Question

Consider the following equations for a cell reaction $$ \begin{gathered} A+B \rightleftharpoons C+D ; \quad E^{\circ}=x \text { volt, } K_{e q}=K_{1} \\\ 2 A+2 B \rightleftharpoons 2 C+2 D ; E^{\circ}=y \text { volt, } K_{e q}=K_{2} \end{gathered} $$ then : (a) \(x=y, K_{1}=K_{2}\) (b) \(x=2 y, K_{1}=2 K_{2}\) (c) \(x=y, K_{1}^{2}=K_{2}\) (d) \(x^{2}=y, K_{1}^{2}=K_{2}\)

Short answer

x = y, K1^2 = K2

Step-by-step solution

Analyzing the equations

The first equation represents a chemical equilibrium with a cell potential of E° = x and an equilibrium constant of Keq = K1. The second equation is the same reaction but doubled. It shows a cell potential of E° = y and an equilibrium constant of Keq = K2.

Understanding electrochemical cell potential

The cell potential (E°) for a reaction is an intensive property, meaning that it does not depend on the amount of substance reacting. Therefore, multiplying the reaction by a coefficient (as in the second reaction where all reactants and products are doubled) does not affect the value of E°.

Comparing the cell potentials

Since the cell potential does not change when the reaction coefficients are multiplied, the cell potentials of the two reactions must be equal: x = y.

Understanding equilibrium constant

The equilibrium constant (Keq) for a reaction is dependent on the stoichiometry of the reaction. When the coefficients of the reaction are doubled, the equilibrium constant for the new reaction is the square of the original equilibrium constant because Keq is calculated based on the concentrations of the reactants and products raised to the power of their coefficients in the balanced equation.

Comparing the equilibrium constants

As the reaction is doubled from K1 to K2, the equilibrium constant for K2 will be the square of K1, written as: K2 = K1^2.

Key concepts

Chemical Equilibrium

The concept of chemical equilibrium is central to understanding how reactions occur in a closed system. At equilibrium, the rate of the forward reaction, wherein reactants convert to products, is equal to the rate of the reverse reaction, where products revert back to reactants. What's fascinating is that even though the rates are equal, the actual amounts of reactants and products don't have to be the same; they just remain constant over time.
This state doesn't mean the reactions have stopped; both the forward and reverse reactions continuously occur, but there is no net change in the concentration of reactants and products. In the context of the exercise, the reaction involving substances A, B, C, and D reaches a point where the rate at which A and B combine to form C and D is the same as the rate at which C and D decompose back into A and B.
To better understand and visualize chemical equilibrium, think of it like a room where two people are adding and removing books from a bookshelf at precisely the same rate. The number of books on the shelf stays the same over time even though there's constant activity.

Equilibrium Constant

Delving deeper, the equilibrium constant (
) is a numerical value that represents the ratio of the concentrations of products to reactants at equilibrium raised to the power of their respective stoichiometric coefficients. It is a vital parameter for predicting the extent of a reaction and is denoted as
). For the equilibrium of the reactions given in the exercise,
defines the relationship between concentrations of A, B, C, and D when the system is at equilibrium.

Significance of Equilibrium Constant

A high value of
indicates that, at equilibrium, the products are favored; the higher the value, the more product is present compared to reactant. Conversely, a low
suggests an equilibrium favoring the reactants. This information is crucial as it helps predict the direction in which a reaction will proceed under given conditions. Importantly, if the equation is altered as seen in the exercise
's value changes. When the reaction is doubled (
), the new equilibrium constant (
) becomes the square of the original constant (
), which aligns with Le Chatelier's principle.

Intensive Property

An intensive property is an intrinsic characteristic of a substance that does not change with the amount of the substance. Common examples include temperature, density, and cell potential (
) as explored in the exercise. In the context of electrochemical cells,
reflects the potential difference between the electrodes when no current is flowing and is a measure of the chemical energy available to do work.

Why Cell Potential is Intensive

The cell potential is dictated by the inherent characteristics of the materials involved and the nature of the reactions, not by the scale of the reaction. Even when the reaction is multiplied, the cell potential remains the same (
), as seen in the exercise where
is identical for both reactions. This property is of significant importance in electrochemistry because it allows for standardization and comparison of different cells under various conditions and provides a basis for calculating other related thermodynamic quantities.