Problem 84 - Chemistry Matter and Change

Question

Analyze Suppose that an equilibrium system at a given temperature has a \(K_{\text { eq equal to } 1.000 . \text { Evaluate the possi- }}\) bility that such a system is made up of 50\(\%\) reactants and 50\(\%\) products. Explain your answer.

Step-by-Step Solution

Answer

It is possible for an equilibrium system with \(K_{eq} = 1.000\) to have a composition of 50% reactants and 50% products if the sum of the stoichiometric coefficients of the reactants equals the sum of the stoichiometric coefficients of the products (i.e., \(a + b = c + d\)).

Step 1: Set up the generic equation

First, let's set up a generic reaction equation involving reactants A and B and products C and D: \[aA + bB \rightleftharpoons cC + dD\]

Step 2: Define the equilibrium constant \(K_{eq}\)

For such a reaction, the equilibrium constant is given by the equation: \[K_{eq} = \frac{[C]^c[D]^d}{[A]^a[B]^b}\] Given that \(K_{eq} = 1.000\), the equation becomes: \[1 = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]

Step 3: Determine the concentrations at 50% composition

If the system has 50% reactants and 50% products, the molar concentrations of each component in the mixture would be evenly split. Let x be the concentration of each reactant and product: \[[A]=[B]=[C]=[D]=x\]

Step 4: Substitute the concentrations in the equilibrium constant equation

Now, we substitute the concentrations from Step 3 into the equilibrium constant equation from Step 2: \[1 = \frac{(x)^c(x)^d}{(x)^a(x)^b}\] which can be simplified to: \[1 = \frac{x^{c+d}}{x^{a+b}}\]

Step 5: Analyze the possibility of such a system

From the equation in Step 4, we can notice a few possibilities: 1. If \(a + b \ne c + d\), the system will not be composed of 50% reactants and 50% products at equilibrium, as the exponents of x are different in the numerator and denominator. 2. If \(a + b = c + d\), the above equation simplifies to: \[1 = \frac{x^{a+b}}{x^{a+b}} = 1\] In this case, the system will be composed of 50% reactants and 50% products at equilibrium. In conclusion, it is only possible for such a system to have a composition of 50% reactants and 50% products if the sum of the stoichiometric coefficients of the reactants equals the sum of the stoichiometric coefficients of the products (i.e., \(a + b = c + d\)).

Key Concepts

Equilibrium Constant

The equilibrium constant, represented as \(K_{eq}\), is a crucial concept in chemistry that describes the ratio of the concentrations of the products to the reactants at equilibrium for a reversible chemical reaction. This ratio is raised to the power of their respective stoichiometric coefficients in the balanced chemical equation. For example, in a generic reaction \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant is given by the formula:
\[K_{eq} = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]
When \(K_{eq}\) equals 1, it implies that the concentrations of the products and reactants are at a state of perfect balance. However, it is important to note that this does not necessarily mean that the amounts of products and reactants are equal; rather, their ratio at equilibrium yields a value of 1 when the stoichiometry of the reaction is considered. In the context of the exercise, it indicates the potency of such a system to have even concentrations of reactants and products when the stoichiometric coefficients of reactants and products are balanced accordingly.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships of the elements and compounds involved in chemical reactions. It is derived from the Greek words \(\text{stoicheion}\) (element) and \(\text{metron}\) (measure). Stoichiometry is applied to calculate the amounts of reactants needed to produce a desired amount of product, as well as the yield from a given amount of reactant. The stoichiometric coefficients, such as \(a, b, c,\) and \(d\) in a balanced chemical equation like \(aA + bB \rightleftharpoons cC + dD\), tell us in what ratio the reactants combine and the products form.
In the exercise, understanding stoichiometry is key to determining whether a system can exist with 50% reactants and 50% products - this condition is met when the sum of the stoichiometric coefficients of the reactants is equal to that of the products.

Reaction Quotient

The reaction quotient, \(Q\), is a measure of the relative amounts of products and reactants present in a reaction mixture at any given point before the state of equilibrium is reached. It is expressed with the same formula as the equilibrium constant, but for non-equilibrium conditions:
\[Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\]
When the reaction quotient is compared to the equilibrium constant, \(K_{eq}\), it can tell us in which direction the reaction will shift to reach equilibrium. If \(Q < K_{eq}\), the reaction will proceed forwards (to the right) to produce more products; if \(Q > K_{eq}\), the reaction will proceed in the reverse direction (to the left) to produce more reactants. At equilibrium, \(Q\) equals \(K_{eq}\), and the reaction will no longer proceed in any direction, indicating a dynamic balance between the forward and reverse reactions.

Chemical Reaction Balance

For a chemical reaction to be balanced, the number of atoms of each element on the reactant side must equal the number of atoms of that element on the product side. This reflects the Law of Conservation of Mass, which states that matter cannot be created nor destroyed in a chemical reaction. Balancing a chemical reaction is the first step toward understanding its stoichiometry and calculating the equilibrium constant.
In the exercise provided, if the sum of the stoichiometric coefficients on the reactant side (\(a + b\)) equals the sum on the product side (\(c + d\)), the reaction is stoichiometrically balanced in a way that would allow for an equilibrium composition of 50% reactants and 50% products when \(K_{eq} = 1\). This alignment of stoichiometry with the equilibrium constant illustrates a perfect theoretical balance in the chemical system.