Problem 74 - Chemistry The Science in Context

Question

The following reaction is carried out in a sealed, rigid vessel at constant temperature. $$ 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g) $$ a. If the change in the partial pressure of \(\mathrm{O}_{2}\) is \(-x,\) what are the changes in the partial pressures of \(\mathrm{NO}\) and \(\mathrm{NO}_{2} ?\) b. As the reaction proceeds, what happens to the total pressure in the reaction vessel?

Step-by-Step Solution

Answer

Answer: The changes in the partial pressures of the gases during the reaction are as follows: - The partial pressure of \(\mathrm{NO}\) decreases by \(2x\). - The partial pressure of \(\mathrm{NO}_{2}\) increases by \(2x\). The total pressure in the reaction vessel will decrease as the reaction proceeds.

Step 1: a. Determine the changes in the partial pressures of \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\)

For the chemical reaction: $$ 2 \mathrm{NO}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}_{2}(g) $$ The stoichiometric coefficients indicate the molar ratios of the reactants and products. We see that 2 moles of \(\mathrm{NO}\) react with 1 mole of \(\mathrm{O}_{2}\) to form 2 moles of \(\mathrm{NO}_{2}\). Since the pressure in the vessel remains constant, we can use the relation between moles and pressure for gases: $$ \frac{n_{NO}}{n_{O_{2}}} = \frac{2}{1} = \frac{-\Delta P_{NO}}{-x} $$ Where \(n_{NO}\) and \(n_{O_{2}}\) are the changes in moles of \(\mathrm{NO}\) and \(\mathrm{O}_{2}\), and \(\Delta P_{NO}\) is the change in the partial pressure of \(\mathrm{NO}\). Solving for \(\Delta P_{NO}\): $$ -\Delta P_{NO} = 2(-x) \Rightarrow \boxed{\Delta P_{NO} = -2x} $$ Similarly, we can find the change in the partial pressure of \(\mathrm{NO}_{2}\): $$ \frac{n_{NO_{2}}}{n_{O_{2}}} = \frac{2}{1} = \frac{\Delta P_{NO_{2}}}{-x} $$ Solving for \(\Delta P_{NO_{2}}\): $$ \boxed{\Delta P_{NO_{2}} = -2x} $$

Step 2: b. Analyze the impact on the total pressure

As the reaction proceeds, we can look at the changes in partial pressures for each gas involved: - The partial pressure of \(\mathrm{O}_{2}\) decreases by \(x\). - The partial pressure of \(\mathrm{NO}\) decreases by \(2x\). - The partial pressure of \(\mathrm{NO}_{2}\) increases by \(2x\). We can observe that the decrease in partial pressure of \(\mathrm{NO}\) is compensated by the increase in partial pressure of \(\mathrm{NO}_{2}\). Therefore, the net change in the total pressure will be given by the change in partial pressure of \(\mathrm{O}_{2}\), which is a decrease of \(x\). Since the vessel is rigid and at constant temperature, the total pressure in the reaction vessel will decrease as the reaction proceeds. The total pressure will be: $$ P_{total} = P_{O_{2}} + P_{NO} + P_{NO_{2}} = (P_{O_{2}} - x) + (P_{NO} - 2x) + (P_{NO_{2}} + 2x) $$ The changes in partial pressures of \(\mathrm{NO}\) and \(\mathrm{NO}_{2}\) cancel out with each other, and we have: $$ P_{new} = P_{O_{2}} + P_{NO} + P_{NO_{2}} - x $$ The total pressure in the reaction vessel will decrease.

Key Concepts

Partial Pressure

Partial pressure is an essential concept in chemical reactions involving gases. It refers to the pressure that a gas in a mixture of gases would exert if it were alone in the entire volume of the mixture. Each gas in the mixture behaves as if it occupies the entire volume by itself.
In the given reaction, oxygen's partial pressure changes as the reaction proceeds. If the partial pressure of oxygen decreases by \(-x\), it tells us how much oxygen is being consumed over time.
According to Dalton’s Law of Partial Pressures, the total pressure of a gas mixture is the sum of the partial pressures of each gas. Understanding partial pressures helps predict changes in the total pressure during reactions.

Chemical Reaction

A chemical reaction occurs when substances interact to form new products. In our example, nitrogen monoxide (\(\mathrm{NO}\)) and oxygen (\(\mathrm{O}_{2}\)) react to form nitrogen dioxide (\(\mathrm{NO}_{2}\)).
This transformation involves breaking and forming chemical bonds, using the principle of stoichiometry. Stoichiometry uses the coefficients in a balanced chemical equation to determine ratios of reactants and products. In this case, the reaction tells us that 2 moles of \(\mathrm{NO}\) react with 1 mole of \(\mathrm{O}_{2}\) to create 2 moles of \(\mathrm{NO}_{2}\).
Understanding these ratios helps us calculate the changes in partial pressures, as these coefficients guide how substances are consumed and produced.

Gas Laws

Gas laws describe the behavior and relationships between pressure, volume, temperature, and quantity of gas. The Ideal Gas Law and Dalton's Law of Partial Pressures are crucial when analyzing gas reactions.
In the context of our reaction, the relationship between the number of moles and pressure is utilized. Because the vessel is sealed and rigid, volume and temperature remain constant, making the changes in moles directly influence pressure.
These laws allow us to calculate how changes in one parameter affect others. For instance, a decrease in moles (from reactants being consumed) leads to a decrease in partial pressures, using the Ideal Gas Law formula:
\[ PV = nRT \]
Where \(P\) is pressure, \(V\) is volume, \(n\) is number of moles, \(R\) is the gas constant, and \(T\) is temperature.

Reaction Kinetics

Reaction kinetics studies the rate of chemical reactions and the factors affecting it. The speed of a reaction and how quickly reactants turn into products vary by factors such as concentration, temperature, and pressure.
In a fixed volume at constant temperature, as our scenario describes, the change in partial pressures helps us understand the reaction's progress.
Kinetics does not just ensure the transformation into products but also provides insight into how optimized conditions, like ideal temperature and pressure, can maximize reaction efficiency. Observing how partial pressures change is crucial for understanding how fast a reaction occurs and reaching equilibrium.