Problem 206 - A Complete Resource Book in Chemistry for JEE Main

Question

The equilibrium constant at \(298 \mathrm{~K}\) for a reaction \(\mathrm{A}+\) \(\mathrm{B} \rightleftharpoons \mathrm{C}+\mathrm{D}\) is \(100 .\) If the initial concentration of all the four species were \(1 \mathrm{M}\) each, then equilibrium concentration of \(\mathrm{D}\left(\right.\) in \(\left.\mathrm{mol} \mathrm{L}^{-1}\right)\) will be: \(\quad[\mathbf{2 0 1 6}]\)(a) \(0.818\) (b) \(1.818\) (c) \(1.182\) (d) \(0.182\)

Step-by-Step Solution

Answer

The equilibrium concentration of \( \mathrm{D} \) is \(1.818 \text{ M}\).

Step 1: Write the expression for the equilibrium constant

For the reaction \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} + \mathrm{D} \), the equilibrium constant \( K_c \) is given by:\[ K_c = \frac{[C][D]}{[A][B]}\]Given that \( K_c = 100 \).

Step 2: Set up the equilibrium concentration changes

Assuming that at equilibrium the change in the concentration of \( \mathrm{A} \) and \( \mathrm{B} \) is \( x \), then the concentration of \( \mathrm{C} \) and \( \mathrm{D} \) will also change by \( +x \).Initial concentrations: \([A]_0 = [B]_0 = [C]_0 = [D]_0 = 1 \text{ M}\).At equilibrium:\[[A] = [B] = 1 - x, \quad [C] = [D] = 1 + x\]

Step 3: Substitute concentrations into the equilibrium constant expression

Substitute the equilibrium concentrations into the equilibrium expression:\[ K_c = \frac{(1+x)^2}{(1-x)^2} = 100\]

Step 4: Solve for x

Start by taking the square root of both sides:\[ \sqrt{\frac{(1+x)^2}{(1-x)^2}} = \sqrt{100} \\frac{1+x}{1-x} = 10\]Cross-multiply to solve for \( x \):\[1+x = 10(1-x) \ 1+x = 10 - 10x \ 11x = 9 \ x = \frac{9}{11} = 0.818\]

Step 5: Find the equilibrium concentration of D

The equilibrium concentration of \( D \) is given by \( [D] = 1 + x \).Substituting the value of \( x \):\[[D] = 1 + 0.818 = 1.818 \text{ M}\]

Key Concepts

Chemical Equilibrium

In chemistry, chemical equilibrium refers to a state in which the concentrations of reactants and products remain constant over time. This doesn't mean that the reactant and product concentrations are equal, rather that their rates of formation and consumption are balanced. At this point, the reaction proceeds in both forward and reverse directions at the same rate.
When a reaction reaches equilibrium, no further changes in concentration occur, and the system's macroscopic properties, such as temperature, pressure, and concentrations, have stabilized. Understanding chemical equilibrium is crucial because it allows for the prediction and manipulation of reactions, especially those that are reversible.

• The reaction can shift back and forth between reactants and products, responding to changes in conditions, such as concentration, temperature, or pressure.
• Equilibrium does not mean that the reactants and products are in equal concentrations; it means that their formation rates are equal.
• Factors such as temperature and pressure can affect the position of equilibrium, according to Le Chatelier's principle.

By analyzing these factors, chemists are able to determine how to shift equilibrium to favor either the products or reactants.

Equilibrium Concentrations

The concentrations of reactants and products at the equilibrium state are termed equilibrium concentrations. These values are essential for evaluating the extent to which a reaction has proceeded. In a chemical reaction, the equilibrium concentrations depend on various factors: initial concentrations, the equilibrium constant, and any changes made to the reaction's conditions.
At equilibrium, the concentrations of the substances involved in the reaction remain constant. To find the equilibrium concentrations, one must set up an equilibrium table (often referred to as an ICE table, for Initial, Change, Equilibrium) where the concentrations are calculated systematically.

• The initial concentrations are noted as the starting point.
• Changes in concentrations are represented by variables, often with an algebraic expression describing their relationship.
• Equilibrium concentrations are then determined by solving these equations, often using the equilibrium constant.

Understanding equilibrium concentrations helps in predicting the amounts of products formed or reactants consumed, leading to practical applications in industrial chemical processes.

Equilibrium Constant Expression

The equilibrium constant, denoted as \( K_c \), is a numerical value that characterizes the position of equilibrium for a reversible reaction at a specific temperature. It is a fundamental concept in chemical equilibrium, representing the ratio of the product concentrations to the reactant concentrations, each raised to the power of their stoichiometric coefficients in the balanced chemical equation.
The expression for the equilibrium constant for a reaction \( aA + bB \rightleftharpoons cC + dD \) is written as:\[ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]This helps in quantifying the relationship between the reactants and products in a chemical equilibrium system.

• If \( K_c \) is large (much greater than 1), the equilibrium favors the products.
• If \( K_c \) is small (much less than 1), the equilibrium favors the reactants.
• The magnitude of \( K_c \) can predict the extent of a reaction but not the speed at which equilibrium is reached.

The equilibrium constant is temperature-dependent and provides valuable insights into the direction and extent of a chemical reaction.