Problem 11 - Chemistry Matter and Change

Question

Calculate Determine the value of \(K_{\text { eq at }} 400\) K for this equation: \(\mathrm{PCl}_{5}(\mathrm{g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{g})+\) \(\mathrm{Cl}_{2}(\mathrm{g})\) if \(\left[\mathrm{PCl}_{5}\right]=0.135 \mathrm{mol} / \mathrm{L}\), \(\left[\mathrm{PCl}_{3}\right]=0.550 \mathrm{mol} / \mathrm{L}\), and \(\left[\mathrm{Cl}_{2}\right]=0.550 \mathrm{mol} / \mathrm{L}\)

Step-by-Step Solution

Answer

The equilibrium constant \(K_{eq}\) at 400 K is approximately 2.241.

Step 1: Identify Given Information

First, extract all the given concentrations from the exercise for the chemical equilibrium. The exercise provides the concentrations as follows: - \([PCl_{5}] = 0.135\) mol/L- \([PCl_{3}] = 0.550\) mol/L- \([Cl_{2}] = 0.550\) mol/L.

Step 2: Write the Equilibrium Expression

Set up the expression for the equilibrium constant, \(K_{eq}\), based on the balanced equation:\[PCl_{5}(g) \rightleftharpoons PCl_{3}(g) + Cl_{2}(g)\]The equilibrium constant expression is\[K_{eq} = \frac{[PCl_{3}][Cl_{2}]}{[PCl_{5}]}\]

Step 3: Substitute Known Values

In the expression for \(K_{eq}\), substitute the known concentrations:\[K_{eq} = \frac{(0.550)(0.550)}{0.135}\]

Step 4: Calculate \(K_{eq}\) Value

Perform the calculation by multiplying the concentrations of products and then dividing by the concentration of the reactant:\[K_{eq} = \frac{0.3025}{0.135} \approx 2.241\]

Key Concepts

Equilibrium Constant

In a chemical reaction, the equilibrium constant, represented by \(K_{eq}\), is a number that provides insight into the balance between the reactants and products at equilibrium. It's a key aspect in the study of chemical equilibrium, showing the ratio of product concentrations to reactant concentrations when the reaction has reached equilibrium.
This constant is specific to a particular reaction and temperature. For instance, in our equation \(\mathrm{PCl}_{5}(\mathrm{g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{g})+ \mathrm{Cl}_{2}(\mathrm{g})\) at \(400\,\text{K}\), the equilibrium constant expression is written as:
\[K_{eq} = \frac{[PCl_{3}][Cl_{2}]}{[PCl_{5}]}\]
From the balanced chemical equation, you can see that for every molecule of \(\mathrm{PCl}_5\) that decomposes, one molecule each of \(\mathrm{PCl}_{3}\) and \(\mathrm{Cl}_{2}\) are formed. By using the expression, students can calculate the equilibrium position of the reaction given the concentrations of the involved species.

Reaction Quotient

The reaction quotient, denoted as \(Q\), serves as a snapshot of the current state of a reaction compared to its equilibrium state. Like the equilibrium constant, it's calculated using concentrations of the reactants and products. However, unlike \(K_{eq}\), \(Q\) can be calculated for any point in time – not just at equilibrium.
It's essential to compare \(Q\) with \(K_{eq}\) to predict the direction in which the reaction will proceed to reach equilibrium.

• If \(Q < K_{eq}\), the reaction will move forward, converting more reactants into products.
• If \(Q > K_{eq}\), the reaction will proceed in reverse, forming more reactants from products.
• If \(Q = K_{eq}\), the reaction is at equilibrium, and no further shift will occur.

This principle helps in understanding the dynamic nature of chemical reactions and how they adjust to maintain balance.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept that helps predict how a system in equilibrium will respond to disturbances. It states that if a change in concentration, temperature, or pressure is applied to a system at equilibrium, the system will adjust itself to counteract, or partially counteract, the effect of the change and reach a new equilibrium state.
In the context of the equation \(\mathrm{PCl}_{5}(\mathrm{g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{g}) + \mathrm{Cl}_{2}(\mathrm{g})\), consider a scenario where additional \(\mathrm{PCl}_5\) is added. According to Le Chatelier's Principle, the equilibrium will shift to the right to produce more \(\mathrm{PCl}_{3}\) and \(\mathrm{Cl}_{2}\), minimizing the effect of the added reactant.
This principle aids in understanding how equilibrium can reestablish itself amidst external modifications, ensuring students grasp how dynamic chemical systems are.

Concentration

Concentration is a crucial factor in chemical equilibrium as it determines the rate and direction of reactions. It refers to the amount of substance in a given volume of solution and is typically expressed in moles per liter (mol/L).
In reactions like \(\mathrm{PCl}_{5}(\mathrm{g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{g}) + \mathrm{Cl}_{2}(\mathrm{g})\), understanding concentration is vital for calculating the equilibrium constant. Here, reactant \([\mathrm{PCl}_{5}] = 0.135\, \text{mol/L}\) and product concentrations \([\mathrm{PCl}_{3}] = [\mathrm{Cl}_{2}] = 0.550\, \text{mol/L}\) are used to find \(K_{eq}\), which provides a stable measure of equilibrium position.

• Higher concentrations of products push the equilibrium to the left.
• Higher concentrations of reactants push the equilibrium to the right.

Through studying concentrations, students realize how these changes can manipulate the direction and extent of reactions at equilibrium, ultimately influencing yields and reaction dynamics.