First Order Reaction Integrated Rate Law

The integrated rate law for a first-order reaction is a cornerstone concept in chemical kinetics, offering a mathematical relationship between reactant concentration and time. It allows us to predict how the concentration of a reactant changes over time, which is essential for understanding and controlling chemical reactions in various fields, from pharmaceuticals to environmental science.

Understanding Reaction Order

Before diving into the integrated rate law, it's crucial to grasp the concept of reaction order. Reaction order describes how the rate of a reaction depends on the concentration of the reactants. It is determined experimentally and cannot be deduced simply from the stoichiometry of the balanced chemical equation.

For a general reaction:

aA + bB → Products

The rate law can be expressed as:

Rate = k[A]^m[B]^n

Where:

• k is the rate constant
• [A] and [B] are the concentrations of reactants A and B
• m and n are the orders of the reaction with respect to reactants A and B, respectively.

The overall order of the reaction is the sum of the individual orders (m + n).

First-Order Reactions: A Detailed Look

A first-order reaction is a reaction whose rate depends linearly on the concentration of only one reactant. This means that if you double the concentration of that reactant, the reaction rate will also double.

Consider the following elementary reaction:

A → Products

The rate law for this first-order reaction is:

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

Where:

• -d[A]/dt represents the rate of disappearance of reactant A with respect to time. The negative sign indicates that the concentration of A is decreasing as the reaction proceeds.
• k is the rate constant, a proportionality constant that reflects the intrinsic speed of the reaction. It is independent of concentration but dependent on temperature.
• [A] is the concentration of reactant A at time t.

Examples of first-order reactions include:

• Radioactive decay: The decay of many radioactive isotopes follows first-order kinetics. For example, the decay of uranium-238 to lead-206.
• Isomerization reactions: Certain isomerization reactions, where a molecule rearranges its structure, can be first-order.
• Decomposition of N₂O₅: The decomposition of dinitrogen pentoxide (N₂O₅) into nitrogen dioxide (NO₂) and oxygen (O₂) in the gas phase.
• Hydrolysis of Aspirin: In certain conditions, the hydrolysis of aspirin follows first-order kinetics.

Derivation of the First-Order Integrated Rate Law

The integrated rate law allows us to determine the concentration of a reactant at any given time during the reaction. It is derived from the differential rate law using calculus. Here's the derivation for a first-order reaction:

Starting with the rate law:

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

1. Separate variables: Rearrange the equation to separate the concentration terms from the time terms:

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

2. Integrate both sides: Integrate both sides of the equation with respect to their respective variables. The limits of integration are from the initial concentration [A]₀ at time t = 0 to the concentration [A] at time t:

   ∫[A]₀[A] d[A]/[A] = ∫0t -k dt

3. Evaluate the integrals: The integral of d[A]/[A] is ln[A], and the integral of -k dt is -kt. Evaluating the integrals at the limits gives:

   ln[A] - ln[A]₀ = -kt

4. Rearrange the equation: Use the properties of logarithms to combine the terms:

   ln([A]/[A]₀) = -kt

5. Solve for [A]: Exponentiate both sides of the equation to solve for [A]:

   [A]/[A]₀ = e^(-kt)

   [A] = [A]₀e^(-kt)

This is the integrated rate law for a first-order reaction. It states that the concentration of reactant A at time t is equal to the initial concentration [A]₀ multiplied by the exponential factor e^(-kt).

Different Forms of the Integrated Rate Law

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

• Exponential Form: [A] = [A]₀e^(-kt) - This form directly relates the concentration of A at time t to its initial concentration and the rate constant.
• Logarithmic Form: ln([A]/[A]₀) = -kt - This form is useful for plotting data to determine the rate constant graphically.
• Linear Form: ln[A] = ln[A]₀ - kt - This form is also useful for graphical determination of the rate constant. If you plot ln[A] versus time, you will obtain a straight line with a slope of -k and a y-intercept of ln[A]₀.

Using the Integrated Rate Law

The integrated rate law is a powerful tool for analyzing first-order reactions. Here's how it can be used:

• Determining the Rate Constant (k): By measuring the concentration of the reactant at different times, you can use the integrated rate law to calculate the rate constant, k. This can be done graphically using the linear form (plotting ln[A] vs. t) or by substituting data points into the logarithmic or exponential form.
• Predicting Concentration at a Given Time: If you know the initial concentration ([A]₀) and the rate constant (k), you can use the integrated rate law to predict the concentration of the reactant ([A]) at any given time (t).
• Determining the Time Required for a Certain Concentration Change: You can rearrange the integrated rate law to solve for the time (t) required for the concentration of the reactant to reach a specific value.
• Determining Half-Life: The half-life (t₁/₂) of a reaction is the time required for the concentration of a reactant to decrease to one-half of its initial concentration. For a first-order reaction, the half-life is constant and independent of the initial concentration.

Half-Life of a First-Order Reaction

The half-life (t₁/₂) is a characteristic property of first-order reactions. To derive the equation for half-life, we set [A] = [A]₀/2 in the integrated rate law:

[A]₀/2 = [A]₀e^(-kt₁/₂)

Divide both sides by [A]₀:

1/2 = e^(-kt₁/₂)

Take the natural logarithm of both sides:

ln(1/2) = -kt₁/₂

Since ln(1/2) = -ln(2):

-ln(2) = -kt₁/₂

Solve for t₁/₂:

t₁/₂ = ln(2)/k ≈ 0.693/k

This equation shows that the half-life of a first-order reaction is inversely proportional to the rate constant k. A faster reaction (larger k) will have a shorter half-life, and a slower reaction (smaller k) will have a longer half-life. Notably, the half-life does not depend on the initial concentration of the reactant. This is a unique characteristic of first-order reactions.

Graphical Representation

The integrated rate law can be visualized graphically, providing a deeper understanding of the reaction kinetics.

• Concentration vs. Time: A plot of [A] vs. t for a first-order reaction will be an exponential decay curve, starting at [A]₀ and approaching zero asymptotically as time increases.
• ln[A] vs. Time: A plot of ln[A] vs. t will be a straight line with a negative slope equal to -k and a y-intercept equal to ln[A]₀. This linear relationship makes it easy to determine the rate constant graphically.

The linear plot of ln[A] versus time is a key identifier of a first-order reaction. If experimental data, when plotted in this manner, yields a straight line, it strongly suggests that the reaction is first order.

Pseudo-First-Order Reactions

Sometimes, a reaction that is not inherently first-order can be made to behave like a first-order reaction by using a large excess of one or more reactants. These are called pseudo-first-order reactions.

For example, consider the reaction:

A + B → Products

If the concentration of B is much larger than the concentration of A ([B] >> [A]), then the change in [B] during the reaction will be negligible. The rate law can then be approximated as:

Rate = k'[A]

Where k' = k[B] (k' is the pseudo-first-order rate constant).

In this case, the reaction appears to be first-order with respect to A, even though the actual rate law may be more complex. This simplification allows us to use the integrated rate law for first-order reactions to analyze the reaction kinetics.

Pseudo-first-order conditions are often used in laboratory experiments to simplify the analysis of complex reactions. By controlling the concentration of one or more reactants, researchers can isolate the effect of a single reactant on the reaction rate.

Factors Affecting the Rate Constant (k)

While the rate constant (k) is independent of concentration, it is highly dependent on temperature. The relationship between the rate constant and temperature is described by the Arrhenius equation:

k = Ae^(-Ea/RT)

Where:

• A is the pre-exponential factor or frequency factor, which represents the frequency of collisions between reactant molecules.
• Ea is the activation energy, the minimum energy required for a reaction to occur.
• R is the ideal gas constant (8.314 J/mol·K).
• T is the absolute temperature in Kelvin.

The Arrhenius equation shows that the rate constant increases exponentially with increasing temperature. This is because at higher temperatures, a greater fraction of molecules have sufficient energy to overcome the activation energy barrier and react.

Taking the natural logarithm of both sides of the Arrhenius equation gives:

ln(k) = ln(A) - Ea/RT

This equation can be used to determine the activation energy experimentally by plotting ln(k) versus 1/T. The slope of the resulting line is -Ea/R, allowing for the calculation of Ea.

Limitations of the Integrated Rate Law

While the integrated rate law is a valuable tool, it is important to be aware of its limitations:

• Applicable only to elementary reactions or reactions with a well-defined rate law: The integrated rate law is derived based on a specific rate law. If the reaction mechanism is complex or the rate law is unknown, the integrated rate law may not be applicable.
• Assumes constant temperature: The rate constant k is temperature-dependent. The integrated rate law assumes that the temperature remains constant throughout the reaction.
• Does not provide information about the reaction mechanism: The integrated rate law only describes the overall kinetics of the reaction. It does not provide information about the individual steps involved in the reaction mechanism.

Examples and Applications

The first-order integrated rate law has numerous applications in various fields:

• Pharmacokinetics: Predicting drug concentrations in the body over time, crucial for determining drug dosage and frequency. Many drugs exhibit first-order elimination kinetics.
• Radioactive Dating: Determining the age of ancient artifacts and geological samples by measuring the decay of radioactive isotopes. Carbon-14 dating is a common example.
• Chemical Engineering: Designing and optimizing chemical reactors, predicting product yields, and controlling reaction rates.
• Environmental Science: Modeling the degradation of pollutants in the environment.
• Food Science: Studying the rates of food spoilage and designing methods to extend shelf life.

Example 1: Radioactive Decay

The half-life of carbon-14 is 5730 years. If a sample initially contains 1.0 g of carbon-14, how much will remain after 10,000 years?

First, calculate the rate constant:

k = ln(2)/t₁/₂ = ln(2)/5730 years = 1.21 x 10⁻⁴ years⁻¹

Then, use the integrated rate law:

[A] = [A]₀e^(-kt) = (1.0 g)e^(-(1.21 x 10⁻⁴ years⁻¹)(10000 years)) = 0.298 g

Therefore, approximately 0.298 g of carbon-14 will remain after 10,000 years.

Example 2: Drug Elimination

A drug has a first-order elimination rate constant of 0.15 h⁻¹. If the initial concentration of the drug in the bloodstream is 2.0 mg/L, how long will it take for the concentration to decrease to 0.5 mg/L?

Use the integrated rate law:

ln([A]/[A]₀) = -kt

ln(0.5 mg/L / 2.0 mg/L) = -(0.15 h⁻¹)t

t = ln(0.25) / (-0.15 h⁻¹) = 9.24 hours

Therefore, it will take approximately 9.24 hours for the drug concentration to decrease to 0.5 mg/L.

FAQ: First-Order Reaction Integrated Rate Law

• Q: How do I know if a reaction is first-order?

  A: Experimentally, you can determine if a reaction is first-order by measuring the concentration of the reactant at different times and plotting ln[A] vs. t. If the plot is a straight line, the reaction is likely first-order. Alternatively, you can calculate the half-life at different initial concentrations. If the half-life is constant and independent of the initial concentration, the reaction is first-order.

• Q: What are the units of the rate constant k for a first-order reaction?

  A: The units of the rate constant k for a first-order reaction are inverse time units, typically s⁻¹, min⁻¹, h⁻¹, or years⁻¹, depending on the time scale of the reaction.

• Q: Can a reaction be zero-order, first-order, and second-order depending on the conditions?

  A: Yes, a reaction can exhibit different orders under different conditions. For example, a reaction may be first-order at low concentrations but become zero-order at high concentrations if the reaction rate becomes limited by the availability of a catalyst or surface area. Also, as discussed earlier, reactions can be pseudo-first order given specific conditions.

• Q: What is the difference between a rate law and an integrated rate law?

  A: The rate law expresses the relationship between the rate of a reaction and the concentrations of the reactants at a particular instant in time. It is a differential equation. The integrated rate law expresses the relationship between the concentration of a reactant and time. It is obtained by integrating the rate law and allows you to calculate the concentration of a reactant at any given time during the reaction.

• Q: Does the integrated rate law apply to reversible reactions?

  A: The simple integrated rate law discussed here applies to irreversible reactions. For reversible reactions, the analysis is more complex and involves considering the equilibrium constant and the rate constants for both the forward and reverse reactions.

Conclusion

The integrated rate law for first-order reactions is a fundamental concept in chemical kinetics that provides a quantitative understanding of how reactant concentrations change over time. By understanding the derivation, different forms, and applications of this law, you can predict reaction rates, determine rate constants, and analyze experimental data. This knowledge is crucial in diverse fields, including chemistry, biology, medicine, and engineering. Mastery of this concept provides a strong foundation for further exploration into more complex reaction kinetics and mechanisms.