Question

It \(\mathrm{N}_{0}\) is the original mass of the substance of halt life 5 years, the amount of substance left after 15 years is (A) \(\left(\mathrm{N}_{0} / 16\right)\) (B) \(\left(\mathrm{N}_{0} / 8\right)\) (C) \(\left(\mathrm{N}_{0} / 4\right)\) (D) \(\left(\mathrm{N}_{0} / 2\right)\)

Short answer

The amount of substance left after 15 years is \(\frac{1}{8}\) of its original mass, which corresponds to answer choice (B).

Step-by-step solution

Understand the Half-Life formula

\ To solve this problem, we need to use the half-life formula, which is: \[N(t) = N_0 * (1/2)^{t/T}\] Where: - \(N(t)\) = remaining mass of the substance at time \(t\) - \(N_0\) = original mass of the substance - \(t\) = time in years - \(T\) = half-life of the substance in years In this exercise, \(T = 5\) years and \(t = 15\) years.

Substitute the given values into the formula

\ Now, we can substitute the given values for \(T\) and \(t\) into the half-life formula: \[N(15) = N_0 * (1/2)^{15/5}\]

Simplify the expression

\ Simplify the exponent 15/5: \[N(15) = N_0 * (1/2)^3\] Now raise the fraction to the power of 3: \[N(15) = N_0 * \frac{1}{8}\] We now have the expression for the remaining mass of the substance after 15 years as a fraction of the original mass.

Compare the result with the answer choices

\ Our result, \(N(15) = N_0 * \frac{1}{8}\), matches answer choice (B): (B) \(\left(N_0 / 8\right)\) Thus, after 15 years, the remaining mass of the substance is \(\frac{1}{8}\) of its original mass.

Key concepts

Radioactive Decay

When a substance undergoes radioactive decay, it transforms into a different element or isotope over time. This transformation is not instantaneous; instead, it happens progressively, with the substance decreasing in mass as atoms emit particles or radiation. This process is natural for unstable isotopes, which seek stability by losing energy.

An essential characteristic of radioactive decay is that it is random at the level of single atoms. However, for large numbers of atoms, it follows a predictable pattern described by the half-life of the substance.

• Radioactive decay is spontaneous, meaning it happens without external influence.
• Each substance has a unique decay rate, expressed in terms of its half-life.
• Common decay particles include alpha particles, beta particles, and gamma rays.

The concept of half-life is crucial in understanding how quickly a substance decays and how much of it remains after a certain period.

Exponential Decay

Exponential decay is a type of decay where the amount of substance decreases by a constant percentage over equal intervals of time. Unlike linear decay, where the decrease is by the same amount each period, exponential decay is characterized by the decay rate being proportional to the quantity that remains.

This mathematical concept is represented by the equation:
\[N(t) = N_0 \cdot e^{-\lambda t}\]
Where:

• \(N(t)\) is the amount remaining at time \(t\).
• \(N_0\) is the initial quantity of the substance.
• \(\lambda\) is the decay constant related to the half-life of the material.
• \(e\) is the base of the natural logarithm, approximately equal to 2.718.

For half-life calculations, we typically simplify using base \(1/2\) rather than \(e\).

By knowing the exponential decay formula, students can determine how long it takes for a quantity to reduce by half and predict how much will remain after several half-lives.

Nuclear Physics

Nuclear physics is the field of study that explores the components and behavior of atomic nuclei. It uncovers the mysterious processes that occur within atoms, particularly during radioactive decay and nuclear reactions.

In the context of half-life and decay, nuclear physics explains how and why nuclei release energy and change form. This has profound implications and uses, ranging from energy production in nuclear plants to medical imaging techniques using radioactive isotopes.

• Nuclear physics helps us understand the forces that hold nuclei together, known as nuclear forces.
• It also explains fundamental concepts like binding energy, which affects the stability of nuclei.
• With nuclear reactions, we investigate how reactions can release or absorb large amounts of energy, such as in nuclear fission and fusion.

Ultimately, nuclear physics provides the basis for technologies that reshape modern-day society and offer insights into the fundamental workings of the universe.