13.5
The Integrated Rate Law: The Dependence of Concentration on Time
While the differential rate law relates the rate and concentrations of reactants, a second form of rate law called the integrated rate law relates con…
A rate law used to determine the reaction rate from reactant concentrations and rate constants can be converted into rate laws demonstrating the dependence of reaction rate on reactant concentration and time.
These rate laws can be used to study how slowly or quickly a reactant is consumed, or how much time is necessary to reach half the concentration of a reactant.
To begin, examine the differential rate law, which expresses the reaction rate as a change in reactant concentration during a specific time interval. Integration of this law leads to the integrated rate law, which expresses the reaction rate as a relation between a reactant’s initial concentration and its concentration after a specific duration.
The integrated rate law is dependent on the overall reaction order and, hence, varies for each reaction type. However, irrespective of the overall order, all integrated rate laws take the form of a standard linear equation with distinct y, m, x, and b components, and can be plotted to generate a straight line.
In a zero-order integrated rate law, [A]t is the reactant concentration at the time t, k is the rate constant, t is the time, and [A]0 is the initial reactant concentration.
For a zero-order reaction, a plot of the reactant concentration as a function of time generates a straight line. The slope is the negative value of the rate constant, and the y-intercept is the initial reactant concentration.
In a first-order reaction, the natural log of reactant concentration plotted as a function of time gives a straight line. The slope corresponds to the negative value of the rate constant, while the y-intercept gives the natural log of the initial reactant concentration.
According to the second-order integrated rate law, a plot of the inverse of the reactant concentration versus time yields a straight line. The slope equates to the rate constant, and the y-intercept represents the inverse of the initial reactant concentration.
The overall reaction order can be identified using experimental kinetic data by plotting the different integrated rate laws. Only the plot with a linear graph corresponds to the correct overall reaction order. Subsequent analysis allows determining the rate constant and reactant concentration at any given time.
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Q1: What is the difference between a differential rate law and an integrated rate law?
A differential rate law expresses reaction rate as a change in reactant concentration over a specific time interval. An integrated rate law, derived by integrating the differential form, expresses the relationship between a reactant's initial concentration and its concentration after a specific duration. Both describe reaction kinetics but at different mathematical levels.
Q2: How do you identify the reaction order using integrated rate laws?
Plot the experimental kinetic data using different integrated rate law forms for zero-order, first-order, and second-order reactions. Only the plot that generates a straight line corresponds to the correct overall reaction order. Once the linear plot is identified, you can determine the rate constant and predict reactant concentration at any given time.
Q3: What does a plot of reactant concentration versus time show for a zero-order reaction?
For a zero-order reaction, plotting reactant concentration against time produces a straight line. The slope equals the negative value of the rate constant, and the y-intercept represents the initial reactant concentration. This linear relationship reflects the constant reaction rate independent of reactant concentration.
Q4: Why is the natural logarithm used in first-order integrated rate laws?
Integration of the first-order differential rate law naturally produces a logarithmic form. Plotting the natural log of reactant concentration versus time yields a straight line, making it easier to identify first-order kinetics and extract the rate constant from the slope and initial concentration from the y-intercept.
Q5: How can integrated rate laws help determine when a radioactive material becomes safe?
Integrated rate laws relate reactant concentration to elapsed time, allowing calculation of how long a radioactive material must decay to reach safe levels. By knowing the initial radioactivity, the rate constant, and the target safe concentration, you can use the appropriate integrated rate law to estimate the required storage duration.
Q6: What is the relationship between reaction order and the form of the integrated rate law?
The integrated rate law depends on the overall reaction order, varying for each reaction type. However, all integrated rate laws take the form of a standard linear equation with distinct y, m, x, and b components. This universal linear structure allows plotting to generate straight lines for kinetic analysis across different reaction orders.
Q7: What does the slope represent in a second-order integrated rate law plot?
In a second-order reaction, plotting the inverse of reactant concentration versus time produces a straight line. The slope of this line equals the rate constant, while the y-intercept represents the inverse of the initial reactant concentration. This relationship allows direct determination of kinetic parameters from experimental data.