Integrated Rate Law First Order Reaction

The integrated rate law for a first-order reaction is a cornerstone concept in chemical kinetics, providing a mathematical relationship between the concentration of a reactant and time. It allows us to predict how the concentration of a reactant changes as a reaction progresses, which is crucial for understanding and controlling chemical processes.

Understanding First-Order Reactions

A first-order reaction is a chemical reaction in which the rate of the reaction is directly proportional to the concentration of only one reactant. Mathematically, this can be expressed as:

rate = -d[A]/dt = k[A]

Where:

• rate: is the rate of the reaction.
• -d[A]/dt: represents the rate of decrease in the concentration of reactant A with respect to time. The negative sign indicates that the concentration of A is decreasing.
• [A]: is the concentration of reactant A at a given time t.
• k: is the rate constant, a proportionality constant that is specific to each reaction at a given temperature. It reflects the intrinsic speed of the reaction.

This equation tells us that as the concentration of A increases, the rate of the reaction also increases proportionally. This is a defining characteristic of first-order reactions.

Examples of First-Order Reactions

Many chemical reactions follow first-order kinetics. Some common examples include:

• Radioactive Decay: The decay of radioactive isotopes, such as uranium-238 or carbon-14, follows first-order kinetics. The rate of decay is proportional to the amount of the radioactive substance present. This principle is the basis of radiometric dating techniques used in archaeology and geology.
• Decomposition of N2O5: The decomposition of dinitrogen pentoxide (N2O5) into nitrogen dioxide (NO2) and oxygen (O2) in the gas phase is a classic example of a first-order reaction.
• Isomerization Reactions: Some isomerization reactions, where a molecule rearranges its structure, can follow first-order kinetics under certain conditions.
• Hydrolysis of Aspirin: The hydrolysis of aspirin (acetylsalicylic acid) in aqueous solution to salicylic acid and acetic acid can be approximated as a first-order reaction.
• Enzyme-Catalyzed Reactions (under specific conditions): Some enzyme-catalyzed reactions can exhibit first-order kinetics when the substrate concentration is much lower than the Michaelis constant (Km).

Derivation of the Integrated Rate Law

To derive the integrated rate law for a first-order reaction, we start with the differential rate law:

rate = -d[A]/dt = k[A]

We can rearrange this equation to separate the variables:

d[A]/[A] = -k dt

Now, we integrate both sides of the equation. The integral of d[A]/[A] is ln[A], and the integral of -k dt is -kt. We also need to include a constant of integration, C:

∫ d[A]/[A] = ∫ -k dt

ln[A] = -kt + C

To determine the constant of integration, C, we use the initial condition: at time t = 0, the concentration of A is equal to its initial concentration, [A]0.

ln[A]0 = -k(0) + C

ln[A]0 = C

Now, substitute this value of C back into the integrated equation:

ln[A] = -kt + ln[A]0

Rearrange the equation to obtain the final form of the integrated rate law for a first-order reaction:

ln[A] - ln[A]0 = -kt

ln([A]/[A]0) = -kt

[A] = [A]0 * e^(-kt)

This equation is the integrated rate law for a first-order reaction. It shows the relationship between the concentration of reactant A at any time t ([A]), the initial concentration of A ([A]0), the rate constant k, and time t.

Different Forms of the Integrated Rate Law

The integrated rate law can be expressed in several different forms, each useful for different purposes:

• Exponential Form: [A] = [A]0 * e^(-kt) - This is the most common and direct form, showing the exponential decay of reactant concentration over time.
• Logarithmic Form: ln([A]/[A]0) = -kt - Useful for plotting data and determining the rate constant k from the slope of the line.
• Linear Form: ln[A] = -kt + ln[A]0 - This form emphasizes the linear relationship between ln[A] and time, allowing for easy graphical analysis.

Applications of the Integrated Rate Law

The integrated rate law for first-order reactions has numerous applications in chemistry, physics, and other scientific fields. Here are some key examples:

• Determining the Rate Constant (k): By measuring the concentration of the reactant at different times, we can use the integrated rate law to calculate the rate constant k. This is often done graphically by plotting ln[A] versus time, where the slope of the line is -k.
• Predicting Reactant Concentration at a Given Time: Knowing the initial concentration [A]0 and the rate constant k, we can use the integrated rate law to predict the concentration of the reactant [A] at any future time t.
• Determining the Time Required for a Certain Amount of Reactant to React: We can rearrange the integrated rate law to solve for the time t required for the reactant concentration to reach a specific value. This is useful for predicting reaction times and optimizing reaction conditions.
• Calculating Half-Life: The half-life (t1/2) of a reaction is the time required for the concentration of the reactant to decrease to half of its initial value. For a first-order reaction, the half-life is constant and can be calculated using the following equation:

t1/2 = ln(2)/k ≈ 0.693/k

The half-life is independent of the initial concentration of the reactant, which is a unique characteristic of first-order reactions. This property is particularly useful in applications such as radioactive dating.

• Radioactive Dating: Radioactive isotopes decay via first-order kinetics. By measuring the ratio of the remaining radioactive isotope to its stable decay product in a sample, we can determine the age of the sample. Carbon-14 dating is a well-known example used for dating organic materials up to around 50,000 years old. Other isotopes with longer half-lives are used for dating geological formations.
• Pharmacokinetics: In pharmacokinetics, the integrated rate law is used to model the elimination of drugs from the body. Many drugs are eliminated following first-order kinetics, meaning that the rate of elimination is proportional to the concentration of the drug in the body. This information is crucial for determining appropriate drug dosages and dosing intervals.
• Chemical Engineering: In chemical engineering, the integrated rate law is used to design and optimize chemical reactors. Understanding the kinetics of a reaction is essential for predicting reactor performance and maximizing product yield.
• Environmental Science: The integrated rate law can be used to model the degradation of pollutants in the environment. Understanding the kinetics of pollutant degradation is important for assessing the impact of pollutants on the environment and developing strategies for remediation.

Graphical Representation

The integrated rate law for a first-order reaction can be graphically represented in several ways:

• Concentration vs. Time Plot: A plot of concentration [A] versus time t shows an exponential decay curve. The concentration decreases rapidly at first and then gradually approaches zero as time increases.
• ln[A] vs. Time Plot: A plot of the natural logarithm of the concentration (ln[A]) versus time t yields a straight line with a negative slope. The slope of the line is equal to -k, the negative of the rate constant. The y-intercept of the line is equal to ln[A]0, the natural logarithm of the initial concentration. This type of plot is particularly useful for determining whether a reaction is first-order and for calculating the rate constant.

The linear relationship observed in the ln[A] vs. time plot is a key indicator of a first-order reaction. If the plot is not linear, the reaction is not first-order.

Determining the Order of a Reaction

While the integrated rate law can be used to analyze first-order reactions, it's also important to be able to determine the order of a reaction in the first place. Several methods can be used to determine the order of a reaction:

• Method of Initial Rates: This method involves measuring the initial rate of the reaction for different initial concentrations of the reactants. By comparing the rates and concentrations, the order of the reaction with respect to each reactant can be determined.
• Integrated Rate Law Method: This method involves plotting the concentration data according to the integrated rate laws for different reaction orders. The order that gives a linear plot is the correct order for the reaction. For a first-order reaction, as we discussed, a plot of ln[A] versus time will be linear.
• Half-Life Method: For first-order reactions, the half-life is independent of the initial concentration. By measuring the half-life for different initial concentrations, one can determine if the reaction is first-order. If the half-life remains constant, the reaction is likely first-order.

Pseudo-First-Order Reactions

In some cases, reactions that are not inherently first-order can be made to behave as if they are first-order by using a large excess of all reactants except one. These are called pseudo-first-order reactions.

For example, consider a reaction that is second order overall:

rate = k[A][B]

If the concentration of reactant B is much larger than the concentration of reactant A ([B] >> [A]), then the concentration of B will remain essentially constant during the reaction. In this case, we can define a pseudo-rate constant k' = k[B], and the rate law becomes:

rate = k'[A]

This rate law is now first-order with respect to A, and the reaction will behave as a first-order reaction. This technique is often used to simplify the analysis of complex reactions.

Limitations

While the integrated rate law for first-order reactions is a powerful tool, it's important to be aware of its limitations:

• Assumes Elementary Reaction: The integrated rate law is derived assuming that the reaction is an elementary reaction, meaning that it occurs in a single step. If the reaction involves multiple steps, the integrated rate law may not be applicable.
• Constant Temperature: The rate constant k is temperature-dependent. The integrated rate law assumes that the temperature remains constant throughout the reaction. If the temperature changes significantly, the integrated rate law may not accurately describe the reaction kinetics.
• Reversibility: The integrated rate law is derived assuming that the reaction is irreversible, meaning that the reverse reaction is negligible. If the reverse reaction is significant, the integrated rate law may not be applicable.
• Complex Reactions: For complex reactions involving multiple reactants and products, the integrated rate law may not be easily applicable. In these cases, more advanced kinetic models may be required.

Examples of Problems and Solutions

Let's look at a couple of examples of how to use the integrated rate law for first-order reactions to solve problems:

Problem 1:

The decomposition of N2O5 at 328 K follows first-order kinetics. The rate constant k is 1.35 x 10-4 s-1. If the initial concentration of N2O5 is 0.100 M, what will be the concentration of N2O5 after 10 minutes?

Solution:

1. Convert time to seconds: 10 minutes * 60 seconds/minute = 600 seconds
2. Use the integrated rate law: [A] = [A]0 * e^(-kt)
3. Plug in the values: [N2O5] = (0.100 M) * e^(-(1.35 x 10-4 s-1)(600 s))
4. Calculate: [N2O5] = (0.100 M) * e^(-0.081) ≈ (0.100 M) * 0.922 ≈ 0.0922 M

Therefore, the concentration of N2O5 after 10 minutes will be approximately 0.0922 M.

Problem 2:

A radioactive isotope has a half-life of 14.3 days. How long will it take for 80% of the isotope to decay?

Solution:

1. Calculate the rate constant: k = ln(2)/t1/2 = ln(2)/14.3 days ≈ 0.0485 days-1
2. Determine the remaining fraction: If 80% has decayed, 20% remains, so [A]/[A]0 = 0.20
3. Use the integrated rate law (logarithmic form): ln([A]/[A]0) = -kt
4. Plug in the values: ln(0.20) = -(0.0485 days-1) * t
5. Solve for t: t = ln(0.20) / -0.0485 days-1 ≈ -1.609 / -0.0485 days-1 ≈ 33.2 days

Therefore, it will take approximately 33.2 days for 80% of the isotope to decay.

Conclusion

The integrated rate law for first-order reactions is a fundamental concept in chemical kinetics, providing a powerful tool for understanding and predicting the behavior of reactions that depend on the concentration of a single reactant. From determining rate constants and half-lives to predicting reactant concentrations and modeling complex processes in various scientific fields, its applications are widespread and essential. Understanding the derivation, applications, and limitations of the integrated rate law is crucial for anyone working in chemistry, physics, biology, or related fields. By mastering this concept, you can gain a deeper understanding of the dynamic world of chemical reactions.