Question

For a zeroth-order reaction: \(\mathrm{C} \rightarrow\) products, \(-\Delta[\mathrm{C}] / \Delta \mathrm{t}\) \(=k\), which of the following graphs would be expected to give a straight line? a. \([\mathrm{C}]^{2}\) vs \(\mathrm{t}\) b. \([\mathrm{C}]\) vs \(\mathrm{t}\) c. In \([\mathrm{C}]\) vs \(\mathrm{t}\) d. \(1 /[\mathrm{C}]\) vs \(\mathrm{t}\)

Short answer

Graph \([\mathrm{C}]\) vs \(\mathrm{t}\) gives a straight line for a zeroth-order reaction.

Step-by-step solution

Understanding Zeroth-Order Reaction

In a zeroth-order reaction, the rate of the reaction is independent of the concentration of the reactants. The rate equation is given by \( -\Delta[\text{C}] / \Delta t = k \), where \( k \) is the rate constant. This means the change in concentration over time is constant.

Identifying the Linear Graph

For a graph to be linear in a zeroth-order reaction, we need the integrated rate law to be in the form of a linear equation. The integrated rate law for a zeroth-order reaction is \( [\text{C}] = [\text{C}]_0 - kt \), which is a linear equation in the form \( y = mx + b \) where \( y = [\text{C}] \), \( x = t \), \( m = -k \), and \( b = [\text{C}]_0 \).

Analyzing the Options

Among the graphs given:- a. \( [\text{C}]^2 \) vs \( t \) will not be linear due to the squared term.- b. \( [\text{C}] \) vs \( t \) matches the integrated rate law \( [\text{C}] = [\text{C}]_0 - kt \), which is linear.- c. \( \ln [\text{C}] \) vs \( t \) is used for first-order reactions, not zeroth-order.- d. \( 1/[\text{C}] \) vs \( t \) is used for second-order reactions, not zeroth-order.

Conclusion

The graph of \( [\text{C}] \) vs \( t \) is the expected graph that would give a straight line for a zeroth-order reaction, according to the integrated rate law.

Key concepts

Rate Equation

In the context of reaction kinetics, the rate equation is essential for understanding how the concentration of reactants changes over time. For a zeroth-order reaction, this rate equation is particularly straightforward. It is expressed as \(-\Delta[\text{C}] / \Delta t = k\). This equation tells us that the rate of change of the concentration of reactant \(\text{C}\) over time \(t\) is constant, and is equal to the rate constant \(k\).
This implies that the rate of the reaction does not depend on the concentration of reactant \([\text{C}]\), differentiating zeroth-order reactions from other reaction orders where concentration plays a critical role in determining reaction speed.
This simplicity in the relationship allows for easy quantification and prediction of how much reactant will remain after a given period.

Integrated Rate Law

The integrated rate law for a zeroth-order reaction provides a linear relationship between reactant concentration and time. From the rate equation, we can derive it as \([\text{C}] = [\text{C}]_0 - kt\). Here, \([\text{C}]_0\) is the initial concentration of the reactant.
This formula resembles the equation of a straight line \(y = mx + b\), making it easier to understand how the concentration of the reactant decreases linearly over time.

• \(y\) corresponds to \([\text{C}]\), the concentration at any time \(t\)
• \(x\) corresponds to \(t\), the time
• \(m = -k\), the negative rate constant, acting as the slope
• \(b = [\text{C}]_0\), the initial concentration, equivalent to the y-intercept

Such a relationship helps in visualizing and confirming the constant rate characteristic of zeroth-order reactions.

Linear Graph

A linear graph is crucial in determining the order of a reaction and verifying the applicable rate equation. For zeroth-order reactions, plotting \([\text{C}]\) versus \(t\) results in a straight line. This occurs because the integrated rate law \([\text{C}] = [\text{C}]_0 - kt\) is already in a linear form.
The slope of the graph represents \(-k\), the rate constant, which is constant through the reaction duration. This feature distinguishes zeroth-order reactions from higher-order reactions, which involve logarithmic or hyperbolic forms for their integrated rate laws and give non-linear plots when \([\text{C}]\) is plotted against \(t\).

• Linear graphs confirm the relationship between concentration and time.
• They make it straightforward to obtain the rate constant \(k\).

The simplicity of these graphs allows for easy experimental verification.

Reaction Kinetics

Reaction kinetics is the branch of chemistry that studies the speed of reactions and how factors affect this speed. Understandably, the zeroth-order reaction exemplifies a unique case in reaction kinetics because the rate does not change with alterations in reactant concentration.
This constancy emphasizes a key characteristic of zeroth-order processes: external factors like catalyst presence or surface area often play significant roles. Such reactions frequently involve scenarios where the reactant is saturated on a surface or involved catalytically.

• Unlike first or second-order reactions, zeroth-order kinetics display consistent behavior regardless of reactant depletion.
• This feature of kinetics allows predictability in reactions over time under consistent conditions.

By thoroughly understanding these principles, one can more fully grasp how different kinetics orders apply to real-world chemical processes.