A radioactive isotope decays at a rate proportional to its...

A radioactive isotope decays at a rate proportional to its concentration. If the concentration of an isotope is $C(\mathrm{mg} / \mathrm{L}),$ then its rate of decay may be expressed as $$r_{\mathrm{d}}[\mathrm{mg} /(\mathrm{L} \cdot \mathrm{s})]=k C$$ where $k$ is a constant. (a) A volume $V(\mathrm{L})$ of a solution of a radioisotope whose concentration is $C_{0}(\mathrm{mg} / \mathrm{L})$ is placed in a closed vessel. Write a balance on the isotope in the vessel and integrate it to prove that the half-life $t_{1 / 2}$ of the isotope $-$ by definition, the time required for the isotope concentration to decrease to half of its initial value- equals ( $\ln 2$ )/ $k$. (b) The half-life of $^{56} \mathrm{Mn}$ is $2.6 \mathrm{h}$. A batch of this isotope that was used in a radiotracing experiment has been collected in a holding tank. The radiation safety officer declares that the activity (which is proportional to the isotope concentration) must decay to $1 \%$ of its present value before the solution can be discarded. How long will this take?

A radioactive isotope decays at a rate proportional to its concentration. If the concentration of an isotope is $C(\mathrm{mg} / \mathrm{L}),$ then its rate of decay may be expressed as $$r_{\mathrm{d}}[\mathrm{mg} /(\mathrm{L} \cdot \mathrm{s})]=k C$$ where $k$ is a constant.
(a) A volume $V(\mathrm{L})$ of a solution of a radioisotope whose concentration is $C_{0}(\mathrm{mg} / \mathrm{L})$ is placed in a closed vessel. Write a balance on the isotope in the vessel and integrate it to prove that the half-life
$t_{1 / 2}$ of the isotope $-$ by definition, the time required for the isotope concentration to decrease to half of its initial value- equals ( $\ln 2$ )/ $k$.
(b) The half-life of $^{56} \mathrm{Mn}$ is $2.6 \mathrm{h}$. A batch of this isotope that was used in a radiotracing experiment has been collected in a holding tank. The radiation safety officer declares that the activity (which
is proportional to the isotope concentration) must decay to $1 \%$ of its present value before the solution can be discarded. How long will this take?

Key Concepts

Exponential Decay

Exponential decay describes a process in which a quantity decreases at a rate that is directly proportional to its current value. This behavior is mathematically modeled by a differential equation, and its solution reveals that the quantity decreases exponentially over time. Such models are fundamental in various fields, including physics, chemistry, biology, and finance, and they are particularly important in describing decay processes in radioactive substances.

First-Order Differential Equations

First-order differential equations are equations involving the first derivative of a function and the function itself. They are commonplace in modeling natural phenomena where the rate of change of a quantity is proportional to the quantity itself. In the context of decay processes, these equations lead to simple exponential solutions and are solved using methods such as separation of variables and integration.

Half-Life

The concept of half-life defines the time required for a quantity undergoing exponential decay to reduce to half its original value. This measure is especially useful in contexts such as radioactivity, where it provides a convenient means of comparing decay rates independent of the initial quantity. The half-life is directly related to the decay constant through a logarithmic relationship, highlighting the interplay between exponential functions and logarithms in decay processes.

Mass Balance Principle

The mass balance principle is a fundamental concept in engineering and the physical sciences, stating that mass in a closed system is conserved over time. When applied to decaying systems, it leads to a differential equation that accounts for changes in mass (or concentration) over time based on input, output, and reaction rates. This principle ensures that all quantities are tracked accurately, forming the foundation upon which decay models are built.

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