Question

\(\mathrm{NiO}\) is to be reduced to nickel metal in an industrial process by use of the reaction $$\mathrm{NiO}(s)+\mathrm{CO}(g) \rightleftharpoons \mathrm{Ni}(s)+\mathrm{CO}_{2}(g)$$ At \(1600 \mathrm{~K}\) the equilibrium constant for the reaction is \(K_{p}=6.0 \times 10^{2}\). If a CO pressure of 150 torr is to be employed in the furnace and total pressure never exceeds 760 torr, will reduction occur?

Short answer

Based on the given reaction and equilibrium conditions, the reduction of \(\mathrm{NiO}\) to \(\mathrm{Ni}\) will occur when the initial \(\mathrm{CO}\) pressure is 150 torr. The total pressure in the furnace at equilibrium will be approximately 150 torr, which is below the maximum allowed pressure of 760 torr.

Step-by-step solution

Write the equilibrium expression

Write the expression for the equilibrium constant \(K_p\) as the ratio of the partial pressures of the products over the reactants. Since all the stoichiometric coefficients are 1, the equilibrium expression will have the following form: $$K_p = \frac{P_{\mathrm{Ni}}P_{\mathrm{CO_2}}}{P_{\mathrm{NiO}}P_{\mathrm{CO}}}$$

Define the change in partial pressures

Let's define the change in partial pressures as the reaction proceeds at equilibrium. Since the initial pressure of \(\mathrm{CO}\) is given, let's assume the initial pressures of the other compounds to be zero: $$P_{\mathrm{CO}} = 150 - x \ \mathrm{torr}$$ $$P_{\mathrm{CO_2}} = x \ \mathrm{torr}$$ $$P_{\mathrm{NiO}} = y \ \mathrm{torr}$$ $$P_{\mathrm{Ni}} = y \ \mathrm{torr}$$ where: - \(x\) represents the decrease in \(\mathrm{CO}\) pressure and the increase in \(\mathrm{CO_2}\) pressure as the reaction progresses; - \(y\) represents the increase in \(\mathrm{Ni}\) and \(\mathrm{NiO}\) pressures as the reaction progresses.

Substitute the pressures into the equilibrium expression

Now, substitute the pressures from Step 2 into the equilibrium expression: $$K_p = \frac{(y)(x)}{(y)(150-x)}$$ Since the equilibrium constant \(K_p = 6.0 \times 10^2\), we can write: $$6.0 \times 10^2 = \frac{(y)(x)}{(y)(150-x)}$$

Simplify the equation and solve for x

Simplify the equation and solve for x: $$6.0 \times 10^2 = \frac{x}{150-x}$$ Now, cross-multiply and solve for \(x\): $$x(6.0 \times 10^2) = (150-x)(x)$$ $$6.0 \times 10^2x = 150x - x^2$$ $$x^2 - 150x + 6.0 \times 10^2x = 0$$ Solve the quadratic equation for x: $$x \approx 144.7 \ \mathrm{torr}$$

Check if the reduction will occur

Since \(x \approx 144.7 \ \mathrm{torr}\), the pressure of \(\mathrm{CO_2}\) will increase from 0 to approximately 144.7 torr at equilibrium, while the pressure of \(\mathrm{CO}\) will decrease from 150 to approximately 5.3 torr. The total pressure in the furnace will be: $$P_{total} = P_{\mathrm{CO}} + P_{\mathrm{CO_2}} = \approx 144.7 + 5.3 = 150 \ \mathrm{torr}$$ The total pressure of 150 torr is less than the maximum allowed pressure of 760 torr. Therefore, the reduction of \(\mathrm{NiO}\) to \(\mathrm{Ni}\) will occur under these conditions.

Key concepts

Understanding Chemical Equilibrium

Chemical equilibrium represents a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction. This balance means that the concentrations of reactants and products remain constant over time, not that the reactions have stopped. Equilibrium can be represented in the equation \[ \text{A} + \text{B} \rightleftharpoons \text{C} + \text{D} \]At equilibrium, the rate at which A and B react to form C and D is equal to the rate at which C and D decompose back into A and B.
In the given exercise, \( \mathrm{NiO} \) is converted to nickel metal, and the equilibrium constant (\( K_p \)) is used to determine if the reaction will proceed under certain conditions. An essential aspect of solving such problems includes setting up the equilibrium expression correctly, which involves the partial pressures of the gases involved, since the reactants and products are in different phases. Understanding this concept is crucial for predicting the behavior of the system under changing conditions and for optimizing industrial processes like the reduction of \( \mathrm{NiO} \) to nickel metal.

Applying Le Chatelier's Principle

Le Chatelier's principle is pivotal for predicting how a system at equilibrium responds to changes in concentration, temperature, or pressure. It states that if an external change is imposed on a system at equilibrium, the system adjusts to minimize that change. For instance, adding more of a reactant to a system will shift the equilibrium to produce more products, whereas an increase in temperature for an exothermic reaction will shift the equilibrium towards the reactants.
In practical terms, when the partial pressure of \( \mathrm{CO} \) in the given reaction is altered, Le Chatelier’s principle helps us understand how the system will shift to re-establish equilibrium. The question regarding the feasibility of reduction refers to whether the equilibrium will favor the formation of nickel metal under the specified conditions. With Le Chatelier's principle, students can grasp why a change in conditions, such as pressure, can influence the direction of the reaction and the success of the industrial process.

Partial Pressure and Its Role in Equilibrium

Partial pressure is a measure of the individual pressure exerted by a specific gas within a mixture of gases. It's critical when dealing with gases in chemical reactions under constant temperature, as it can influence the position of equilibrium. According to Dalton’s Law of Partial Pressures, the total pressure exerted by a gas mixture is the sum of the partial pressures of each individual gas.
In the context of the exercise, understanding partial pressures is necessary to solve for \( x \)—the change in pressure of the gases as the reaction reaches equilibrium. By manipulating these pressures, one can determine the conditions necessary for a reaction to proceed favorably. For example, as shown in the problem, maintaining the total pressure below 760 torr ensures the reaction's progression toward the desired product, within the safety limits of the furnace.