Problem 22 - Calculus: Graphical, Numerical, Algebraic

Question

Half-Life An isotope of neptunium \((\mathrm{Np}-240)\) has a half-life of 65 minutes. If the decay of \(\mathrm{Np}-240\) is modeled by the differential equation \(d y / d t=-k y,\) where \(t\) is measured in minutes, what is the decay constant \(k\) ?

Step-by-Step Solution

Answer

The decay constant \(k\) of the isotope neptunium-240 is approximately \(0.0107 min^{-1}\).

Step 1: Understand the Relationship Between Half-Life and the Decay Constant

From the theory of first-order reactions, the decay constant \(k\) is related to the half-life \(t_{1/2}\) by the formula \(t_{1/2} = \ln(2) / k\). From this, we can express the decay constant \(k\) in terms of half-life: \(k = \ln(2) / t_{1/2}\). Now, given that the half-life is 65 minutes, we can replace \(t_{1/2}\) with 65 in this equation to determine the value of the decay constant \(k\).

Step 2: Calculate the Decay Constant

The decay constant \(k\) is calculated as follows: \(k = \ln(2) / 65 \approx 0.0107\) \(min^{-1}\). This is the rate of radioactive decay per minute.

Key Concepts

Understanding First-Order Reactions

When studying chemical kinetics, the term 'first-order reaction' refers to a process where the rate at which a reactant is transformed into a product is directly proportional to the concentration of that reactant.

In simpler terms, for every constant time interval, the amount of reactant decreases by a consistent fraction. A common example of a first-order reaction is radioactive decay, where isotopes break down over time. The mathematical representation of this type of reaction is an important differential equation: \( \dfrac{d[Reactant]}{dt} = -k[Reactant] \), where \(k\) is the decay constant and \( [Reactant] \) is the concentration of the reactant.

The decay constant, \(k\), stands as a marker of the reaction's speed. A higher value indicates a faster reaction rate. In educational exercises, this concept allows students to quantify and understand the dynamic progressions in chemical reactions, bridging the gap between theoretical chemical principles and real-world applications.

Radioactive Decay and its Measurement

Radioactive decay is an intriguing natural process by which an unstable atomic nucleus loses energy by emitting radiation. This can happen through various modes, such as alpha decay, beta decay, and gamma emission. In educational settings, understanding this process is critical for disciplines like nuclear physics, chemistry, and environmental science.

The half-life of a radioactive isotope, expressed in the exercise about \(\mathrm{Np}-240\), is the time required for half of the isotopes in a sample to decay. It serves as a key characteristic of each radioactive substance.

Practical Application

Knowledge of half-lives is applied in fields such as radiometric dating, medical diagnostics, and nuclear power generation. Students learning about half-life gain insights into not only the stability of isotopes but also the practical considerations in managing radioactive materials and understanding geologic or archaeological time scales.

Applying Differential Equations in Kinetics

Differential equations play a fundamental role in modeling a wide variety of natural phenomena, including chemical kinetics and nuclear physics. These are mathematical equations that relate a function with its derivatives, indicating how a system evolves over time.

In the context of radioactive decay, a simple differential equation \( \frac{dy}{dt} = -ky \) captures the decay process. The variable \(y\) represents the number of radioactive nuclei present at time \(t\), and the decay constant \(k\) signifies the proportionality between the rate of decay and the number of remaining nuclei.

Why Differential Equations Matter

By solving this equation, one can predict the future behavior of the radioactive sample, a tool of immense value in safety planning for radiation exposure and waste management. For students, grasping differential equations opens doors to understanding complex systems beyond the classroom, such as ecosystems, financial markets, and weather patterns.