Problem 45 - Chemistry An Atoms First Approach

Question

A \(1.00-\mathrm{L}\) flask was filled with 2.00 moles of gaseous \(\mathrm{SO}_{2}\) and 2.00 moles of gaseous \(\mathrm{NO}_{2}\) and heated. After equilibrium was reached, it was found that 1.30 moles of gaseous NO was present. Assume that the reaction $$\mathrm{SO}_{2}(g)+\mathrm{NO}_{2}(g) \rightleftharpoons \mathrm{SO}_{3}(g)+\mathrm{NO}(g)$$, occurs under these conditions. Calculate the value of the equilibrium constant, \(K\), for this reaction.

Step-by-Step Solution

Answer

The equilibrium constant, \(K\), for the given reaction can be calculated using the formula: \( K = \frac{[\mathrm{SO_3}][\mathrm{NO}]}{[\mathrm{SO_2}][\mathrm{NO_2}]} \). By determining the change in concentrations and finding the equilibrium concentrations, we calculate the value of \(K \approx 3.45\).

Step 1: Write the initial concentrations for each reactant and product

We know the initial amounts of SO2 and NO2 are 2.00 moles and the volume of the flask is 1.00 L. The initial concentrations can be calculated as follows: \[ [\mathrm{SO_2}]_{initial} = \frac{2.00 \ \text{moles}}{1.00 \ \text{L}} = 2.00 \ \mathrm{M} \] \[ [\mathrm{NO_2}]_{initial} = \frac{2.00 \ \text{moles}}{1.00 \ \text{L}} = 2.00 \ \mathrm{M} \] Initially, there is no SO3 and NO in the flask, hence: \[ [\mathrm{SO_3}]_{initial} = [\mathrm{NO}]_{initial} = 0 \ \mathrm{M} \]

Step 2: Calculate the change in concentrations

Given that 1.30 moles of NO are present at equilibrium, we can calculate x, the change in concentration. For every mole of NO formed, one mole of SO2 and NO2 each are consumed. \[ x = \frac{1.30 \ \text{moles}}{1.00 \ \text{L}} = 1.30 \ \mathrm{M} \]

Step 3: Determine the equilibrium concentrations

Using the calculated value of x, we can find the concentrations at equilibrium: \[ [\mathrm{SO_2}]_{eq} = [\mathrm{SO_2}]_{initial} - x = 2.00 \ \mathrm{M} - 1.30 \ \mathrm{M} = 0.70 \ \mathrm{M} \] \[ [\mathrm{NO_2}]_{eq} = [\mathrm{NO_2}]_{initial} - x = 2.00 \ \mathrm{M} - 1.30 \ \mathrm{M} = 0.70 \ \mathrm{M} \] \[ [\mathrm{SO_3}]_{eq} = [\mathrm{SO_3}]_{initial} + x = 0 \ \mathrm{M} + 1.30 \ \mathrm{M} = 1.30 \ \mathrm{M} \] \[ [\mathrm{NO}]_{eq} = [\mathrm{NO}]_{initial} + x = 0 \ \mathrm{M} + 1.30 \ \mathrm{M} = 1.30 \ \mathrm{M} \]

Step 4: Calculate the equilibrium constant, K

Now that we have the equilibrium concentrations, we can calculate K: \[ K = \frac{[\mathrm{SO_3}]_{eq}[\mathrm{NO}]_{eq}}{[\mathrm{SO_2}]_{eq}[\mathrm{NO_2}]_{eq}} = \frac{(1.30 \ \mathrm{M})(1.30 \ \mathrm{M})}{(0.70 \ \mathrm{M})(0.70 \ \mathrm{M})} = \frac{1.69}{0.49} \] Thus, \[ K \approx 3.45 \]

Key Concepts

Chemical Equilibrium

Chemical equilibrium is a fundamental concept in chemistry that describes a state where the concentrations of all reactants and products remain constant over time. This occurs when the rates of the forward and reverse reactions are equal, meaning there is no net change in the amounts of substances involved.
Imagine a busy highway with equal numbers of cars moving in both directions. While cars continuously enter and exit at each end, the number of cars on the road remains unchanged. Similarly, in a chemical reaction at equilibrium, reactants convert to products at the same rate as products convert back into reactants.
When a system reaches equilibrium, it may not appear dynamic, but reactions continue to occur. It is this constant activity that maintains equilibrium. Understanding the dynamic nature of equilibrium helps in various fields including chemical engineering and pharmaceuticals where control over reaction conditions is crucial.

Reaction Quotient

The reaction quotient, denoted as \( Q \), is a tool used to determine the direction in which a reaction will proceed to reach equilibrium. It is calculated using the same expression as the equilibrium constant \( K \), but with the initial concentrations of reactants and products instead of their equilibrium values.
The expression for the reaction quotient of a reaction \( aA + bB \rightleftharpoons cC + dD \) is given by:

• \( Q = \frac{[C]^c[D]^d}{[A]^a[B]^b} \)

By comparing the value of \( Q \) with \( K \):

• If \( Q < K \): The reaction will proceed in the forward direction to form more products.
• If \( Q > K \): The reaction will proceed in the reverse direction to form more reactants.
• If \( Q = K \): The system is already at equilibrium.

By using \( Q \), we can predict whether a system will shift toward the products or the reactants, providing vital insight into reaction dynamics.

Equilibrium Concentrations

Equilibrium concentrations refer to the amounts of reactants and products in a reaction mixture when the system is at equilibrium. For the reaction provided in the original exercise, we can determine these using initial amounts and the change experienced as the system reaches equilibrium.
The process typically involves:

• Determining initial concentrations.
• Calculating the change in concentration over time as the system moves towards equilibrium.
• Using these changes to find the concentrations at equilibrium.

For instance, in the problem, initial concentrations of \( \[ \mathrm{SO}_2 \]
\) and \( \[ \mathrm{NO}_2 \]
\) started at 2.00 M, but the equilibrium concentrations decreased due to their consumption to form \( \[ \mathrm{SO}_3 \]
\) and \( \[ \mathrm{NO} \]
\). Understanding these changes provides insights into how far a reaction progresses and the conditions that control reaction behavior.

Le Chatelier's Principle

Le Chatelier's Principle is a crucial concept for predicting how a change in conditions affects a chemical equilibrium. It states that if a system at equilibrium is subjected to a disturbance, such as a change in concentration, temperature, or pressure, it will adjust to counteract that change and a new equilibrium will be established.
The principle can be understood through several examples:

• If the concentration of a reactant is increased, the system shifts towards the products to reduce the added reactant.
• Increasing the pressure will shift the equilibrium to the side with fewer gas molecules.
• If the temperature is increased, the endothermic direction of the reaction will be favored as it absorbs the excess heat.

Using Le Chatelier's Principle helps in predicting the direction and extent of shifts in equilibrium due to external influences. It is a vital tool for chemists and engineers in optimizing reactions and processes in various industrial applications.