Problem 5 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

An important reaction in the formation of smog is: \\[ \mathrm{O}_{3}(\mathrm{g})+\mathrm{NO}(\mathrm{g}) \rightleftharpoons \mathrm{O}_{2}(\mathrm{g})+\mathrm{NO}_{2}(\mathrm{g}) \\] Under certain conditions, the equilibrium constant for this reaction is \(K=6.0 \times 10^{34}\). If the partial pressures of each gas in the air over your home town were \(1.0 \times 10^{-6} \mathrm{bar} \mathrm{O}_{3}\) \(1.0 \times 10^{-5} \mathrm{bar} \mathrm{NO}, 2.5 \times 10^{-4} \mathrm{bar} \mathrm{NO}_{2},\) and \(8.2 \times 10^{-3} \mathrm{bar} \mathrm{O}_{2}\) what could you say about the course of the reaction as it moves to equilibrium? (Section 15.2 )

Step-by-Step Solution

Answer

The reaction will shift towards the products to reach equilibrium.

Step 1: Write the Reaction Quotient Expression

The reaction quotient \(Q\) is given by the expression \(Q = \frac{[\text{products}]}{[\text{reactants}]}\). For the given reaction: \[ \mathrm{O}_{3}( ext{g})+\mathrm{NO}( ext{g}) \rightleftharpoons \mathrm{O}_{2}( ext{g})+\mathrm{NO}_{2}( ext{g}) \]\, the expression becomes: \[ Q = \frac{P_{\mathrm{O}_2} \cdot P_{\mathrm{NO}_2}}{P_{\mathrm{O}_3} \cdot P_{\mathrm{NO}}} \] where \(P\) indicates the partial pressure of each gas.

Step 2: Substitute Partial Pressures into the Expression

Substitute the given partial pressures into the expression for \(Q\): \[ Q = \frac{(8.2 \times 10^{-3})(2.5 \times 10^{-4})}{(1.0 \times 10^{-6})(1.0 \times 10^{-5})} \] Calculate \(Q\) by solving the math.

Step 3: Calculate the Reaction Quotient

Calculate the numerical value for \(Q\):\[ Q = \frac{(8.2 \times 10^{-3})(2.5 \times 10^{-4})}{(1.0 \times 10^{-6})(1.0 \times 10^{-5})} = \frac{2.05 \times 10^{-6}}{1.0 \times 10^{-11}} = 2.05 \times 10^{5} \]

Step 4: Compare Reaction Quotient \(Q\) to Equilibrium Constant \(K\)

With \(Q = 2.05 \times 10^{5}\) and \(K = 6.0 \times 10^{34}\), compare the two values. Since \(Q \ll K\), the reaction is not at equilibrium and will shift towards the products to reach equilibrium.

Key Concepts

Reaction Quotient

In chemical reactions, the reaction quotient, represented as \(Q\), helps to determine the direction in which a reaction will proceed to reach equilibrium. It is calculated with the same expression as the equilibrium constant \(K\), but uses the current concentrations or partial pressures of reactants and products. The formula is:

• \(Q = \frac{P_{\text{products}}}{P_{\text{reactants}}}\)

For the reaction \( \text{O}_3 (\text{g}) + \text{NO} (\text{g}) \rightleftharpoons \text{O}_2 (\text{g}) + \text{NO}_2 (\text{g}) \), it becomes \(Q = \frac{P_{\text{O}_2} \cdot P_{\text{NO}_2}}{P_{\text{O}_3} \cdot P_{\text{NO}}}\).
When \(Q < K\), the reaction shifts towards the products to reach equilibrium. Conversely, if \(Q > K\), the reaction will shift towards the reactants. In our exercise, with \(Q = 2.05 \times 10^{5}\) against \(K = 6.0 \times 10^{34}\), \(Q\) is significantly smaller, meaning the reaction will proceed towards forming more products.

Equilibrium Constant

The equilibrium constant \(K\) is a value that expresses the ratio of product concentrations to reactant concentrations at chemical equilibrium. For a reaction of the form \(aA + bB \leftrightharpoons cC + dD\), the equilibrium constant \(K\) is expressed as:

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

Equilibrium is reached when the rates of the forward and reverse reactions are equal, and thus the concentrations remain constant.
In the case of our smog-forming reaction, \(K\) is extraordinarily high at \(6.0 \times 10^{34}\), indicating that the products \(\text{O}_2\) and \(\text{NO}_2\) are heavily favored at equilibrium under these conditions. This value of \(K\) suggests that the forward reaction proceeds almost to completion before it reaches equilibrium.

Partial Pressure

Partial pressure refers to the pressure that a single gas in a mixture of gases contributes to the total pressure. It is a crucial concept in understanding how gases behave in reaction mixtures. The partial pressure of a gas is directly proportional to its concentration in the mixture and is related through the equation of state for gases.

• The formula: \(P_i = n_iRT / V\), where \(P_i\) is the partial pressure, \(n_i\) is the number of moles, \(R\) is the gas constant, and \(V\) is the volume.

In our given reaction scenario, calculating \(Q\) involves using the partial pressures of \(\text{O}_3\), \(\text{NO}\), \(\text{O}_2\), and \(\text{NO}_2\) rather than their concentrations since they are all in the gaseous state. These pressures represent the conditions of the gases in the air at the site of interest.

Smog Formation

Smog is a type of air pollution mainly formed by the reaction of sunlight with pollutants such as nitrogen oxides and volatile organic compounds. These compounds, when present in the atmosphere, undergo chemical reactions that yield ozone and other secondary pollutants that we perceive as smog.
In urban areas, the reaction \(\text{O}_3 (\text{g}) + \text{NO} (\text{g}) \rightleftharpoons \text{O}_2 (\text{g}) + \text{NO}_2 (\text{g})\) plays an essential role in smog formation. The presence of vehicles and industrial activities increase the emission of these nitrogen oxides, and with the presence of ozone, it sets the stage for harmful smog.

• Key conditions for smog formation:
  • High concentrations of pollutants
  • Sunlight
  • Stagnant atmospheric conditions

Understanding the chemical behavior and equilibrium dynamics of such reactions can help in developing strategies to reduce and control air pollution.

Gas Reactions

Gas reactions involve substances in the gaseous state reacting with each other to form products, which can also be gases. Such reactions are characterized by the use of partial pressures instead of concentrations, due to the nature of gases spreading out to fill their container.
In gas reactions like \(\text{O}_3 (\text{g}) + \text{NO} (\text{g}) \rightleftharpoons \text{O}_2 (\text{g}) + \text{NO}_2 (\text{g})\), understanding the partial pressures and their changes is crucial. Le Châtelier's principle applies, where any change in pressure, concentration, or temperature can shift the equilibrium position.

• Key points about gas reactions:
  • Influenced by temperature, as gases expand or contract with heat.
  • Pressure changes can shift equilibrium positions.

Through studying these reactions, chemists can predict how atmospheric and reaction conditions impact the formation and consumption of gases, applying this to areas such as environmental science and industrial chemistry.