Question

For the following reaction, \(K_{\mathrm{p}}=6.5 \times 10^{4}\) at \(308 \mathrm{~K}\) $$ 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{NOCl}(g) $$ At equilibrium, \(P_{\mathrm{NO}}=0.35 \mathrm{~atm}\) and \(P_{\mathrm{Cl}_{2}}=0.10 \mathrm{~atm} .\) What is the equilibrium partial pressure of \(\operatorname{NOCl}(g) ?\)

Short answer

The equilibrium partial pressure of NOCl is approximately 28.21 atm.

Step-by-step solution

Write the equilibrium expression

For the given reaction, the equilibrium expression in terms of partial pressures (Kp) is:\[ K_{\text{p}} = \frac{(P_{\text{NOCl}})^2}{(P_{\text{NO}})^2 (P_{\text{Cl}_2})} \]

Plug in the given values

We are given the following equilibrium values:\[ K_{\text{p}} = 6.5 \times 10^4 \]\[ P_{\text{NO}} = 0.35 \text{ atm} \]\[ P_{\text{Cl}_2} = 0.10 \text{ atm} \]Substitute these values into the equilibrium expression:\[ 6.5 \times 10^4 = \frac{(P_{\text{NOCl}})^2}{(0.35)^2 (0.10)} \]

Solve for \( (P_{\text{NOCl}})^2 \)

First, calculate the denominator:\[ (0.35)^2 = 0.1225 \]\[ 0.1225 \times 0.10 = 0.01225 \]Then rearrange to solve for \( (P_{\text{NOCl}})^2 \):\[ (P_{\text{NOCl}})^2 = 6.5 \times 10^4 \times 0.01225 \]\[ (P_{\text{NOCl}})^2 = 796.25 \]

Find \( P_{\text{NOCl}} \) by taking the square root

Take the square root of both sides to find the partial pressure of \( \text{NOCl} \):\[ P_{\text{NOCl}} = \sqrt{796.25} \]\[ P_{\text{NOCl}} \approx 28.21 \text{ atm} \]

Key concepts

Partial Pressure

Partial pressure is the pressure exerted by an individual gas in a mixture of gases. In a mixture, each gas behaves independently and exerts pressure proportionate to its amount fraction, much like the entire gas mixture itself. The concept relates to Dalton's Law of Partial Pressures, which states that the total pressure of a mixture of ideal gases is the sum of the partial pressures of each component gas. For example, if we have gases A, B, and C in a container:
\( P_{total} = P_A + P_B + P_C \).
This principle is very useful in chemical equilibrium calculations. For instance, in the given exercise, we calculate the equilibrium constant for the reaction based on the partial pressures. Understanding partial pressures allows us to understand better how each gas contributes to the system's total pressure.

Chemical Equilibrium

Chemical equilibrium occurs when a reversible reaction's forward and backward reactions happen at the same rate. As a result, the concentrations of reactants and products remain constant over time. This dynamic balance does not mean the reactions have stopped but that their rates are equal.
In the given exercise, the reaction
\( 2 \text{NO}(g) + \text{Cl}_2(g) \rightleftharpoons 2 \text{NOCl}(g) \)
reaches equilibrium when the rate of formation of \( \text{NOCl} \) equals the rate of dissociation back into \( \text{NO} \) and \( \text{Cl}_2 \). The equilibrium constant (\( K_{\text{p}} \)) for this reaction shows the ratio of the product of the partial pressures of the products to the reactants when the system is at equilibrium. In our case, we have:
\( K_{\text{p}} = \frac{(P_{\text{NOCl}})^2}{(P_{\text{NO}})^2 (P_{\text{Cl}_2})} \).
This expression allows us to determine unknown quantities if the equilibrium constant and some partial pressures are known.

Le Chatelier's Principle

Le Chatelier's Principle states that if a dynamic equilibrium is disturbed by changing the conditions, the system will adjust itself to partially counteract the change and restore a new equilibrium. This principle can predict how changes in pressure, temperature, and concentration affect the system.
Consider the given reaction:
\( 2 \text{NO}(g) + \text{Cl}_2(g) \rightleftharpoons 2 \text{NOCl}(g) \).
If the pressure of the system increases, the equilibrium will shift towards the side with fewer gas molecules. Here, we have three gas molecules (2 NO and 1 \( \text{Cl}_2 \)) on the left and two molecules of \( \text{NOCl} \) on the right. This means an increase in pressure will favor the formation of \( \text{NOCl} \). Conversely, increasing the concentration of \( \text{NO} \) or \( \text{Cl}_2 \) will shift the equilibrium towards the right to produce more \( \text{NOCl} \) to offset the change. The principle helps understand how external changes influence the balance of the reaction, optimizing processes, such as synthesis and chemical manufacturing.