Question

100\. A reaction \(A(g) \rightleftharpoons B(s)\) has an equilibrium constant of \(1.0 \times 10^{-4}\). For which of the inital reaction mixtures is the \(x\) is small approximation most likely to apply? a. \([A]-0.0010 M ;[B]-0.00 M\) b. \([A]-0.00 \mathrm{M} ;[\mathrm{B}]-0.10 \mathrm{M}\) \begin{tabular}{l} c. \([A]-0.10 M ;[B]-0.10 M\) \\ \hline \end{tabular} d. \([\mathrm{A}]-0.10 \mathrm{M} ;[\mathrm{B}]-0.00 \mathrm{M}\)

Short answer

The 'x is small' approximation is most likely to apply to option d) [A]-0.10 M; [B]-0.00 M because the initial concentration of A is significantly higher than the equilibrium constant.

Step-by-step solution

Understanding the 'x is small' approximation

The 'x is small' approximation is used in equilibrium calculations when the change in concentration of reactants and products during the reaction ('x') is very small compared to the initial concentrations. This approximation simplifies calculations by considering that the change does not significantly affect the initial concentrations.

Analyze Initial Concentrations

To determine when the 'x is small' approximation is most applicable, examine the initial concentrations. A very low equilibrium constant (in this case, 1.0 x 10^-4) suggests that at equilibrium, the concentration of A will not change significantly, making 'x' small relative to its initial concentration. The approximation is most likely to be valid when the initial concentration of A is much larger than the value of the equilibrium constant.

Evaluate Options

a) Initial [A] is 0.0010 M, which is much greater than the equilibrium constant, making the 'x is small' approximation potentially valid. For option b) [A] starts at 0.00 M, the reaction cannot proceed in the forward direction. Option c) has equal initial concentrations of A and B, leaving no preference for the direction of the reaction; this situation makes the 'x is small' approximation less likely to hold. Option d) has a relatively high initial concentration of A (0.10 M), which enables the 'x is small' to be a reasonable approximation since the change in concentration will be small compared to the initial amount.

Comparing Options a and d

While both a) and d) could potentially make the 'x is small' approximation valid, option d) with a highest initial concentration of 0.10 M is more likely to validate the approximation because 'x' will be a much smaller fraction of the initial concentration compared to option a).

Key concepts

Equilibrium Constant

Understanding the equilibrium constant is crucial when studying chemical reactions at a state of balance. The equilibrium constant, denoted as K, is a value that expresses the ratio of the concentrations of products to reactants at equilibrium, raised to the power of their stoichiometric coefficients. It's essential to know that the constant is only applicable to reversible reactions at a given temperature, as temperature changes can alter the value of K.

For a simple reaction such as \( A(g) \rightleftharpoons B(s) \), the equilibrium constant expression would be written as \( K = [B]/[A] \), where concentrations are typically measured in molarity (M). A high equilibrium constant indicates a greater concentration of products at equilibrium, while a low value suggests a predominance of reactants. In the exercise provided, the equilibrium constant of \(1.0 \times 10^{-4}\) implies that, at equilibrium, reactants are favored over products.

X is Small Approximation

The 'x is small' approximation is a simplification method used in equilibrium calculations. When the change in concentration (x) during the reaction is very small compared to the initial concentrations, we can assume that the change is negligible. This approximation allows us to simplify the algebra involved in solving for equilibrium concentrations.

In our example, the equilibrium constant is quite small, indicating that the formation of products is limited so that x, representing the change in concentration, would indeed be small. This approximation is more valid when the initial concentration of the reactant is high compared to the equilibrium constant. In the case of option d) from the exercise, the initial concentration of \( A \) of 0.10 M justifies using the 'x is small' approximation because any change in concentration during the establishment of equilibrium would have a minimal effect on the initial concentration.

Reactant and Product Concentrations

At chemical equilibrium, the concentrations of reactants and products remain constant. However, this does not necessarily mean they are equal; their relationship is defined by the equilibrium constant. In our example, determining which initial mixture best lends itself to using the 'x is small' approximation involves examining reactant A and product B concentrations.

The equilibrium position is influenced by these concentrations. Starting with a higher concentration of A, as in option d), means it's less likely for the small change in concentration upon reaching equilibrium to significantly impact the initial amount. Comparatively, starting with a lower concentration of A or a high concentration of B changes the dynamic and may not allow for the 'x is small' approximation to be accurately applied.

Equilibrium Calculations

Equilibrium calculations often involve setting up an ICE (Initial, Change, Equilibrium) table to track the concentrations of reactants and products. For the reaction \( A(g) \rightleftharpoons B(s) \), the process would start with initial concentrations, apply the changes during the reaction (represented by ±x), and arrive at the concentrations at equilibrium.

When using the 'x is small' approximation, the Change row in the ICE table simplifies to a small value that we can often subtract from the initial concentration without materially affecting it. This dramatically reduces the complexity of the calculation. The validity of this approximation, as shown in our example, often hinges on the size of the initial concentrations relative to the equilibrium constant, thereby ensuring that x is indeed 'small'. In conclusion, option d) with an initial concentration of 0.10 M for reactant A allows us to confidently use this approximation in our equilibrium calculations.