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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?

Step by step solution

01

Derive the Half-life Formula

The rate of decay \(r_d\) is given by the relation \( r_d=kC\), or \(\frac{dC}{dt}=-kC\). This is a first order differential equation. On separating the variables and integrating, we get \(\int_{C_0}^{C}\frac{dC'}{C'}=-k\int_{0}^{t}dt'\). Solving the integrals we have \( \ln{C} - \ln{C_0} = -kt\). Dividing through by -1, \(\ln\left(\frac{C_0}{C}\right) = kt\). For half-life \(C = \frac{C_0}{2}\), therefore, \(\ln2 = kt_{1/2}\). Hence, \(t_{1/2} = \frac{ln2}{k}\).

02

Calculate the Time Required

The given half-life is \(2.6h\). We know that \(t_{1/2} = \frac{\ln2}{k}\), from which we get \(k = \frac{\ln2}{2.6h}\). Now we need to find the time \(t\) when the isotope decays to 1% of its current concentration. Substituting \(C=\frac{C_0}{100}\) in the equation \(\ln\left(\frac{C_0}{C}\right) = kt\), we get \(t = \frac{\ln100}{k}\). Substituting \(k\) from the previous step, \(t = \frac{\ln100}{\ln2 / 2.6h}\), we get the time required.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-life

Understanding half-life is crucial in the study of radioactive decay. It refers to the time required for half of a radioactive isotope's quantity to decay. This measure is important because it helps predict how long a substance will remain active and hazardous.
In our context, the half-life is calculated using the formula:\[t_{1/2} = \frac{\ln 2}{k}\]Here, \(k\) is the decay constant, a value that indicates the likelihood of decay per unit time. The formula shows that the half-life is inversely proportional to \(k\), meaning the faster the decay (larger \(k\)), the shorter the half-life.
Understanding half-life allows scientists to gauge how long radioactive substances will remain dangerous and helps in planning safe handling and disposal. It also has a big role in radiotracing, archaeological dating, and nuclear medicine.

Differential Equations

Differential equations often describe how quantities change over time, and they play a vital role in modeling the decay process of radioactive isotopes. The decay rate given by:\[\frac{dC}{dt} = -kC\]This is a first-order differential equation, indicating that the rate of change of the concentration depends directly on the concentration itself. "Separating variables," we can integrate both sides to solve for the concentration function over time.
The general solution to this differential equation gives:\[\ln \frac{C_0}{C} = kt\]This equation allows calculation of the concentration of isotopes at any given time \(t\). Understanding and solving differential equations provide insights into the behavior and prediction of isotopic concentration over time, a crucial step for applications like radiotracing and determining safe levels of exposure.

Isotope Concentration

Isotope concentration levels indicate the amount of a radioactive substance present in a solution. It's crucial to manage due to the potential hazards of radioactivity. The concentration \(C\) decreases over time as the isotope decays, and this decrease follows a logarithmic pattern as described by the equation:\[\frac{dC}{dt} = -kC\]Understanding concentration is important for numerous reasons:

• Ensuring safety in handling radioactive materials, as exposure must be minimized.
• Accurate measurement helps in scientific research, including radiotracing techniques in medicine and environmental studies.
• Managing waste and disposal, as the concentration needs to be reduced to safe levels prior to disposal.

Therefore, being aware of how concentration levels change over time can greatly aid in effective and safe management of radioactive substances. Calculating when the isotope concentration reduces to a safe level such as 1% of its original amount is vital in various industrial and research settings.