Deriving the Integrated Rate Law for a First-Order Reaction

📚 Deriving the Integrated Rate Law for a First-Order Reaction

The integrated rate law expresses the concentration of a reactant as a function of time. This is incredibly useful for predicting the amount of reactant remaining after a certain period or determining the time required for a reaction to reach a specific stage. For first-order reactions, the derivation is relatively straightforward and relies on basic calculus. Let's dive in!

📜 Background

Before we start, let's recall some key concepts:

• ⚛️ Rate Law: The rate law expresses the rate of a reaction in terms of the concentrations of reactants. For a first-order reaction, the rate law is given by: $rate = -\frac{d[A]}{dt} = k[A]$, where [A] is the concentration of reactant A, t is time, and k is the rate constant.
• 🧪 Differential Rate Law: The rate law we just described is also known as the differential rate law.
• 📈 Integrated Rate Law: The integrated rate law tells us how the concentration of a reactant changes *over time*.

⚗️ Derivation Steps

Here's a step-by-step guide to deriving the integrated rate law:

1. ➗ Step 1: Separate Variables: Start with the differential rate law: $\frac{d[A]}{dt} = -k[A]$. Rearrange the equation to separate the variables [A] and t: $\frac{d[A]}{[A]} = -k dt$.
2. ∫ Step 2: Integrate Both Sides: Integrate both sides of the equation with respect to their respective variables. The limits of integration for [A] are from $[A]_0$ (the initial concentration of A) to $[A]_t$ (the concentration of A at time t). The limits of integration for t are from 0 to t. So, we have: $\int_{[A]_0}^{[A]_t} \frac{d[A]}{[A]} = \int_0^t -k dt$.
3. 📏 Step 3: Evaluate the Integrals: The integral of $\frac{d[A]}{[A]}$ is $ln[A]$, and the integral of $-k dt$ is $-kt$. Evaluating these integrals with the limits gives: $ln[A]_t - ln[A]_0 = -kt$.
4. 💡 Step 4: Rearrange the Equation: Use the properties of logarithms to simplify the equation: $ln(\frac{[A]_t}{[A]_0}) = -kt$.
5. 📈 Step 5: Solve for [A]t: Exponentiate both sides of the equation to remove the natural logarithm: $\frac{[A]_t}{[A]_0} = e^{-kt}$. Finally, solve for $[A]_t$: $[A]_t = [A]_0 e^{-kt}$.

✅ The Integrated Rate Law

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

$[A]_t = [A]_0 e^{-kt}$

Where:

• ⏱️ $[A]_t$ is the concentration of reactant A at time t
• 🚀 $[A]_0$ is the initial concentration of reactant A
• 🔑 $k$ is the rate constant
• ⌛ $t$ is the time
• ℯ is the base of the natural logarithm (approximately 2.71828)

➗ Half-Life

A useful concept related to first-order reactions is half-life ($t_{1/2}$), which is the time required for the concentration of the reactant to decrease to one-half of its initial value. For a first-order reaction, the half-life is constant and can be derived from the integrated rate law:

1. 🎯 Let $[A]_t = \frac{1}{2}[A]_0$ at $t = t_{1/2}$.
2. ✍️ Substituting into the integrated rate law: $\frac{1}{2}[A]_0 = [A]_0 e^{-kt_{1/2}}$.
3. ➗ Divide both sides by $[A]_0$: $\frac{1}{2} = e^{-kt_{1/2}}$.
4. ln Take the natural logarithm of both sides: $ln(\frac{1}{2}) = -kt_{1/2}$.
5. 🔑 Solve for $t_{1/2}$: $t_{1/2} = \frac{ln(2)}{k} \approx \frac{0.693}{k}$.

☢️ Real-World Examples

• 📅 Radioactive Decay: Many radioactive isotopes decay via first-order kinetics. This is used in carbon dating to determine the age of ancient artifacts.
• 💊 Drug Metabolism: The elimination of many drugs from the body follows first-order kinetics. This is crucial for determining appropriate dosages and dosing intervals.
• 🔥 Chemical Reactions in Solution: Certain chemical reactions in solution, such as the decomposition of dinitrogen pentoxide ($N_2O_5$), follow first-order kinetics.

💡 Conclusion

Deriving and understanding the integrated rate law for first-order reactions is fundamental in chemistry. It allows us to predict reactant concentrations over time and determine reaction half-lives. With the integrated rate law, we can analyze a wide array of chemical phenomena, from radioactive decay to drug metabolism. Keep practicing, and you'll master it! 🎉