Question

\(1.25\) moles of NOCl were placed in a \(2.50 \mathrm{~L}\) reaction chamberat \(427^{\circ} \mathrm{C}\). After equilibrium was reached, 1.10 molesofNOClremained. Calculatetheequilibrium constant \(\mathrm{K}_{\mathrm{c}}\) for the reaction, \(2 \mathrm{NOC} 1(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}):\) (a) \(1.6 \times 10^{-3}\) (b) \(5.6 \times 10^{-4}\) (c) \(2.6 \times 10^{-3}\) (d) \(4.6 \times 10^{-4}\)

Short answer

The equilibrium constant \( K_c \) is closest to option (b) \( 5.6 \times 10^{-4} \).

Step-by-step solution

Write the Balanced Equation and Set Up Initial Concentrations

The balanced equation for the reaction is:\[ 2 \text{NOCl} (\text{g}) \rightleftharpoons 2 \text{NO} (\text{g}) + \text{Cl}_2 (\text{g}) \]Start by calculating the initial concentration of NOCl.\[ \text{Initial moles of NOCl} = 1.25 \text{ moles} \]\[ \text{Volume of reaction chamber} = 2.50 \text{ L} \]Initial concentration of NOCl:\[ \text{Initial } [\text{NOCl}] = \frac{1.25}{2.50} = 0.50 \text{ M} \]

Determine Change in Concentration

Since at equilibrium, 1.10 moles of NOCl remain, calculate the change in moles and then the change in concentration of NOCl:\[ \text{Change in moles of NOCl} = 1.25 - 1.10 = 0.15 \text{ moles} \]Change in concentration of NOCl:\[ \Delta [\text{NOCl}] = \frac{0.15}{2.50} = 0.06 \text{ M} \]

Calculate Equilibrium Concentrations

Use the change in concentration to find the equilibrium concentrations:For NOCl:\[ [\text{NOCl}]_\text{equilibrium} = 0.50 - 0.06 = 0.44 \text{ M} \]For NO and Cl_2:\[ \Delta [\text{NO}] = 2 \times \frac{0.06}{2} = 0.06 \text{ M} \] (since each mole of NOCl produces a mole each of NO and \( \frac{1}{2} \) mole of Cl_2)\[ [\text{NO}]_\text{equilibrium} = 0.06 \text{ M} \]\[ \Delta [\text{Cl}_2] = \frac{0.06}{2} = 0.03 \text{ M} \]\[ [\text{Cl}_2]_\text{equilibrium} = 0.03 \text{ M} \]

Substitute into the Expression for Kc

The equilibrium constant expression for the reaction is:\[ K_c = \frac{[\text{NO}]^2 [\text{Cl}_2]}{[\text{NOCl}]^2} \]Substituting the equilibrium concentrations:\[ K_c = \frac{(0.06)^2 (0.03)}{(0.44)^2} \]\[ K_c = \frac{0.000108}{0.1936} \]

Calculate Kc and Compare with Options

Calculate \( K_c \):\[ K_c = 5.58 \times 10^{-4} \]Compare with the given options to find the closest value:The equilibrium constant \( K_c \) is closest to option (b) \( 5.6 \times 10^{-4} \).

Key concepts

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction in a chemical system. This balance implies that the concentrations of reactants and products remain constant over time.
At equilibrium, although reactions continue to occur, there is no net change in the concentration of either reactants or products. This concept is crucial for understanding various chemical processes and is represented by the equilibrium constant, denoted as \( K_c \).

In our example, the equilibrium was reached after starting with 1.25 moles of NOCl, and at equilibrium, 1.10 moles of NOCl remained. This allows us to calculate the equilibrium concentrations of all species involved.

• Initial NOCl concentration was determined based on the moles and volume of the chamber.
• Concentration changes help track the system's approach to equilibrium.
• Equilibrium concentrations of NO and Cl\(_2\) are derived from the changes in NOCl.

Reaction Quotient

The reaction quotient, \( Q \), is a measure used to determine the direction in which a chemical reaction will proceed. It is calculated using the same expression as \( K_c \) but with the initial concentrations of reactants and products, rather than those at equilibrium.

By comparing \( Q \) to \( K_c \), it is possible to predict whether a reaction will move towards the formation of products (forward) or revert to forming reactants (reverse).

• If \( Q < K_c \), the reaction proceeds forward, producing more products.
• If \( Q > K_c \), the reaction moves in reverse, favoring reactants.
• If \( Q = K_c \), the system is at equilibrium.

In calculations similar to our example, comparing \( Q \) with \( K_c \) helps understand how far a system is from achieving equilibrium and in which direction the reaction will proceed to reach it.

Le Chatelier's Principle

Le Chatelier's Principle is a central concept in predicting how a system at equilibrium will react to external changes, such as changes in concentration, pressure, or temperature.
When a system at equilibrium experiences a disturbance, it will adjust in such a way as to counteract that change, thus moving towards a new state of equilibrium.

For our reaction:

• If more NOCl was added, the system would adjust by forming more products (NO and Cl\(_2\)).
• Removing NO or Cl\(_2\) results in the reaction producing more to replace the removed substances.
• Changing the pressure by altering the volume can affect gas-phase reactions.

This principle is critical for control in industrial chemical processes, ensuring maximum yield of desired products by managing reaction conditions effectively.

Mole Calculations

Mole calculations are fundamental in determining the quantities of substances involved in chemical reactions. Understanding mole relations allows you to calculate concentrations, track changes, and interpret equilibrium conditions.

In the exercise:

• We first calculated the initial concentration of NOCl using its molar quantity and the volume of the reaction chamber, \( 0.50 \, \text{M} \).
• The change in moles (0.15 moles decreased in NOCl) was used to find the changes in other species.
• From this change, we calculated the concentration changes for NO and Cl\(_2\) at equilibrium, respecting stoichiometry.

These calculations show how the interrelatedness of concentrations in chemical systems can be systematically decoded to understand chemical equilibria more thoroughly. Knowing how to compute these values is key to mastering chemical reaction analysis.