E$ = cell potential in volts (V)
• Positive $E$ gives negative $\Delta G$, so spontaneous galvanic cells have positive voltages.

Under standard conditions, this becomes $\Delta G° = -nFE°$. You can combine this with $\Delta G° = -RT \ln K$ to get a useful relationship:

$E° = \frac{RT}{nF} \ln K$

This equation connects the standard cell potential directly to the equilibrium constant. At 298 K, it simplifies to:

$E° = \frac{0.02569 \text{ V}}{n} \ln K$

Here's the typical workflow for these chain problems:

1. Use $\Delta G° = -nFE°$ to convert a given $E°$ to $\Delta G°$
2. Use $\Delta G° = -RT \ln K$ to solve for K
3. Or go directly with $E° = \frac{RT}{nF} \ln K$

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Phase Transitions and Equilibrium Conditions

At phase boundaries and equilibrium points, the system has minimized its Gibbs free energy. Competing phases or reactions are perfectly balanced.

Equilibrium Criterion

$\Delta G = 0$ (at equilibrium)

• Minimum Gibbs free energy defines the equilibrium state. The system has no thermodynamic driving force to change.
• Phase transitions (melting, boiling, sublimation) occur when $\Delta G = 0$ between phases at specific T and P. For example, water boils at 100°C (at 1 atm) because that's where $\Delta G$ for the liquid-to-gas transition equals zero.
• Forward and reverse rates are equal, connecting thermodynamic equilibrium to kinetic equilibrium.

Temperature Dependence: Gibbs-Helmholtz Equation

$\left(\frac{\partial(\Delta G/T)}{\partial T}\right)_P = -\frac{\Delta H}{T^2}$

• Relates the temperature dependence of $\Delta G$ to enthalpy.
• Exothermic reactions ($\Delta H < 0$) become less favorable as temperature increases.
• Enables calculation of $\Delta G$ at new temperatures when you know $\Delta H$ and $\Delta G$ at a reference temperature.

Note: This equation is more commonly tested in physical chemistry than in Gen Chem II. Your course may or may not require it. If your professor hasn't covered it, focus on the simpler approach of using $\Delta G = \Delta H - T\Delta S$ with the assumption that $\Delta H$ and $\Delta S$ are roughly constant over the temperature range.

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Multicomponent Systems and Mixtures

For systems with multiple species, Gibbs free energy depends on both the amounts and the chemical potentials of each component. Most Gen Chem II courses only touch on these ideas briefly, so check whether your course covers them before spending too much time here.

Chemical Potential and Total Free Energy

$G = \sum_i \mu_i n_i$

Chemical potential ($\mu_i$) is the partial molar Gibbs free energy. Think of it as how much the total G changes when you add one mole of component i while holding everything else constant. Each component contributes to total G proportionally to its amount ($n_i$) and its chemical potential ($\mu_i$).

At equilibrium, the chemical potential of each species is equal in all phases it occupies. This is what drives phase distribution.

Gibbs Free Energy of Mixing (Ideal Solutions)

$\Delta G_{mix} = nRT(x_1 \ln x_1 + x_2 \ln x_2)$

• Always negative for ideal solutions because mole fractions are between 0 and 1, making the logarithms negative. Mixing is always spontaneous for ideal solutions.
• $n$ is the total number of moles, and $x_1$ and $x_2$ are mole fractions of the two components.
• This equation assumes no enthalpy change upon mixing ($\Delta H_{mix} = 0$), so the spontaneity is driven entirely by entropy. The increase in disorder from mixing is the only driving force.

Gibbs-Duhem Equation

$\sum_i x_i \, d\mu_i = 0$

This equation says that chemical potentials in a mixture are interdependent. You can't change one component's $\mu$ without affecting the others. It's essential for non-ideal solution thermodynamics and activity coefficient calculations, but it's more commonly tested in physical chemistry than in Gen Chem II.

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Quick Reference Table

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Self-Check Questions

1. A reaction has $\Delta H > 0$ and $\Delta S > 0$. Under what temperature conditions will it be spontaneous? Which equation would you use to find the crossover temperature?

2. If $\Delta G° = -30$ kJ/mol, is K greater than, less than, or equal to 1? Now suppose you're told that current concentrations give Q > K. Is the reaction spontaneous in the forward direction?

3. Compare $\Delta G°$ and $\Delta G$. Why is it critical to distinguish between them when solving equilibrium problems with non-standard concentrations?

4. An electrochemical cell has $E° > 0$. What does this tell you about $\Delta G°$ and K for the cell reaction? Write the equation that connects all three quantities.

5. Why is $\Delta G_{mix}$ always negative for ideal solutions, even though there's no enthalpy change? Which term in $\Delta G = \Delta H - T\Delta S$ is responsible?