Question

An equilibrium mixture of the following reaction is found to have \(\left[\mathrm{SbCl}_{3}\right]=0.0255 \mathrm{M}\) and \(\left[\mathrm{Cl}_{2}\right]=0.135 \mathrm{M}\) at \(248^{\circ} \mathrm{C}\). What is the concentration of \(\mathrm{SbCl}_{5}\) ? $$ \begin{gathered} \mathrm{SbCl}_{5}(g) \rightleftharpoons \mathrm{SbCl}_{3}(g)+\mathrm{Cl}_{2}(g) \\ K_{\mathrm{eq}}=4.9 \times 10^{-4} \text { at } 248^{\circ} \mathrm{C} \end{gathered} $$

Short answer

[SbCl_5] = 7.03 \times 10^{-3} M

Step-by-step solution

Write the equilibrium expression

The first step is to write the equilibrium constant expression (K_eq) for the given reaction: $$ K_{eq} = \frac{[SbCl_3][Cl_2]}{[SbCl_5]} $$

Rearrange the equilibrium expression to solve for [SbCl_5]

Rearrange the expression to solve for the concentration of SbCl_5: $$ [SbCl_5] = \frac{[SbCl_3][Cl_2]}{K_{eq}} $$

Plug the given concentrations into the rearranged equation

Substitute the given concentrations and the value of K_eq into the equation to find the concentration of SbCl_5: $$ [SbCl_5] = \frac{(0.0255)(0.135)}{4.9 \times 10^{-4}} $$

Calculate [SbCl_5]

Perform the calculation to find [SbCl_5]: $$ [SbCl_5] = \frac{0.0034425}{4.9 \times 10^{-4}} \approx 7.03 \times 10^{-3} M $$

Key concepts

Equilibrium Constant Expression

The equilibrium constant expression, denoted as K_eq, is vital in understanding the balance between reactants and products in a chemical reaction at equilibrium. For the reaction \[\mathrm{SbCl}_{5}(g) \rightleftharpoons \mathrm{SbCl}_{3}(g) + \mathrm{Cl}_{2}(g)\], the equilibrium expression is \[K_{\mathrm{eq}} = \frac{[\mathrm{SbCl}_3][\mathrm{Cl}_2]}{[\mathrm{SbCl}_5]}\].
This expression relates the concentrations of the products to the reactants. When the concentration values are substituted into the expression, it helps determine whether the system is at equilibrium.
Furthermore, solving for an unknown concentration involves rearranging the equilibrium expression for the missing component. For instance, to find the concentration of \(\mathrm{SbCl}_5\), the expression is rearranged as \[[\mathrm{SbCl}_5] = \frac{[\mathrm{SbCl}_3][\mathrm{Cl}_2]}{K_{\mathrm{eq}}}\].
Knowing each concentration and the equilibrium constant allows solving for the unknown concentration, making equilibrium expressions fundamental in predicting the extent of a reaction.

Reaction Quotient

The reaction quotient, Q, is a way to gauge a reaction's progress towards equilibrium. It is similar to the equilibrium constant but uses the initial concentrations instead of the equilibrium concentrations. For the same equation \[\mathrm{SbCl}_{5}(g) \rightleftharpoons \mathrm{SbCl}_{3}(g) + \mathrm{Cl}_{2}(g)\], the reaction quotient's expression is the same as K_eq; however, it reflects the system's state at any point, not just at equilibrium. \[Q = \frac{[\mathrm{SbCl}_3][\mathrm{Cl}_2]}{[\mathrm{SbCl}_5]}\].
Comparing Q to the K_eq value determines the direction the reaction will proceed to reach equilibrium. If \(Q < K_{\mathrm{eq}}\), the forward reaction is favored, producing more products. Conversely, if \(Q > K_{\mathrm{eq}}\), the reaction will shift towards the reactants. Understanding Q helps predict how changes in concentration, pressure, or temperature will affect the reaction progress.

Le Chatelier's Principle

Le Chatelier's principle provides insight into how a chemical reaction at equilibrium responds to external changes. It states that if a dynamic equilibrium is disturbed by changing the conditions, such as concentration, temperature, or pressure, the position of equilibrium will shift in a way that counteracts the disturbance.
For instance, increasing the concentration of \(\mathrm{SbCl}_3\) or \(\mathrm{Cl}_2\) for our reaction \[\mathrm{SbCl}_{5}(g) \rightleftharpoons \mathrm{SbCl}_{3}(g) + \mathrm{Cl}_{2}(g)\] would shift the equilibrium to the left, producing more \(\mathrm{SbCl}_5\) as the system seeks to reduce the impact of the added substances.
Similarly, a change in temperature would affect the equilibrium depending on the reaction's exothermic or endothermic nature. Utilizing Le Chatelier's principle, chemists can predict and control the outcome of reactions, making it a powerful tool in chemical manufacturing and various industrial processes.