Question

In equilibrium problems involving equilibrium constants that are small relative to the initial concentrations of reactants, we can often assume that the quantity \(x\) (which represents how far the reaction proceeds to-.ward products) is small. When this assumption is made, we can ignore the quantity \(x\) when it is subtracted from a large number but not when it is multiplied by a large number. In other words, \(2.5-x \approx 2.5,\) but \(2.5 x \neq 2.5 .\) Explain why we can ignore a small \(x\) in the first case but not in the second.

Short answer

A small quantity \(x\) can be ignored when subtracted from a large number because the result is approximately the same as the large number. However, in multiplication, the small quantity has a scaling effect and cannot be ignored.

Step-by-step solution

Understanding Subtracting a Small Quantity

When we have a small quantity represented by \(x\), subtracting it from a much larger number, like \(2.5\), results in a value that is very close to the original large number. This is because the relative difference made by subtracting a small number from a large number is insignificant, and the larger number dominates the value. Thus, \(2.5 - x\) is approximately equal to \(2.5\) when \(x\) is small.

Understanding Multiplying by a Small Quantity

On the other hand, when we multiply a small quantity \(x\) by a large number, the product is affected significantly by the small number. This is because multiplication is a scaling operation, and the small number \(x\) acts as a scaling factor for the larger number, changing the magnitude of the result. Therefore, \(2.5x\) will vary significantly with \(x\), even if \(x\) is small, and cannot be approximated as \(2.5\).

Applying the Concept to Equilibrium Problems

In the context of equilibrium problems, when the equilibrium constant is much smaller than the initial concentrations of reactants, we assume that the change in concentration, represented by \(x\), is negligible. Therefore, when subtracting \(x\) from the initial concentration, we can use the approximation that the concentration is unchanged. However, if \(x\) were to be multiplied by a concentration to calculate the product formed or relate to the equilibrium constant, the exact value of \(x\) would be significant and cannot be ignored.

Key concepts

Chemical Equilibrium

Chemical equilibrium refers to the state of a reversible chemical reaction where the rates of the forward and reverse reactions are equal, resulting in no overall change in the concentrations of reactants and products over time. At this point, both reactants and products are present in concentrations that have no further tendency to change with time.

Understanding the concept of equilibrium is crucial, as it dictates that for a reaction at equilibrium, the system’s conditions are stable. For example, if we consider the reaction \(A + B \leftrightarrow C + D\), at chemical equilibrium the rate at which \(A\) and \(B\) react to form \(C\) and \(D\) equals the rate at which \(C\) and \(D\) revert back to \(A\) and \(B\). This dynamic balance does not imply that the reactants and products are in equal concentrations, but that their ratios remain constant over time.

Reaction Quotient

The reaction quotient, denoted as \(Q\), is a measure used to predict the direction a reaction will shift to reach equilibrium. It is calculated in the same way as the equilibrium constant \(K\), but with the initial concentrations or partial pressures of the reactants and products before the system reaches equilibrium. For a general reaction \(aA + bB \leftrightarrow cC + dD\), the reaction quotient is \(Q = \frac{[C]^c[D]^d}{[A]^a[B]^b}\).

Comparing \(Q\) to the equilibrium constant \(K\) allows us to determine the reaction's direction: if \(Q < K\), the reaction will proceed forward to produce more products, and if \(Q > K\), the reaction will move in the reverse direction to produce more reactants. If \(Q = K\), the system is at equilibrium, and no net change will occur.

Le Chatelier's Principle

Le Chatelier's principle states that if an external change is applied to a system at equilibrium, the system adjusts itself to partially counteract the change and a new equilibrium is established. This principle helps predict how a shift in conditions (concentration, pressure, temperature) will influence the direction of the reaction.

For example, increasing the concentration of reactants will typically shift the equilibrium towards the products to reduce the effect of the change, whereas a temperature increase will favor an endothermic reaction or hinder an exothermic one, based on whether the system absorbs or releases heat. Through this principle, we learn how dynamic the equilibrium state truly is, as it can be manipulated by altering the reaction conditions.

Equilibrium Concentration Approximation

The equilibrium concentration approximation is a simplification method often used when dealing with small equilibrium constants. It allows us to make reasonable assumptions regarding the concentration changes during a reaction. As mentioned in the exercise, when an equilibrium constant is small relative to the initial concentrations of reactants, the change in concentration (represented by \(x\)) is tiny. In such cases, we can assume that the subtraction of \(x\) from a large concentration will have a negligible impact, providing a simpler way to solve equilibrium problems.

However, it is essential to recognize when \(x\) cannot be ignored—specifically, when multiplied by a large number, or when calculating the concentration of the products, as the small value of \(x\) might substantially affect the concentration. These approximations are not applicable to every equilibrium scenario but they are helpful in economizing time and computational resources while still yielding reasonably accurate results.