Equilibrium Constant from Nernst Equation — Core Principles

Core Principles

The equilibrium constant ($K_c$) for an electrochemical reaction in a galvanic cell can be determined directly from its standard cell potential ($E^circ_{cell}$) using a modified form of the Nernst equation.

At equilibrium, a galvanic cell's net potential ($E_{cell}$) becomes zero, and the reaction quotient ($Q$) equals the equilibrium constant ($K_c$). Substituting these conditions into the Nernst equation, $E_{cell} = E^circ_{cell} - \frac{RT}{nF} ln Q$, yields $0 = E^circ_{cell} - \frac{RT}{nF} ln K_c$.

Rearranging this gives the crucial relationship: $E^circ_{cell} = \frac{RT}{nF} ln K_c$. At $298,\text{K}$ ($25^circ\text{C}$), this simplifies to $E^circ_{cell} = \frac{0.0592}{n} log K_c$. This equation allows us to calculate $K_c$ if $E^circ_{cell}$ and the number of electrons transferred ($n$) are known, or vice versa.

A larger $E^circ_{cell}$ corresponds to a larger $K_c$, indicating a more spontaneous reaction that proceeds further towards products at equilibrium. This connection is vital for predicting the feasibility and extent of redox reactions in various applications.

Important Differences

vs Reaction Quotient (Q) vs. Equilibrium Constant (K_c)

Aspect	This Topic	Reaction Quotient (Q) vs. Equilibrium Constant (K_c)
Definition	Reaction Quotient (Q): A measure of the relative amounts of products and reactants present in a reaction at any given time, not necessarily at equilibrium.	Equilibrium Constant (K_c): A specific value of the reaction quotient when the system has reached chemical equilibrium, where net reaction ceases.
Calculation	Calculated using concentrations/pressures at any point during the reaction: $Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}$ (for $aA+bB \rightleftharpoons cC+dD$).	Calculated using equilibrium concentrations/pressures: $K_c = \frac{[C]_{eq}^c[D]_{eq}^d}{[A]_{eq}^a[B]_{eq}^b}$.
Value	Its value changes as the reaction proceeds towards equilibrium.	Its value is constant for a given reaction at a specific temperature, regardless of initial concentrations.
Predictive Power	Compares to $K_c$ to predict the direction of net reaction: If $Q < K_c$, reaction proceeds forward; if $Q > K_c$, reaction proceeds backward.	Indicates the extent of reaction at equilibrium: A large $K_c$ means products are favored, a small $K_c$ means reactants are favored.
Nernst Equation Context	Used in the general Nernst equation to calculate $E_{cell}$ under non-standard conditions: $E_{cell} = E^\circ_{cell} - \frac{RT}{nF} \ln Q$.	Used when $E_{cell} = 0$ (at equilibrium) to relate to $E^\circ_{cell}$: $E^\circ_{cell} = \frac{RT}{nF} \ln K_c$.

The reaction quotient ($Q$) is a dynamic measure of reactant and product ratios at any point in a reaction, used to predict the direction a reaction will shift to reach equilibrium. In contrast, the equilibrium constant ($K_c$) is a fixed value for a given reaction at a specific temperature, representing the ratio of products to reactants once equilibrium has been established. In the context of the Nernst equation, $Q$ is used to calculate the cell potential under non-standard conditions, while $K_c$ is specifically used when the cell potential is zero, linking it directly to the standard cell potential.