Question

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

Short answer

Option (d), with an initial concentration of \(\mathrm{A}\)=0.10 \mathrm{M} and \(\mathrm{B}\)=0.00 \mathrm{M}, is the most likely scenario where the 'x is small' approximation applies.

Step-by-step solution

Understand the x is small approximation

The 'x is small' approximation is used in equilibrium calculations where the change in concentration (x) of reactants or products is very small compared to their initial concentrations. It simplifies the calculation by assuming that the equilibrium concentration of the reactants remains approximately the same as the initial concentration.

Analyze the given options

To determine when the x is small approximation is most applicable, we need to look for the scenario where the initial concentration of reactants is high relative to the equilibrium constant, as this would mean that the change in concentration upon reaching equilibrium (x) will be small.

Compare initial concentrations against equilibrium constant

Option (a) starts with a low concentration of A, which means x could be a significant fraction of 0.0010 M, hence this is not the best candidate. Option (b) starts with no A, so we can't form more B, meaning x is irrelevant in this context. Option (c) has equal concentrations of A and B, which means the system might already be at equilibrium or that x is not small compared to the concentrations. Option (d) has a relatively high concentration of A and no B, which is favorable for the x is small approximation because the initial concentration of A (0.10 M) is much larger relative to the equilibrium constant (1.0 x 10^-4).

Determine the most likely scenario for the x is small approximation to apply

The x is small approximation is most likely to apply to the mixture where the initial concentration of A is high, and the initial concentration of B is zero. This allows the approximation that the change in concentration of A upon reaching equilibrium will be negligible in comparison to its initial concentration. Therefore, option (d) is the most likely scenario where the x is small approximation would apply.

Key concepts

Equilibrium Constant

The equilibrium constant (\(K_{eq}\)) is a numerical value that describes the ratio of the concentrations of products to reactants at chemical equilibrium for a reversible reaction. It is a measure of how far a reaction will proceed before reaching a state of balance, where the rate of the forward reaction equals the rate of the reverse reaction. For the reaction \(\mathrm{A}(g) \rightleftharpoons \mathrm{B}(g)\), with \(K_{eq} = 1.0 \times 10^{-4}\), this constant indicates a reaction favoring the reactants over the products under standard conditions.

It's crucial to note that the equilibrium constant only depends on the temperature and is exclusive to the specific reaction at hand. In the context of the 'x is small' approximation, we look for scenarios where the change in concentration of products or reactants at equilibrium (often denoted by \(x\)) is minor compared to the initial concentrations. Hence, if the initial concentration of a reactant is significantly greater than the equilibrium constant, it implies a reaction where \(x\) will likely be small compared to the initial concentrations. In our example, option (d), with a higher initial concentration of reactant A and none of product B, fits this description well.

Equilibrium Concentration

Equilibrium concentration refers to the amount of reactants and products present when a chemical reaction reaches a state of equilibrium. At this point, the rate of the forward reaction equals the rate of the backward reaction, and the concentrations of all species remain constant over time. Understanding equilibrium concentrations is key to predicting how changes in conditions (like concentration, pressure, or temperature) will affect the reaction's position.

In using the 'x is small' approximation, we operate under the assumption that the equilibrium concentrations of reactants will be nearly the same as their initial concentrations. This approximation greatly simplifies equilibrium calculations. For example, in option (d) from our exercise, since the initial concentration of A is relatively high, \(0.10 \mathrm{M}\), and there is no initial B, the change in concentration of A in forming B is expected to be so small that the equilibrium concentration of A will be close to \(0.10 \mathrm{M}\).

Chemical Reaction Equilibrium

Chemical reaction equilibrium is a dynamic state where the concentrations of reactants and products no longer change because the rates of the forward and reverse reactions are equal. This does not mean the reactions have stopped; rather, they are proceeding at the same rate in both directions. The point of equilibrium is characterized by a constant equilibrium constant (\(K_{eq}\)), which does not imply that the concentrations of reactants and products are equal, but that they have reached a certain ratio that will remain constant unless the reaction conditions change.

In our discussion of the 'x is small' approximation, we're focusing on the initial steps towards reaching chemical equilibrium. By determining that option (d) is most likely to fit this approach, we imply that the starting point of the reaction—with a high initial concentration of reactant A—will lead to a negligible change as the system shifts to reach equilibrium. This ties back to the understanding that in equilibrium, the overall system is in balance, but the specifics of that balance depend on the reaction conditions, including the initial concentrations of reactants and products.