First-Order Reactions and Radioactive Decay: The Connection

📚 Understanding First-Order Reactions

In chemistry, a first-order reaction is a reaction whose rate depends on the concentration of only one reactant. This means the rate of the reaction is directly proportional to the amount of that reactant present. Mathematically, this is expressed as:

$\text{Rate} = k[A]$

Where:

• 🔍 $k$ is the rate constant, a value specific to the reaction and temperature.
• ⏱️ $[A]$ is the concentration of the reactant.

This leads to an integrated rate law that describes how the concentration of the reactant changes over time:

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

Where:

• 📈 $[A]_t$ is the concentration of reactant A at time t.
• 🌡️ $[A]_0$ is the initial concentration of reactant A.
• ⏰ $t$ is the time elapsed.
• ℯ is Euler's number (approximately 2.71828).

☢️ Radioactive Decay: A First-Order Process

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This is a spontaneous process, meaning it happens without any external influence. The crucial connection is that radioactive decay *always* follows first-order kinetics.

The rate of decay is proportional to the number of radioactive nuclei present. This is expressed similarly to first-order reactions:

$\text{Rate of decay} = \lambda N$

Where:

• ⚛️ $N$ is the number of radioactive nuclei.
• ⚠️ $\lambda$ is the decay constant, representing the probability of decay per nucleus per unit time.

The integrated rate law for radioactive decay is:

$N_t = N_0 e^{-\lambda t}$

Where:

• 📊 $N_t$ is the number of radioactive nuclei at time t.
• 🔬 $N_0$ is the initial number of radioactive nuclei.

⏳ Half-Life: A Key Parameter

A vital concept for both first-order reactions and radioactive decay is the half-life ($t_{1/2}$). It's the time required for the concentration of the reactant (in a first-order reaction) or the number of radioactive nuclei to reduce to half of its initial value.

For first-order reactions:

$t_{1/2} = \frac{0.693}{k}$

For radioactive decay:

$t_{1/2} = \frac{0.693}{\lambda}$

• 💡 Notice that the half-life is independent of the initial concentration or number of nuclei, a key characteristic of first-order processes.

⚗️ Real-World Examples

Radioactive Decay:

• 📅 Carbon-14 dating: Archaeologists use the decay of carbon-14 (a radioactive isotope of carbon) to determine the age of ancient organic materials.
• 🩺 Medical imaging: Radioactive isotopes like Technetium-99m are used in medical imaging to diagnose various conditions.
• ⚡ Nuclear Power: Radioactive decay of uranium and plutonium provides the heat necessary for nuclear power generation.

First-Order Reactions:

• 💊 Drug degradation: The breakdown of many medications follows first-order kinetics, which is crucial for determining shelf life and dosage.
• 🔥 Thermal decomposition of certain compounds: Some compounds break down into simpler substances at a rate proportional to their concentration.

🧪 Practice Quiz

1. A radioactive isotope has a half-life of 10 years. What fraction of the initial amount will remain after 30 years?
2. The rate constant for a first-order reaction is $0.05 \text{ s}^{-1}$. What is the half-life of the reaction?
3. If the initial concentration of a reactant in a first-order reaction is 1.0 M and after 60 seconds the concentration is 0.25 M, what is the rate constant?
4. Technetium-99m ($^{99m}\text{Tc}$) is a radioactive isotope used in medical imaging with a half-life of approximately 6 hours. If a patient is injected with a dose containing 10 mCi of $^{99m}\text{Tc}$, how much activity will remain after 24 hours?
5. A first-order reaction has a half-life of 45 minutes. How long will it take for the reactant concentration to decrease to 1/8 of its original value?
6. The decay constant for a radioactive isotope is $1.5 \times 10^{-10} \text{ s}^{-1}$. Calculate its half-life in years.
7. Compound A decomposes in a first-order reaction. The concentration of A decreases from 0.50 M to 0.125 M in 40 minutes. Calculate the rate constant for the reaction.

🔑 Conclusion

Understanding the link between first-order reactions and radioactive decay is fundamental in chemistry and physics. Both follow the same mathematical principles, with the rate of change being proportional to the amount of the substance present. Grasping these concepts allows you to solve problems related to reaction kinetics, radioactive dating, and much more! 🚀