Question

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products: $$ \mathrm{A}(\mathrm{g})+2 \mathrm{~B}(\mathrm{~g}) \longrightarrow \mathrm{C}(\mathrm{g}) $$ When [B] is decreasing at \(0.5 \mathrm{~mol} / \mathrm{L}\) -s, how fast is \([\mathrm{A}]\) decreasing?

Short answer

The concentration of [A] is decreasing at a rate of 0.25 mol/L·s.

Step-by-step solution

Write the General Rate Law Expression

For the reaction \[\mathrm{A} (\mathrm{g}) + 2 \mathrm{~B} (\mathrm{g}) \rightarrow \mathrm{C} (\mathrm{g})\], we can express the rate of reaction in terms of the concentration changes of each species:\[\text{Rate} = - \frac{1}{\Delta t} \frac{\Delta [\mathrm{A}]}{1} = - \frac{1}{\Delta t} \frac{\Delta [\mathrm{B}]}{2} = \frac{1}{\Delta t} \frac{\Delta [\mathrm{C}]}{1} \]

Insert Given Values

From the problem, we know that the concentration of \(\mathrm{B}\) is decreasing at a rate of \(0.5 \text{ mol/L} \cdot \text{s}\). Therefore:\[- \frac{\Delta [\mathrm{B}]}{\Delta t} = 0.5 \mathrm{~mol} / \mathrm{L} \, \text{s} \]

Relate Rates of B and A

Using the relationship between the rates: \[- \frac{\Delta [\mathrm{B}]}{2 \Delta t} = \frac{\Delta [\mathrm{A}]}{\Delta t}\] Since \(\Delta [\mathrm{B}] / \Delta t = 0.5 \mathrm{~mol} / \mathrm{L} \, \text{s}\), we can rewrite it as: \[-\frac{0.5}{2} \text{ mol/L}s = \frac{\Delta [\mathrm{A}]}{\Delta t}\]

Solve for the Rate of Change of [A]

Calculate \[\frac{\Delta [\mathrm{A}]}{\Delta t} = -\frac{0.5}{2} \mathrm{~mol} / \mathrm{L} \cdot \text{s} = -0.25 \mathrm{~mol} / \mathrm{L} \cdot \text{s}\] Therefore, \(\mathrm{A}\) is decreasing at a rate of \(0.25 \mathrm{~mol} / \mathrm{L} \, \text{s}\).

Key concepts

rate law

The rate law of a chemical reaction is a mathematical expression that describes the rate of the reaction in relation to the concentration of its reactants. It’s essential because it helps predict how a reaction behaves over time.
For the reaction \[\text{A (g) + 2B (g) } \rightarrow \text{ C (g)}\], we can determine the rate by looking at how quickly each reactant's concentration changes. In general, the rate law is given by:
\[ \text{Rate = k [A]^m [B]^n}\]
Here, \( k \) is the rate constant, and \( m \) and \( n \) are the orders of the reaction with respect to \( A \) and \( B \), respectively. For our purpose, we need to relate it to the given changes in concentrations.

concentration change

Concentration change in a reaction refers to how the amount of reactants decreases and products increase over time. This is directly linked to the reaction's rate. If we know how fast one substance changes, we can use stoichiometry to find out about others.
In our exercise, we know that \( [B] \) is decreasing at a rate of \( 0.5 \text{ mol/L} \text{s} \). Using the balanced reaction \[ \text{A (g) + 2B (g) } \rightarrow \text{ C (g)}\], we see that for every 2 molecules of B that react, 1 molecule of A reacts.
This means the rate of change in \( [A] \) is half of that of \( [B] \). So if \( [B] \) is decreasing by \( 0.5 \text{ mol/L} \text{s}\), \( [A] \) would decrease by \( 0.25 \text{ mol/L} \text{s}\).

reaction kinetics

Reaction kinetics is the study of the speed or rate at which chemical reactions occur. It's about understanding how different factors like concentration, temperature, and the presence of a catalyst affect the reaction rates.
The central part of reaction kinetics is to quantify how reactants transform into products over time. In our example, we calculated the rate at which \( [A] \) is decreasing by using the given rate of \( [B] \). The relationship is derived from the stoichiometry of the reaction, which aligns with the principles of kinetics.

chemical reactions

Chemical reactions involve the rearrangement of atoms to transform reactants into products. They are represented by balanced equations, detailing the substances involved and their proportions.
In our given reaction \[\text{A (g) + 2B (g) } \rightarrow \text{ C (g)}\], it indicates that 1 molecule of \( \text{A} \) reacts with 2 molecules of \( \text{B} \) to form 1 molecule of \( \text{C} \). Understanding this allows us to relate the rate of change of \( \text{B} \) to \( \text{A} \).
By learning the stoichiometry, we can predict that if \( [B] \) decreases by \( 0.5 \text{ mol/L} \text{s} \), then \( [A] \) will decrease by half that rate, which is \( 0.25 \text{ mol/L} \text{s} \). This fundamental grasp helps in predicting the dynamics of the reaction.