Problem 4 - Chemistry and Chemical Reactivity

Question

The reaction $$ 2 \mathrm{NO}_{2}(\mathrm{g}) \rightleftarrows \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g}) $$ has an equilibrium constant, \(K_{c},\) of 170 at \(25^{\circ} \mathrm{C} .\) If \(2.0 \times 10^{-3} \mathrm{mol}\) of \(\mathrm{NO}_{2}\) is present in a \(10 .-\mathrm{L}\). flask along with \(1.5 \times 10^{-3}\) mol of \(\mathrm{N}_{2} \mathrm{O}_{4},\) is the system at equilibrium? If it is not at equilibrium, does the concentration of \(\mathrm{NO}_{2}\) increase or decrease as the system proceeds to equilibrium?

Step-by-Step Solution

Answer

The system is not at equilibrium; \([\mathrm{NO}_2]\) will increase as the system proceeds to equilibrium.

Step 1: Write the Equilibrium Expression

For the reaction \(2 \mathrm{NO}_{2}(\mathrm{g}) \rightleftarrows \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{g})\), the equilibrium constant \(K_c\) is defined as \[ K_c = \frac{[\mathrm{N}_{2} \mathrm{O}_{4}]}{[\mathrm{NO}_{2}]^2} \] where \([\mathrm{NO}_2]\) and \([\mathrm{N}_2O_4]\) are the molar concentrations of \(\mathrm{NO}_2\) and \(\mathrm{N}_2O_4\), respectively.

Step 2: Calculate Initial Concentrations

Calculate the initial concentrations of \(\mathrm{NO}_2\) and \(\mathrm{N}_2O_4\) in the flask. The concentration \([A]\) is calculated as \([A] = \frac{\text{moles of } A}{\text{volume in liters}}\). Thus, \[ [\mathrm{NO}_2] = \frac{2.0 \times 10^{-3} \text{ mol}}{10 \text{ L}} = 2.0 \times 10^{-4} \text{ M} \] and \[ [\mathrm{N}_2O_4] = \frac{1.5 \times 10^{-3} \text{ mol}}{10 \text{ L}} = 1.5 \times 10^{-4} \text{ M} \].

Step 3: Calculate the Reaction Quotient \(Q_c\)

Determine the reaction quotient \(Q_c\), which uses the same formula as the equilibrium constant but with initial concentrations: \[ Q_c = \frac{[\mathrm{N}_2O_4]}{[\mathrm{NO}_2]^2} = \frac{1.5 \times 10^{-4}}{(2.0 \times 10^{-4})^2} = \frac{1.5 \times 10^{-4}}{4.0 \times 10^{-8}} = 3750 \].

Step 4: Compare \(Q_c\) with \(K_c\)

Compare \(Q_c\) to \(K_c\) to determine if the system is at equilibrium. Since \(Q_c = 3750 > K_c = 170\), the concentration of the product is too high, indicating that the reaction will shift to the left (toward the formation of \(\mathrm{NO}_2\)) to reach equilibrium.

Key Concepts

Equilibrium Constant

The equilibrium constant, denoted as \(K_c\), is a vital concept in chemical equilibrium. It quantifies the ratio of the concentrations of the products to the reactants at equilibrium for a reversible chemical reaction.
For a general reaction like \(aA + bB \rightleftarrows cC + dD\), the equilibrium constant expression is given by:

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

Where [A], [B], [C], and [D] are the molar concentrations of the respective species, and a, b, c, d are their stoichiometric coefficients.
In the context of the given reaction:

• \(\mathrm{2NO_2(g)} \rightleftarrows \mathrm{N_2O_4(g)}\),
• \(K_c = \frac{[\mathrm{N_2O_4}]}{[\mathrm{NO_2}]^2}\)

Here, the equilibrium constant, \(K_c = 170\), is used to determine when the system has reached equilibrium at a given temperature, typically 25°C in this scenario.
The value of \(K_c\) helps predict the extent of a reaction, indicating whether the products or reactants are favored. A large \(K_c\) signifies more products at equilibrium, while a small \(K_c\) implies more reactants.

Reaction Quotient

The reaction quotient, denoted as \(Q_c\), resembles the equilibrium constant but is calculated using the initial concentrations of the reactants and products. It provides a snapshot of the system's state and helps determine which direction a reaction will proceed to achieve equilibrium.
The expression for \(Q_c\) for a reaction remains the same as for \(K_c\):

• For the reaction \(\mathrm{2NO_2(g)} \rightleftarrows \mathrm{N_2O_4(g)}\), the formula is \(Q_c = \frac{[\mathrm{N_2O_4}]}{[\mathrm{NO_2}]^2}\)

After determining the initial concentrations:

• \([\mathrm{NO_2}] = 2.0 \times 10^{-4}\; M\)
• \([\mathrm{N_2O_4}] = 1.5 \times 10^{-4}\; M\)

We compute \(Q_c = 3750\). To determine the current state of equilibrium, compare \(Q_c\) with \(K_c\). If \(Q_c > K_c\), as in this problem, the system has too many products, and the reaction will shift toward the reactants to reach equilibrium. Conversely, if \(Q_c < K_c\), the reaction will proceed in the forward direction.

Gaseous Reactions

Gaseous reactions, like the one in this problem, involve reactants and products in the gaseous state. These reactions often have equilibrium constants expressed in terms of concentrations (\(K_c\)) or partial pressures (\(K_p\)).
For the reaction \(2\mathrm{NO_2(g)} \rightleftarrows \mathrm{N_2O_4(g)}\), the equilibrium constant \(K_c\) is used based on concentration, given the volumes that express concentrations rather than pressure.
In gaseous reactions, temperature and pressure dramatically influence the state of equilibrium. This is because a change in volume can alter concentrations substantially. The provided system specifies reactions at 25°C, and any deviation from this set condition would potentially change \(K_c\) and thus the system's equilibrium position.
In such cases, it is crucial to calculate initial concentrations correctly and ensure conditions are controlled, which directly ties back to Le Chatelier's Principle, discussed in the next section.

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental guideline for predicting how a chemical system at equilibrium responds to changes in concentration, temperature, or pressure. It states that if a dynamic equilibrium is disturbed by changing the conditions, the system shifts in the direction that counteracts the change.
In the context of the current reaction, where \(Q_c = 3750 > K_c = 170\), it suggests that the reaction is product-heavy and must shift towards the reactants (left shift) to restore equilibrium.
This principle can be simplified as follows:

• Adding reactants or removing products shifts the equilibrium towards products.
• Removing reactants or adding products shifts the equilibrium towards reactants.
• A change in temperature or pressure also causes the system to adjust, in a way that reduces the impact of this change.

Through this principle, chemists can predict how a system might change when conditions alter, allowing for controlled manipulations to obtain desired concentrations of reactants or products.