Question

In a reaction \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\), the equilibrium concentrations of \(\mathrm{PC}_{5}\) and \(\mathrm{PCl}_{3}\) are \(0.4\) and \(0.2\) moles/litre respectively. If the value of \(K_{c}\) is \(0.5\), what is the concentration of \(\mathrm{C} 1_{2}\) in mole/litre? (a) \(1.0\) (b) \(1.5\) (c) \(0.5\) (d) \(2.0\)

Short answer

The concentration of \(\mathrm{Cl}_2\) is 1.0 mole/litre.

Step-by-step solution

Substitute the known values into the equilibrium expression

Replace the values of \(K_c, [\mathrm{PCl}_5], [\mathrm{PCl}_3]\) into the expression to get the equation \(0.5 = (0.2[\mathrm{Cl}_2])/0.4\). Solve this equation for \([\mathrm{Cl}_2]\).

Solve for \([\mathrm{Cl}_2]\)

Solving the above equation gives \([\mathrm{Cl}_2] = 0.5 * 0.4 / 0.2 = 1.0\) (mole/litre).

Key concepts

Equilibrium Constant

Understanding the equilibrium constant (\(K_c\)) is crucial in grasping chemical equilibrium. It's a way to quantify the position of equilibrium in a reaction that occurs in a closed system. Think of the equilibrium constant as a number that tells you how much products and reactants are present once the system has settled down into a state of balance.
For instance, consider the formula \[ K_c = \frac{[\text{products}]}{[\text{reactants}]} \] where the products and reactants are raised to the power of their stoichiometric coefficients in the balanced equation and their concentrations are expressed in moles per liter. The value of the equilibrium constant is not affected by the initial concentrations but instead is influenced by the temperature.
In our exercise example, the equilibrium constant for the reaction \( \mathrm{PCl}_{5}(\mathrm{~g}) \rightleftharpoons \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) \) is given as 0.5. With the concentrations of \( \mathrm{PCl}_5 \) and \( \mathrm{PCl}_3 \) known, we can calculate the concentration of \( \mathrm{Cl}_2 \) using the value of \( K_c \) to complete the ratio.

Reaction Quotient

The reaction quotient (\(Q\)) is like a snapshot of a reaction at any point before reaching equilibrium. It has the same formula as the equilibrium constant, except it uses the current concentrations of the reactants and products, not the concentrations at equilibrium. Here’s its expression: \[ Q = \frac{[\text{current products}]}{[\text{current reactants}]} \] It's important to compare \( Q \) with \( K_c \) to predict the direction the reaction will shift to reach equilibrium. If \( Q < K_c \) the reaction will proceed forward, increasing the concentration of products. Conversely, if \( Q > K_c \) the reaction will proceed in reverse, reducing the concentration of products.
In solving the given exercise, we don't calculate \( Q \) because the system is already at equilibrium where \( Q = K_c \) by definition. However, understanding the concept of reaction quotient is essential because it allows us to predict the behavior of reactions under non-equilibrium conditions.

Le Chatelier's Principle

Le Chatelier's principle offers insight into how a chemical system at equilibrium responds to disturbances. It simply states that if an external change is applied to a system at equilibrium, the system will adjust itself to partially counteract the change. Some disturbances can include changes in concentration, pressure, volume, or temperature.
For example, if we increase the concentration of a reactant, the system shifts to produce more products. Likewise, when the concentration of a product is increased, the system will shift to produce more reactants.

Applying Le Chatelier's Principle

To see how you might apply this to the exercise presented, imagine if the concentration of \( \mathrm{PCl}_3 \) was increased. According to Le Chatelier’s principle, the equilibrium would shift to the left to reduce the concentration of \( \mathrm{PCl}_3 \) by converting it back to \( \mathrm{PCl}_5 \). Such shifts would also affect the concentration of \( \mathrm{Cl}_2 \) and the equilibrium constant would remain the same, because it's a constant at a given temperature regardless of concentration changes.