Given the Equilibrium Reaction

Oct 13, 2025 by wbrum.org 146 views

Introduction

In chemistry, equilibrium constants play a crucial role in understanding the behavior of chemical reactions. The equilibrium constant, denoted by $K_c$ , is a measure of the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium. In this article, we will explore the concept of equilibrium constants and how they can be used to determine the equilibrium concentrations of reactants and products in a chemical reaction.

What is an Equilibrium Constant?

The equilibrium constant, $K_c$ , is a numerical value that describes the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium. It is defined as:

${ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} }$

where $[C]$ , $[D]$ , $[A]$ , and $[B]$ are the equilibrium concentrations of the products and reactants, and $a$ , $b$ , $c$ , and $d$ are the stoichiometric coefficients of the reactants and products in the balanced chemical equation.

Given the Equilibrium Reaction

The given equilibrium reaction is:

${ 2A(aq) + B(aq) \rightleftharpoons C(aq) + D(aq) }$

We are given that the equilibrium constant, $K_c$ , for this reaction is 44.9. Our goal is to determine the equilibrium concentrations of $A$ , $B$ , and $C$ .

Using the Equilibrium Constant to Determine Equilibrium Concentrations

To determine the equilibrium concentrations of $A$ , $B$ , and $C$ , we can use the equilibrium constant expression:

${ K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b} }$

Substituting the given values, we get:

${ 44.9 = \frac{[C][D]}{[A]^2[B]} }$

We can rearrange this equation to get:

${ [C][D] = 44.9[A]^2[B] }$

Assuming Initial Concentrations

Let's assume that the initial concentrations of $A$ , $B$ , and $C$ are $[A]_0$ , $[B]_0$ , and $[C]_0$ , respectively. We can express the equilibrium concentrations of $A$ , $B$ , and $C$ in terms of their initial concentrations and the stoichiometric coefficients of the reactants and products.

For $A$ , we have:

${ [A] = [A]_0 - 2x }$

For $B$ , we have:

${ [B] = [B]_0 - x }$

For $C$ , we have:

${ [C] = x }$

For $D$ , we have:

${ [D] = x }$

Substituting the Equilibrium Concentrations into the Equilibrium Constant Expression

Substituting the equilibrium concentrations of $A$ , $B$ , $C$ , and $D$ into the equilibrium constant expression, we get:

${ 44.9 = \frac{x^2}{([A]_0 - 2x)^2([B]_0 - x)} }$

Simplifying the Equation

We can simplify this equation by canceling out the $x^2$ term:

${ 44.9 = \frac{1}{([A]_0 - 2x)^2([B]_0 - x)} }$

Assuming Small Changes in Concentration

Since the equilibrium constant is a large number, we can assume that the changes in concentration are small. This allows us to make the following approximations:

${ [A]_0 - 2x \approx [A]_0 }$

${ [B]_0 - x \approx [B]_0 }$

Substituting the Approximations into the Equation

Substituting these approximations into the equation, we get:

${ 44.9 = \frac{1}{[A]_0^2[B]_0} }$

Solving for the Initial Concentrations

We can solve for the initial concentrations of $A$ and $B$ by rearranging this equation:

${ [A]_0^2[B]_0 = \frac{1}{44.9} }$

${ [A]_0[B]_0 = \sqrt{\frac{1}{44.9}} }$

Calculating the Equilibrium Concentrations

Now that we have the initial concentrations of $A$ and $B$ , we can calculate the equilibrium concentrations of $A$ , $B$ , and $C$ .

For $A$ , we have:

${ [A] = [A]_0 - 2x }$

For $B$ , we have:

${ [B] = [B]_0 - x }$

For $C$ , we have:

${ [C] = x }$

Numerical Values

Let's assume that the initial concentrations of $A$ and $B$ are $[A]_0 = 1.0$ M and $[B]_0 = 1.0$ M, respectively. We can calculate the equilibrium concentrations of $A$ , $B$ , and $C$ using the equations above.

For $A$ , we have:

${ [A] = [A]_0 - 2x }$

${ [A] = 1.0 - 2x }$

For $B$ , we have:

${ [B] = [B]_0 - x }$

${ [B] = 1.0 - x }$

For $C$ , we have:

${ [C] = x }$

Calculating the Equilibrium Constant

We can calculate the equilibrium constant using the equilibrium concentrations of $A$ , $B$ , and $C$ .

${ K_c = \frac{[C][D]}{[A]^2[B]} }$

${ K_c = \frac{x^2}{(1.0 - 2x)^2(1.0 - x)} }$

Numerical Value of the Equilibrium Constant

We can calculate the numerical value of the equilibrium constant using the equilibrium concentrations of $A$ , $B$ , and $C$ .

${ K_c = 44.9 }$

Conclusion

Q: What is an equilibrium constant?

A: An equilibrium constant, denoted by $K_c$ , is a numerical value that describes the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium.

Q: How is the equilibrium constant calculated?

A: The equilibrium constant is calculated using the equilibrium concentrations of the products and reactants, and the stoichiometric coefficients of the reactants and products in the balanced chemical equation.

Q: What is the significance of the equilibrium constant?

A: The equilibrium constant is a measure of the favorability of a reaction. A large equilibrium constant indicates that the reaction favors the products, while a small equilibrium constant indicates that the reaction favors the reactants.

Q: How is the equilibrium constant used in chemical reactions?

A: The equilibrium constant is used to determine the equilibrium concentrations of reactants and products in a chemical reaction. It is also used to predict the direction of a reaction and the extent of a reaction.

Q: What are the units of the equilibrium constant?

A: The units of the equilibrium constant depend on the units of the concentrations of the reactants and products. For example, if the concentrations are in moles per liter (M), the equilibrium constant is dimensionless.

Q: Can the equilibrium constant be negative?

A: No, the equilibrium constant cannot be negative. The equilibrium constant is always a positive number.

Q: How is the equilibrium constant affected by temperature?

A: The equilibrium constant is affected by temperature. An increase in temperature can cause the equilibrium constant to increase or decrease, depending on the reaction.

Q: Can the equilibrium constant be used to predict the rate of a reaction?

A: No, the equilibrium constant cannot be used to predict the rate of a reaction. The rate of a reaction is determined by the rate constant, which is a separate parameter.

Q: How is the equilibrium constant related to the rate constant?

A: The equilibrium constant and the rate constant are related through the Arrhenius equation, which describes the relationship between the rate constant and the temperature.

Q: Can the equilibrium constant be used to predict the direction of a reaction?

A: Yes, the equilibrium constant can be used to predict the direction of a reaction. If the equilibrium constant is large, the reaction favors the products, while if the equilibrium constant is small, the reaction favors the reactants.

Q: How is the equilibrium constant used in industrial applications?

A: The equilibrium constant is used in a variety of industrial applications, including the design of chemical reactors, the optimization of reaction conditions, and the prediction of reaction yields.

Q: Can the equilibrium constant be used to predict the extent of a reaction?

A: Yes, the equilibrium constant can be used to predict the extent of a reaction. If the equilibrium constant is large, the reaction will proceed to completion, while if the equilibrium constant is small, the reaction will not proceed to completion.

Q: How is the equilibrium constant related to the concept of equilibrium?

A: The equilibrium constant is a measure of the equilibrium state of a reaction. It describes the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium.

Q: Can the equilibrium constant be used to predict the stability of a reaction?

A: Yes, the equilibrium constant can be used to predict the stability of a reaction. If the equilibrium constant is large, the reaction is stable, while if the equilibrium constant is small, the reaction is unstable.

Q: How is the equilibrium constant used in environmental applications?

A: The equilibrium constant is used in a variety of environmental applications, including the prediction of the fate and transport of pollutants in the environment, the design of wastewater treatment systems, and the prediction of the effects of climate change on ecosystems.

Q: Can the equilibrium constant be used to predict the toxicity of a substance?

A: Yes, the equilibrium constant can be used to predict the toxicity of a substance. If the equilibrium constant is large, the substance is likely to be toxic, while if the equilibrium constant is small, the substance is likely to be non-toxic.

Q: How is the equilibrium constant related to the concept of thermodynamics?

A: The equilibrium constant is a measure of the thermodynamic state of a reaction. It describes the ratio of the concentrations of the products to the concentrations of the reactants at equilibrium, which is a fundamental concept in thermodynamics.