Question

The equilibrium \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NOCl}(g)\) is established at \(500 \mathrm{~K}\). An equilibrium mixture of the three gases has partial pressures of \(0.095 \mathrm{~atm}, 0.171 \mathrm{~atm}\), and \(0.28\) atm for \(\mathrm{NO}, \mathrm{Cl}_{2}\), and \(\mathrm{NOCl}\), respectively. Calculate \(K_{p}\) for this reaction at \(500 \mathrm{~K}\).

Short answer

The equilibrium constant \(K_p\) for the reaction \(2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2\mathrm{NOCl}(g)\) at 500K is approximately 50.87.

Step-by-step solution

Write the balanced chemical equation

The given balanced chemical equation is: \[ 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2\mathrm{NOCl}(g) \]

Write the equilibrium expression

For the given balanced chemical equation, the equilibrium constant expression is: \[K_p = \frac{\mathrm{[NOCl]}^2}{\mathrm{[NO]}^2 \cdot \mathrm{[Cl}_2]} \] Here, [NO], [Cl₂], and [NOCl] represent the partial pressures of the respective gases at equilibrium.

Substitute the given partial pressures

We are given the partial pressures of the three gases at equilibrium: - [NO] = 0.095 atm - [Cl₂] = 0.171 atm - [NOCl] = 0.28 atm Substitute these values into the equilibrium constant expression: \[ K_p = \frac{(0.28~\text{atm})^2}{(0.095~\text{atm})^2 \cdot (0.171~\text{atm})} \]

Calculate the value of \(K_p\)

Now, calculate the value of \(K_p\): \[ K_p \approx \frac{0.0784}{0.001541375} \] \[ K_p \approx 50.87 \] So, the equilibrium constant \(K_p\) for this reaction at 500K is approximately 50.87.

Key concepts

Chemical Equilibrium

Chemical equilibrium is a state in a closed system where the concentrations of reactants and products remain constant over time. At this point, the rate of the forward reaction equals the rate of the reverse reaction, meaning that the amounts of reactants and products no longer change.

While it might seem as if no reactions are occurring, in reality, reactions continue at an equal pace in both directions. This dynamic status means that equilibrium is not static but a balancing act of ongoing processes. In our exercise, the equilibrium between nitrogen monoxide, chlorine gas, and nitrosyl chloride demonstrates this principle.

Partial Pressure

When discussing gas mixtures, partial pressure is the pressure that each gas in the mixture would exert if it alone occupied the entire volume. In a mixture of gases, each gas contributes to the total pressure. Hence, the total pressure of the system is the sum of the partial pressures of all the individual gases.

This concept is crucial when dealing with chemical equilibria involving gases, as it helps us deduce the amount of each gas present. In the exercise, the different substances involved in the equilibrium have different partial pressures within the mixture.

Equilibrium Expression

The equilibrium expression is a formula that relates the concentrations (or partial pressures) of the reactants and products involved in a reversible reaction at equilibrium. For reactions involving gases, as we have in our exercise, the equilibrium expression is written in terms of partial pressures, which is designated as \( K_p \).

The general form of the equilibrium expression for a reaction like \( aA + bB \rightleftharpoons cC + dD \) would be \( K_p = \frac{[C]^c[D]^d}{[A]^a[B]^b} \), taking into account the balancing coefficients from the chemical equation. Identifying this correct form and substituting the provided partial pressures will give us the equilibrium constant \( K_p \).

Le Chatelier's Principle

Le Chatelier's principle is a guiding concept in chemistry that qualitatively predicts how a change in conditions can shift the position of equilibrium. If a system at equilibrium is subjected to a change in concentration, pressure, or temperature, the system will adjust itself to counteract the effect of the change and restore a new equilibrium.

For example, increasing the pressure of the system by decreasing the volume would shift the position of equilibrium towards the side with fewer moles of gas. It essentially says that equilibria want to maintain balance and will shift accordingly when they are disturbed. This principle allows chemists to control reactions and optimize conditions for desired products.