Problem 80 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

An aqueous solution of volume \(500 \mathrm{ml}\), when the reaction: \(2 \mathrm{Ag}^{+}(\mathrm{aq})+\mathrm{Cu}(\mathrm{s}) \rightleftharpoons\) \(\mathrm{Cu}^{2+}(\mathrm{aq})+2 \mathrm{Ag}(\mathrm{s})\), reached equilibrium, the concentration of \(\mathrm{Cu}^{2+}\) ions was \(x \mathrm{M}\). To this solution, \(500 \mathrm{~m}\) of water is added. At the new equilibrium, the concentration of \(\mathrm{Cu}^{2+}\) ions would be (a) \(2 x \mathrm{M}\) (b) \(x \underline{M}\) (c) between \(x\) and \(0.5 x \mathrm{M}\) (d) less than \(0.5 x\) M

Step-by-Step Solution

Answer

The new concentration of \text{Cu}^{2+} ions will be less than 0.5x M.

Step 1: Understand the Effect of Dilution

Diluting a solution will decrease the concentration of the solutes because the amount of solute stays the same while the volume of the solution increases. In this case, adding 500 ml of water to another 500 ml solution will double the total volume, and hence, the concentration of the ions will be halved.

Step 2: Calculate New Concentration of Cu2+ Ions

The initial concentration of Cu2+ ions is given as 'x' M in 500 ml. Upon adding 500 ml of water, the total volume becomes 500 ml + 500 ml = 1000 ml (or 1 L). The new concentration, therefore, is the initial concentration divided by the dilution factor (which is 2). The new concentration becomes x / 2 or 0.5x M.

Step 3: Select the Correct Option

Based on the calculation in Step 2, the new concentration of Cu2+ ions after dilution is 0.5x M. Therefore, the correct answer is that the concentration of Cu2+ ions will be less than 0.5x M, since the system will shift to re-establish equilibrium after dilution, but it will still be based on the new diluted concentration.

Key Concepts

Le Chatelier's Principle

Understanding Le Chatelier's principle is crucial for predicting how a chemical system at equilibrium reacts to changes in concentration, temperature, volume, or pressure. This principle states that if an external stress is applied to a system at equilibrium, the system will adjust in a way that counteracts the stress.

For instance, when the concentration of a reactant or product in a chemical reaction is changed, the equilibrium of the reaction will shift to restore balance. In the context of our exercise involving the reaction between silver ions and copper, adding water dilutes the concentration of the copper ions. According to Le Chatelier's principle, the reaction will shift to the right to increase the concentration of copper ions and re-establish the equilibrium. We can predict this shift because the reaction requires fewer moles of gaseous or aqueous reactants than products, a general rule for predicting equilibrium shifts.

Dilution and Concentration

Dilution involves adding solvent to a solution, which decreases the concentration of the solutes. The concept of dilution is expressed mathematically as \( C_1V_1 = C_2V_2 \), where \( C_1 \) and \( C_2 \) are the initial and final concentrations, and \( V_1 \) and \( V_2 \) are the initial and final volumes.

In our textbook exercise, we saw how adding 500 ml of water to a 500 ml solution results in a new, larger volume. The amount of solute, in this case, the \( \text{Cu}^{2+} \) ions, remains constant while the solution's volume increases. Thus, the concentration decreases. After dilution, the concentration is not just halved, as initially calculated, because the equilibrium will shift, affecting the concentration further.

Equilibrium Constant Calculation

The equilibrium constant, denoted as \( K_c \) for reactions in solution, is a quantitative measure of the extent of a chemical reaction at equilibrium. It is calculated from the concentrations of the products and reactants at equilibrium, raised to the power of their stoichiometric coefficients. The formula for a generic reaction \( aA + bB \rightleftharpoons cC + dD \) is given by \( K_c = \frac{[C]^c \cdot [D]^d}{[A]^a \cdot [B]^b} \).

For the reaction in our exercise, the equilibrium constant would involve the concentrations of silver ions and copper. When dilution occurs, it's important to understand that although the numerical value of \( K_c \) does not change, the concentrations of reactants and products do change as the system moves towards a new equilibrium point. This calculation is essential when determining concentrations at equilibrium after a disturbance such as dilution.