Integrated Rate Law For 0 Order Reaction

The integrated rate law for a zero-order reaction describes how the concentration of a reactant changes over time in a chemical reaction where the rate is independent of the reactant's concentration. Understanding this law is crucial for predicting reaction progress and determining reaction mechanisms.

Unveiling the Zero-Order Reaction

A zero-order reaction is a chemical reaction in which the rate of reaction is independent of the concentration of the reactant(s). This means that the reaction proceeds at a constant rate, regardless of how much reactant is present. This might seem counterintuitive, as we often think of reactions speeding up with higher concentrations of reactants. However, zero-order reactions are common in situations where the reaction rate is limited by a factor other than reactant concentration, such as surface area of a catalyst or intensity of light.

Characteristics of Zero-Order Reactions

• Rate Independence: The defining characteristic is that the reaction rate doesn't change with reactant concentration.
• Constant Rate: The reaction proceeds at a steady pace until the reactant is completely consumed.
• Rate Law: The rate law for a zero-order reaction is expressed as: rate = k, where k is the rate constant.

Examples of Zero-Order Reactions

• Catalytic Reactions: Reactions occurring on the surface of a catalyst can be zero-order if the catalyst surface is saturated with reactant. For example, the decomposition of ammonia ($NH_3$) on a hot tungsten ($W$) surface.
• Photochemical Reactions: Reactions initiated by light, where the rate depends on the intensity of light rather than the concentration of the reactant. An example is the photochemical decomposition of hydrogen iodide ($HI$).
• Enzyme-Catalyzed Reactions: When an enzyme is saturated with substrate, the reaction becomes zero-order with respect to the substrate concentration.
• Combustion of Alcohol: The combustion of alcohol ($C_2H_5OH$) on a hot platinum ($Pt$) surface is a zero-order reaction.

Deriving the Integrated Rate Law for a Zero-Order Reaction

The integrated rate law provides a mathematical relationship between the concentration of reactants and time. For a zero-order reaction, the derivation is relatively straightforward.

Step-by-Step Derivation

Let's consider a simple zero-order reaction:

$A \rightarrow Products$

1. Differential Rate Law: The differential rate law expresses the rate of the reaction in terms of the change in concentration of reactant A with respect to time (t):

   $- \frac{d[A]}{dt} = k$

   Where:

   • $[A]$ is the concentration of reactant A at time t.
   • $k$ is the rate constant.
   • The negative sign indicates that the concentration of A is decreasing as the reaction proceeds.

2. Separation of Variables: Rearrange the equation to separate the variables $[A]$ and $t$:

   $d[A] = -k dt$

3. Integration: Integrate both sides of the equation. We'll integrate from the initial concentration $[A]_0$ at time $t = 0$ to the concentration $[A]_t$ at time $t$:

   $\int_{[A]_0}^{[A]t} d[A] = -k \int{0}^{t} dt$

4. Evaluating the Integrals: The integrals are simple:

   $[A]_t - [A]_0 = -kt$

5. Integrated Rate Law: Rearrange the equation to solve for $[A]_t$:

   $[A]_t = [A]_0 - kt$

The Final Integrated Rate Law

The integrated rate law for a zero-order reaction is:

$[A]_t = [A]_0 - kt$

Where:

• $[A]_t$ is the concentration of reactant A at time $t$.
• $[A]_0$ is the initial concentration of reactant A at time $t = 0$.
• $k$ is the rate constant.
• $t$ is the time elapsed.

Graphical Representation

The integrated rate law can be visually represented by plotting the concentration of the reactant $[A]$ against time $t$.

Linear Relationship

The equation $[A]_t = [A]_0 - kt$ has the form of a linear equation: $y = mx + b$

Where:

• $y = [A]_t$ (concentration at time t)
• $x = t$ (time)
• $m = -k$ (slope, which is the negative of the rate constant)
• $b = [A]_0$ (y-intercept, which is the initial concentration)

Graph

When you plot $[A]_t$ on the y-axis and $t$ on the x-axis, you will obtain a straight line with:

• Slope: $-k$ (negative rate constant)
• Y-intercept: $[A]_0$ (initial concentration)

This linear relationship is a key characteristic of zero-order reactions and can be used to identify them experimentally. If a plot of concentration versus time yields a straight line, the reaction is likely zero-order.

Half-Life of a Zero-Order Reaction

The half-life ($t_{1/2}$) of a reaction is the time required for the concentration of a reactant to decrease to one-half of its initial concentration. The half-life is a useful parameter for characterizing the rate of a reaction.

Deriving the Half-Life Equation

For a zero-order reaction, we can derive the half-life equation as follows:

1. Definition of Half-Life: At $t = t_{1/2}$, $[A]_t = \frac{1}{2}[A]_0$

2. Substitute into Integrated Rate Law: Substitute these values into the integrated rate law:

   $\frac{1}{2}[A]_0 = [A]0 - kt{1/2}$

3. Solve for $t_{1/2}$: Rearrange the equation to solve for $t_{1/2}$:

   $kt_{1/2} = [A]_0 - \frac{1}{2}[A]_0$

   $kt_{1/2} = \frac{1}{2}[A]_0$

   $t_{1/2} = \frac{[A]_0}{2k}$

The Half-Life Equation

The half-life of a zero-order reaction is:

$t_{1/2} = \frac{[A]_0}{2k}$

Key Observation

Notice that the half-life of a zero-order reaction is directly proportional to the initial concentration $[A]_0$. This means that as the initial concentration increases, the half-life also increases. This is a unique characteristic of zero-order reactions that distinguishes them from first-order and second-order reactions.

Applications and Implications

Understanding the integrated rate law and half-life for zero-order reactions has significant implications in various fields.

Chemical Kinetics

• Reaction Mechanism Determination: By analyzing the concentration-time data and determining the order of a reaction, chemists can propose and validate reaction mechanisms.
• Rate Constant Determination: The integrated rate law allows for the experimental determination of the rate constant $k$ by measuring the change in concentration over time.
• Predicting Reaction Progress: The integrated rate law can be used to predict the concentration of reactants and products at any given time during the reaction.

Pharmaceutical Science

• Drug Degradation: Many drug degradation processes follow zero-order kinetics, especially in solid dosage forms. Understanding the rate of degradation is crucial for determining the shelf life of a drug.
• Drug Release: The release of a drug from a transdermal patch or a controlled-release tablet can sometimes follow zero-order kinetics, ensuring a constant drug delivery rate.

Environmental Science

• Pollutant Degradation: The degradation of certain pollutants in the environment can be zero-order under specific conditions, such as when the degradation rate is limited by the availability of a catalyst.
• Atmospheric Chemistry: Some atmospheric reactions, particularly those involving photochemical processes, can exhibit zero-order kinetics.

Industrial Chemistry

• Catalysis: Many industrial processes rely on catalytic reactions. If the catalyst surface is saturated, the reaction can be zero-order, simplifying the reactor design and optimization.
• Polymerization: Some polymerization reactions can exhibit zero-order kinetics, particularly when the initiation step is rate-limiting.

Examples and Practice Problems

To solidify your understanding, let's work through some examples and practice problems.

Example 1: Decomposition of a Drug

A drug degrades by zero-order kinetics. The initial concentration of the drug is 5.0 mg/mL, and the rate constant for the degradation is 0.2 mg/mL/day.

a) What is the concentration of the drug after 10 days?

b) What is the half-life of the drug?

Solution:

a) Using the integrated rate law: $[A]_t = [A]_0 - kt$

$[A]_{10} = 5.0 \, \text{mg/mL} - (0.2 \, \text{mg/mL/day})(10 \, \text{days})$

$[A]_{10} = 5.0 \, \text{mg/mL} - 2.0 \, \text{mg/mL}$

$[A]_{10} = 3.0 \, \text{mg/mL}$

Therefore, the concentration of the drug after 10 days is 3.0 mg/mL.

b) Using the half-life equation: $t_{1/2} = \frac{[A]_0}{2k}$

$t_{1/2} = \frac{5.0 \, \text{mg/mL}}{2(0.2 \, \text{mg/mL/day})}$

$t_{1/2} = \frac{5.0}{0.4} \, \text{days}$

$t_{1/2} = 12.5 \, \text{days}$

Therefore, the half-life of the drug is 12.5 days.

Example 2: Catalytic Reaction

A reaction occurs on a catalyst surface and follows zero-order kinetics. The rate constant is 0.05 mol/L/s. If the initial concentration of the reactant is 2.0 mol/L, how long will it take for the reactant concentration to decrease to 0.5 mol/L?

Solution:

Using the integrated rate law: $[A]_t = [A]_0 - kt$

We want to find $t$ when $[A]_t = 0.5 , \text{mol/L}$, $[A]_0 = 2.0 , \text{mol/L}$, and $k = 0.05 , \text{mol/L/s}$.

Rearrange the equation to solve for $t$:

$t = \frac{[A]_0 - [A]_t}{k}$

$t = \frac{2.0 , \text{mol/L} - 0.5 , \text{mol/L}}{0.05 , \text{mol/L/s}}$

$t = \frac{1.5}{0.05} , \text{s}$

$t = 30 , \text{s}$

Therefore, it will take 30 seconds for the reactant concentration to decrease to 0.5 mol/L.

Practice Problems

1. A substance decomposes with a rate constant of 0.015 M/min. If the initial concentration is 1.20 M, how long will it take for the concentration to decrease to 0.45 M? Assume zero-order kinetics.
2. The half-life for the zero-order reaction $A \rightarrow Products$ is 50 minutes when the initial concentration of A is 0.20 M. What is the rate constant for this reaction?
3. A photochemical reaction has a rate constant of 5.0 x 10^-3 M/s. If the initial concentration of the reactant is 0.50 M, what will the concentration be after 2 minutes?

Contrasting with Other Reaction Orders

Understanding zero-order reactions is enhanced by comparing them to first-order and second-order reactions.

First-Order Reactions

• Rate Law: Rate = k[A]
• Integrated Rate Law: ln[A]t = ln[A]0 - kt
• Half-Life: t1/2 = 0.693/k (independent of initial concentration)
• Concentration vs. Time Plot: Exponential decay

Key Difference: The rate of a first-order reaction is directly proportional to the concentration of the reactant. The half-life is constant and independent of the initial concentration.

Second-Order Reactions

• Rate Law: Rate = k[A]^2 or Rate = k[A][B]
• Integrated Rate Law: 1/[A]t = 1/[A]0 + kt (for Rate = k[A]^2)
• Half-Life: t1/2 = 1/(k[A]0) (inversely proportional to initial concentration)
• Concentration vs. Time Plot: Non-linear, slower decrease than first-order

Key Difference: The rate of a second-order reaction is proportional to the square of the concentration of one reactant or the product of the concentrations of two reactants. The half-life is inversely proportional to the initial concentration.

Summary Table

Limitations and Considerations

While the integrated rate law for zero-order reactions provides a valuable tool for understanding reaction kinetics, it's important to be aware of its limitations.

Ideal Conditions

The integrated rate law is derived under ideal conditions, such as constant temperature and pressure. Deviations from these conditions can affect the reaction rate and invalidate the predictions of the integrated rate law.

Reaction Complexity

Many real-world reactions involve multiple steps and complex mechanisms. The zero-order kinetics may only apply under specific conditions or over a limited range of concentrations.

Catalyst Saturation

In catalytic reactions, the zero-order kinetics are observed only when the catalyst surface is saturated with the reactant. If the reactant concentration is low, the reaction may follow first-order kinetics instead.

Experimental Error

Experimental errors in measuring concentrations and time can affect the accuracy of the rate constant and half-life values determined from the integrated rate law.

Conclusion

The integrated rate law for zero-order reactions is a fundamental concept in chemical kinetics. It describes the relationship between reactant concentration and time for reactions where the rate is independent of concentration. By understanding the derivation, graphical representation, half-life, and applications of this law, you can gain valuable insights into the behavior of chemical reactions and their applications in various fields. Remember to consider the limitations and ideal conditions when applying this law to real-world scenarios.