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The equilibrium constant, often represented as \( K_c \), is a number that expresses the relationship between the amounts of products and reactants present at equilibrium in a reversible chemical reaction at a given temperature. It is crucial to understand that \( K_c \) is affected only by temperature and not by changes in concentration, pressure, or volume.
To calculate the equilibrium constant \( K_c \), you use the formula:

• \( K_c = \frac{[ ext{products}]}{[ ext{reactants}]} \)

For the given reaction \(\mathrm{PCl}_{3} + \mathrm{Cl}_{2} \rightleftharpoons \mathrm{PCl}_{5} \), the equilibrium expression would be \( K_c = \frac{[ ext{PCl}_{5}]}{[ ext{PCl}_{3}][\text{Cl}_{2}]} \). This tells us the reaction proportions but does not mean the concentrations of all species are equal. They vary in a way that satisfies this ratio.
What this means is that while the system at equilibrium is stable, the actual concentration values of each species can differ greatly depending on initial starting points and reaction conditions.

In the context of chemical equilibrium, the reaction rates play a fundamental role. For a system at equilibrium, the rate of the forward reaction equals the rate of the reverse reaction. This balance of rates is why the concentrations of reactants and products remain constant over time. It's crucial to clarify that this balance does not imply that the concentrations of reactants (like \( \mathrm{PCl}_{3} \) and \( \mathrm{Cl}_{2} \)) and products (like \( \mathrm{PCl}_{5} \)) are equal. Rather, it implies stability in concentration levels, where no net change is observed.
Under this balanced condition:

• The reaction from reactants to products proceeds at the same pace as the reaction from products back to reactants.
• The equilibrium does not indicate or require equal amounts of reactants and products.

This distinction is important for understanding why in the problem, the concentrations of \( \mathrm{PCl}_{3} \) and \( \mathrm{Cl}_{2} \) are not required to be equal.

Chemical concentrations refer to the amount of a substance present in a unit volume of solution. In reactions at equilibrium, these concentrations reach a state where they no longer change, although they are not inherently equal. In our given reaction, \( \mathrm{PCl}_{3} + \mathrm{Cl}_{2} \rightleftharpoons \mathrm{PCl}_{5} \), it's possible for one reactant to be in excess or for different initial amounts to be present, leading to differing equilibrium concentrations. This real-life scenario emphasizes that equilibrium balances the rates of the forward and backward reactions, rather than making all components equal.
Key points about chemical concentrations at equilibrium:

• They must satisfy the equilibrium constant expression, \( K_c \).
• Differences in initial concentrations lead to different proportions at equilibrium, but the ratio defined by \( K_c \) remains constant.
• Stability at equilibrium does not equal uniformity in concentration levels.

Thus, while concentrations stabilize at equilibrium, they do so in a manner dictated by the reaction nature and initial conditions, not by default equality.