Question

Pure NOCl gas was heated at \(240^{\circ} \mathrm{C}\) in a 1.00 - \(\mathrm{L}\) container. At equilibrium the total pressure was 1.00 atm and the NOCl pressure was 0.64 atm. $$ 2 \mathrm{NOCl}(g) \rightleftharpoons 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) $$ (a) Calculate the partial pressures of \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) in the system. (b) Calculate the equilibrium constant \(K_{P}\).

Short answer

The partial pressures of NO and Cl2 are 0.24 atm and 0.12 atm respectively. The value of the equilibrium constant \(K_{P}\) is 0.75.

Step-by-step solution

Determine Partial Pressures

At equilibrium, we know that the pressure of NOCl is 0.64 atm and the total pressure is 1 atm. Therefore, the partial pressures of NO and Cl2 combined is \(1 - 0.64 = 0.36 atm\). Since the reaction produces 2 moles of NO and 1 mole of Cl2, we would expect twice as much NO as Cl2. Therefore, the partial pressure of NO is \(2/3 * 0.36 = 0.24 atm\) and the partial pressure of Cl2 is \(1/3 * 0.36 = 0.12 atm\).

Determine the Pressure New Mixture

Use the equation for the equilibrium constant for pressure (Kp), which is equal to the products of the partial pressures of the products raised to their stoichiometric coefficients, divided by the product of the partial pressures of the reactants raised to their stoichiometric coefficients. So, according to the balanced chemical equation, we see that Kp should be equal to (P(NO)^2 * P(Cl2)) / (P(NOCl)^2).

Calculate the Equilibrium Constant

Substitute the determined pressures into the \(K_{P}\) equation: \(K_{P} = (0.24^{2} * 0.12) / 0.64^{2}= 0.75\).

Key concepts

Partial Pressure

Partial pressure is the pressure that each gas in a mixture contributes to the total pressure. It reflects how much of the overall pressure is due to a particular gas. In chemical reactions involving gas mixtures, understanding partial pressures helps us determine concentrations of various gases at equilibrium.
In the provided exercise, we're considering the decomposition of \(2 \, \mathrm{NOCl}(g) \rightleftharpoons 2 \, \mathrm{NO}(g) + \mathrm{Cl}_{2}(g)\)within a confined space. The equilibrium condition specifies a total pressure of 1 atm and a partial pressure of \(\mathrm{NOCl}\) at 0.64 atm.
This scenario implies the remaining partial pressure (from the total) is from the products, namely \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\).

• Since the total pressure is 1 atm and \(\mathrm{NOCl}\) alone contributes 0.64 atm, the contribution of \(\mathrm{NO}\) and \(\mathrm{Cl}_{2}\) is \(1 - 0.64 = 0.36\,\mathrm{atm}\).
• According to the balanced reaction, every two moles of \(\mathrm{NOCl}\), produces 2 moles of \(\mathrm{NO}\) and 1 mole of \(\mathrm{Cl}_{2}\), signifying twice as much \(\mathrm{NO}\) present as \(\mathrm{Cl}_{2}\).
• Therefore, if these 0.36 atm is divided such that \(\mathrm{NO}\) gets two parts and \(\mathrm{Cl}_{2}\) one part, we ascertain that the partial pressure of \(\mathrm{NO}\) is \(0.24\,\mathrm{atm}\), and \(\mathrm{Cl}_{2}\) is \(0.12\,\mathrm{atm}\).

This clear understanding of partial pressures lets us confidently predict gas behavior under given conditions.

Equilibrium Constant

The equilibrium constant \(K_{P}\) for a reaction in the gas phase expresses the relationship between the partial pressures of the reactants and the products at equilibrium.
It allows us to quantitatively analyze the position of equilibrium and predict how changes in conditions might shift it.
In this equation, the components' pressures at equilibrium are raised to their stoichiometric coefficients as they appear in the balanced reaction equation. For our specific reaction:
\[K_{P} = \frac{(P(\mathrm{NO})^{2} \times P(\mathrm{Cl}_{2}))}{(P(\mathrm{NOCl})^{2})}\]
We use this relationship to effectively link the reactant and product concentrations. The given partial pressures are:

• \(P(\mathrm{NO}) = 0.24\,\mathrm{atm}\),
• \(P(\mathrm{Cl}_{2}) = 0.12\,\mathrm{atm}\),
• \(P(\mathrm{NOCl}) = 0.64\,\mathrm{atm}\).

Substituting these values into the equation yields a calculated \(K_{P}\) of 0.75.

• The value of \(K_{P}\) provides insight into the equilibrium position, where a smaller value suggests that, at equilibrium, reactants predominate over products.

Understanding the equilibrium constant is crucial for predicting the behavior of a chemical system.

Le Chatelier's Principle

Le Chatelier's Principle describes how a chemical system at equilibrium responds to a change in conditions. It offers a qualitative perspective on how the system moves to alleviate stress, maintaining equilibrium. Various factors can affect equilibrium, such as pressure, concentration, or temperature.
In our problem, the changes specifically revolve around pressure adjustments.
Given the reaction:

• \(2\, \mathrm{NOCl}(g) \rightleftharpoons 2\, \mathrm{NO}(g) + \mathrm{Cl}_{2}(g)\)

We need to consider how the system might react if the total pressure was altered. According to Le Chatelier:

• If pressure is increased, the system tends to favor the side with fewer moles of gas to counterbalance that pressure increase.
• Conversely, if pressure is lowered, the system can shift towards the side with more moles of gas.
• In our case, an increase in pressure could cause a shift toward forming more \(\mathrm{NOCl}\), as it comprises fewer moles (2 moles) compared to the products (3 moles total).

This principle is invaluable for predicting how systems at equilibrium might adjust with varying conditions, ensuring it remains in balance.