Question

(a) Consider the combustion of ethylene, \(\mathrm{C}_{2} \mathrm{H}_{4}(g)+\) \(3 \mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) .\) If the concentration of \(\mathrm{C}_{2} \mathrm{H}_{4}\) is decreasing at the rate of \(0.025 \mathrm{M} / \mathrm{s}\), what are the rates of change in the concentrations of \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O} ?\) (b) The rate of decrease in \(\mathrm{N}_{2} \mathrm{H}_{4}\) partial pressure in a closed reaction vessel from the reaction \(\mathrm{N}_{2} \mathrm{H}_{4}(g)+\mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{NH}_{3}(g)\) is 63 torr \(/ \mathrm{h}\). What are the rates of change of \(\mathrm{NH}_{3}\) partial pressure and total pressure in the vessel?

Short answer

In part (a), the rates of change for both \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) are \(0.05\frac{\mathrm{M}}{\mathrm{s}}\). In part (b), the rate of change for \(\mathrm{NH}_{3}\) partial pressure is \(126\frac{\text{torr}}{\mathrm{h}}\), while the rate of change of total pressure is \(0\frac{\text{torr}}{\mathrm{h}}\).

Step-by-step solution

Write down the given information

The rate of \(\mathrm{C}_{2} \mathrm{H}_{4}\) decreasing is \(0.025\mathrm{M} / \mathrm{s}\).

Determine the relationship between rate of change in concentrations of reactants and products

According to the stoichiometry of the reactions, for every 1 mole of \(\mathrm{C}_{2} \mathrm{H}_{4}\) consumed, 2 moles of \(\mathrm{CO}_{2}\) and 2 moles of \(\mathrm{H}_{2} \mathrm{O}\) are produced.

Calculate the rates of change for \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\)

The rate of change for \(\mathrm{CO}_{2}\) is $$\frac{2\text{ moles }\mathrm{CO}_{2}}{1\text{ mole }\mathrm{C}_{2}\mathrm{H}_{4}}\times0.025\frac{\mathrm{M}}{\mathrm{s}}=0.05\frac{\mathrm{M}}{\mathrm{s}}$$ The rate of change for \(\mathrm{H}_{2} \mathrm{O}\) is $$\frac{2\text{ moles }\mathrm{H}_{2}\mathrm{O}}{1\text{ mole }\mathrm{C}_{2}\mathrm{H}_{4}}\times0.025\frac{\mathrm{M}}{\mathrm{s}}=0.05\frac{\mathrm{M}}{\mathrm{s}}$$ Part B:

Write down the given information

The rate of decrease in \(\mathrm{N}_{2} \mathrm{H}_{4}\) partial pressure is 63 torr \(/ \mathrm{h}\).

Determine the relationship between rate of change in concentrations of reactants and products

According to the stoichiometry of the reactions, for every 1 mole of \(\mathrm{N}_{2} \mathrm{H}_{4}\) consumed, 2 moles of \(\mathrm{NH}_{3}\) are produced.

Calculate the rate of change for \(\mathrm{NH}_{3}\) partial pressure

The rate of change for \(\mathrm{NH}_{3}\) partial pressure is $$\frac{2\text{ moles }\mathrm{NH}_{3}}{1\text{ mole }\mathrm{N}_{2}\mathrm{H}_{4}}\times63\frac{\text{torr}}{\mathrm{h}}=126\frac{\text{torr}}{\mathrm{h}}$$

Calculate the rate of change for total pressure

We know that 1 mole of \(\mathrm{N}_{2} \mathrm{H}_{4}\) and 1 mole of \(\mathrm{H}_{2}\) are consumed, and 2 moles of \(\mathrm{NH}_{3}\) are produced. So the total rate of change in the number of moles is: \((-1-1+2)=0\) Since the rate of change in the number of moles is 0 and the volume is constant (closed vessel), the total pressure in the vessel remains constant as well. Thus, the rate of change of total pressure is: \(0\frac{\text{torr}}{\mathrm{h}}\)

Key concepts

Stoichiometry

Stoichiometry is a key concept in chemistry that involves the quantitative relationship between reactants and products in a chemical reaction. It helps us understand how much of each substance is involved and produced during the reaction.
In the context of reaction rates, stoichiometry allows us to determine how the rate of disappearance of a reactant relates to the rate of appearance of a product. For example, in the combustion of ethylene reaction:

• For every 1 mole of ethylene (\(\mathrm{C}_{2} \mathrm{H}_{4}\)) consumed, 2 moles of carbon dioxide (\(\mathrm{CO}_{2}\)) and 2 moles of water (\(\mathrm{H}_{2} \mathrm{O}\)) are produced.

This relationship can be used to calculate the rates of formation of products given the rate of consumption of a reactant. In our problem, the rate at which \(\mathrm{C}_{2} \mathrm{H}_{4}\) decreases is \(0.025 \mathrm{M/s}\). Thus, the rates for \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) are both twice that value: \(0.05 \mathrm{M/s}\). This shows how stoichiometric coefficients in balanced equations guide calculations of reaction rates.

Partial Pressure

Understanding partial pressure is crucial in reactions involving gases. It refers to the pressure exerted by a single type of gas in a mixture of gases. In a closed container, the total pressure exerted by the gas mixture equals the sum of the partial pressures of each gas.
Partial pressures of reactants and products in a gas-phase reaction can change over time as the reaction proceeds. In the reaction:

• \(\mathrm{N}_{2} \mathrm{H}_{4}(g) + \mathrm{H}_{2}(g) \rightarrow 2 \mathrm{NH}_{3}(g)\), the partial pressure of \(\mathrm{N}_{2} \mathrm{H}_{4}\) is decreasing at a rate of 63 torr/h.

Using the stoichiometric ratios from the balanced equation, we find that the rate of increase in the partial pressure of \(\mathrm{NH}_{3}\) is 126 torr/h. This doubling effect is because one mole of \(\mathrm{N}_{2} \mathrm{H}_{4}\) produces two moles of \(\mathrm{NH}_{3}\). Additionally, since the overall change in moles of gas is zero when moving from reactants to products, the total pressure in the system remains constant.

Combustion Reactions

Combustion reactions are a type of exothermic chemical reaction that usually occur when a hydrocarbon reacts with oxygen to produce carbon dioxide and water. They are common in processes such as burning fuels for energy.
The ethylene combustion reaction, \(\mathrm{C}_{2} \mathrm{H}_{4}(g) + 3 \mathrm{O}_{2}(g) \rightarrow 2 \mathrm{CO}_{2}(g) + 2 \mathrm{H}_{2} \mathrm{O}(g)\), represents a typical combustion reaction. Here, ethylene acts as the hydrocarbon fuel, and it reacts with oxygen to form carbon dioxide and water.

• The reaction is balanced so that all of the atoms in the reactants are accounted for in the products.
• The reaction is defined by a stoichiometric ratio of reactants to products (1:3:2:2 for \(\mathrm{C}_{2} \mathrm{H}_{4}\):\(\mathrm{O}_{2}\):\(\mathrm{CO}_{2}\):\(\mathrm{H}_{2} \mathrm{O}\)).

Studying combustion reactions is important for understanding the principles of energy conversion and impacts on environmental pollution.

Concentration Change

To understand reaction rates, one must consider how the concentration of reactants and products changes over time. Concentration change is a direct measure of how quickly a reaction proceeds.
In the exercise about ethylene combustion, the concentration of \(\mathrm{C}_{2} \mathrm{H}_{4}\)is decreasing at 0.025 M/s, indicating that this is the speed at which ethylene is consumed. This decrease can tell us how fast the products, \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\), are formed.

• The rate of change in concentration for a product is positive, as it increases with the reaction's progress.
• For reactants, the rate is negative because their concentration diminishes over time.

By using stoichiometry, you can determine that the rates of change for \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}\) are both 0.05 M/s. Reaction rates are fundamental for studying kinetics and understanding how different conditions affect the speed of reactions.