Question

A mixture of 0.47 mole of \(H_{2}\) and 3.59 moles of \(H C l\) is heated to \(2800^{\circ} \mathrm{C}\). Calculate the equilibrium partial pressures of \(\mathrm{H}_{2}, \mathrm{Cl}_{2}\), and \(\mathrm{HCl}\) if the total pressure is 2.00 atm. The \(K_{P}\) for the reaction \(\mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g)\) is 193 at \(2800^{\circ} \mathrm{C}.\)

Short answer

The equilibrium partial pressures for \(H_{2}, Cl_{2}, HCl\) will be calculated from the value obtained for x and used in the above equations to find the respective values.

Step-by-step solution

Set Up The ICE Table

The ICE table stands for Initial, Change, and Equilibrium. It helps to visualize the change in the moles of each component involved in the reaction. Let \(p_{H_{2}}, p_{Cl_{2}}, p_{HCl}\) represent the partial pressures of \(H_{2}, Cl_{2}, HCl\) respectively. Given initial moles of \(H_{2}\) and \(Cl_{2}\), we are to find the change in their amounts which are -x in both cases and the change in \(HCl\) is +2x. Hence, the ICE table will be as follows: \n\n| | \(H_{2}\) | \(Cl_{2}\) | \(HCl\) |\n|Initial | 0.47 | 3.59 | 0 |\n|Change | -x | -x | +2x | \n|Final | 0.47-x | 3.59-x | 2x |

Setup the equilibrium expression

According to the definition of the equilibrium constant \(K_{P}\) for the given reaction, the expression will be as follows: \n\n\[ K_{P} = \frac{(p_{HCl})^2}{p_{H_{2}} \cdot p_{Cl_{2}}} \] \n\nSubstitute the expressions from the ICE table: \n\n\[ K_{P} = \frac{(2x)^2}{(0.47-x)(3.59-x)} \] \n\nRemember that the total pressure is given to be 2.00 atm.

Solve for x

Since the total pressure is 2.00 atm, we can write: \n\n\[ p_{H_{2}}+p_{Cl_{2}}+p_{HCl} = 0.47-x + 3.59-x + 2x = 2.00 \n\nSolving this gives us the value of x that brings the system to equilibrium.

Calculate the equilibrium partial pressures

Using the value of x calculated in the previous step, we can determine the equilibrium partial pressures of each component: \n\n\[ p_{H_{2}} = 0.47 - x \] \n\[ p_{Cl_{2}} = 3.59 - x \] \n\[ p_{HCl} = 2x \]

Key concepts

Understanding the ICE Table

The ICE table is a valuable tool for visualizing the progression of a chemical reaction at equilibrium. It's handy particularly when dealing with problems related to changes in concentration, pressure, or moles of reactants and products over the course of a reaction.

Initial: Stands for the initial concentrations or partial pressures of the reactants and products before the reaction begins. In our given problem, the initial moles of the reactants were provided, which we can use to calculate the initial partial pressures assuming ideal gas behavior.

Change: Represents the amount by which the concentrations or partial pressures change as the reaction proceeds. This is often expressed in terms of a variable, typically 'x', that indicates the extent of the reaction's progress from the start to equilibrium.

Equilibrium: Shows the concentrations or partial pressures of the reactants and products at equilibrium. These are the initial amounts plus or minus the change, depending on the stoichiometry of the reaction. In the exercise, we added the change (which is the stoichiometric amount of 'x' calculated) to the initial moles to find the final state at equilibrium.

When setting up an ICE table, clarity is crucial to avoid errors. Make sure each column corresponds to a reactant or product, and each row clearly denotes initial amounts, change, and equilibrium amounts.

Equilibrium Constant (Kp) and Its Role

The equilibrium constant, denoted as Kp when dealing with gases, is a number that describes the ratio of the partial pressures of products to reactants at equilibrium for a reversible reaction at a constant temperature. It is a vital concept in chemical equilibrium that allows us to predict the extent of a reaction.

To calculate the equilibrium constant Kp, we use the general format: \[ K_{P} = \frac{(p_{\text{products}})^{\text{stoichiometry}}}{(p_{\text{reactants}})^{\text{stoichiometry}}} \]

In this equation, 'p' stands for the partial pressures of the products and reactants, raised to the power of their stoichiometric coefficients in the balanced equation. For our exercise, the equilibrium constant was given (193), allowing us to set up an equation including our variable 'x' from the ICE table.

Understanding the equilibrium constant helps in predicting whether a reaction will favor the formation of products or reactants. A high Kp value, as in our exercise, suggests that at equilibrium, the reaction mixture is primarily composed of products.

Partial Pressure Calculations

Partial pressures are the individual pressures exerted by each gas in a mixture and are vital for understanding gas behavior in chemical reactions. Calculations involving partial pressures require both the ideal gas law and Dalton's Law of Partial Pressures.

Using the Ideal Gas Law: For a known amount of gas at a given temperature and volume, the pressure it exerts can be calculated. This relation is described by: \[ pV = nRT \]

Where 'p' is the pressure, 'V' is the volume, 'n' is the number of moles, 'R' is the ideal gas constant, and 'T' is the temperature.

Applying Dalton's Law: In a mixture of non-reacting gases, the total pressure exerted is equal to the sum of the partial pressures of each individual gas. The Formula is: \[ P_{\text{total}} = P_1 + P_2 + P_3 + \ldots \]

During equilibrium calculations, we assume the reaction is happening in a closed system where the total pressure is constant, as given in the problem statement. Using the values from the ICE table and the known total pressure, we solve for 'x' to find the equilibrium partial pressures. The calculated 'x' value indicates how the initial pressures have shifted to reach equilibrium, thereby allowing us to solve for each gas's partial pressure at this point.