Question

Iodine- 131 is used in the diagnosis and treatment of thyroid disease and has a half-life of $8.0$ days. If a patient with thyroid disease consumes a sample of ${\mathrm{Na}}^{131}$ I containing $10.\mu \mathrm{g}{}^{131}\mathrm{I}$, how long will it take for the amount of ${}^{131}\mathrm{I}$ to decrease to $1/100$ of the original amount?

Short answer

It will take approximately 53.3 days for the amount of Iodine-131 to decrease to 1/100 of the original amount.

Step-by-step solution

Identify the known and unknown variables

We are given the following: - Half-life of I-131 (T) = 8.0 days - Initial amount of I-131 (A₀) = 10 μg - Final amount of I-131 (A) is 1/100 of the initial amount = A₀/100 Our task is to find the time t in days.

Set up the decay equation

Use the decay equation to set up the problem: $A=A_0\ast (1/2{)}^{t/T}$ Plug in the known values: $\frac{A_0}{100}=A_0\ast (1/2{)}^{t/8}$

Solve for time t

First, cancel the A₀ terms from both sides of the equation: $\frac{1}{100}=(1/2{)}^{t/8}$ Now, we need to find t from the equation. To do this, we will take the natural logarithm on both sides: $ln(\frac{1}{100})=ln((1/2{)}^{t/8})$ Using the property of logarithms, we can bring the exponent, t/8, in front of the logarithm: $ln(\frac{1}{100})=\frac{t}{8}\ast ln(\frac{1}{2})$ Now it's easier to solve for t. Divide both sides by $ln(\frac{1}{2})$: $\frac{ln(\frac{1}{100})}{ln(\frac{1}{2})}=\frac{t}{8}$ To find the value of t, multiply both sides by 8: $t=8\ast \frac{ln(\frac{1}{100})}{ln(\frac{1}{2})}$ Calculate for t: $t\approx 53.3days$ It will take approximately 53.3 days for the amount of I-131 to decrease to 1/100 of the original amount.

Key concepts

Half-life

Half-life is a key concept when studying radioactive decay. It represents the time it takes for half of a sample of radioactive material to decay. During this time, half of the radioactive atoms transform into a more stable form. For example, if we start with 10 micrograms of a radioactive isotope and its half-life is 8 days, only 5 micrograms will remain after those 8 days.

In real-world terms:

• Half-life helps us understand the speed of decay for a radioactive element.
• Shorter half-lives mean the material decays quickly, while longer half-lives indicate slower decay.

Half-life is not affected by external conditions like temperature or pressure. It remains constant for any given isotope, making it a very reliable measure in scientific calculations.

Iodine-131

Iodine-131 is a radioactive isotope of iodine used in medical applications, especially in treatments and diagnostics of thyroid diseases. It is valued in medical fields because of its ability to destroy excess thyroid tissue effectively.

Key properties of Iodine-131 include:

• It has a half-life of 8 days, which means it becomes roughly half as potent after each 8-day period.
• Its decay emits beta and gamma radiation, which are used to target and treat thyroid ailments.

The relatively short half-life of Iodine-131 makes it suitable for medical applications as it becomes inactive quickly, reducing long-term radiation exposure for patients.

Decay Equation

The decay equation is a mathematical expression used to predict the quantity of a radioactive material that remains after a certain period. A common form of this equation is:$$A=A_0\times {\left(\frac{1}{2}\right)}^{\frac{t}{T}}$$Where:

• $A$ is the amount of substance remaining.
• $A_0$ is the initial amount of substance.
• $t$ is the time elapsed.
• $T$ is the half-life of the substance.

This decay equation indicates how the amount of radioactive substance drops as time progresses. The factor $(1/2{)}^{t/T}$ reflects the repetitive halving of the material in each interval equaling the half-life. It helps scientists and medical professionals calculate how long a sample will remain effective, which is crucial in both research and clinical settings.

Logarithms in Decay Calculations

Logarithms play a crucial role in solving decay equations, especially when we are trying to determine how long it takes for a material to decay to a certain level. By using logarithms, we can solve equations that start in an exponential form.

The natural logarithm helps us manage the equation:$$\ln \left(\frac{1}{100}\right)=\frac{t}{8}\times \ln \left(\frac{1}{2}\right)$$ Using logarithms to solve:

• Helps bring down the exponent, simplifying the equation to find time $t$.
• Translates exponential growth or decay into a linear form for easier calculations.

Logarithms transform complex multiplicative processes into simpler additive ones, making them indispensable in the realm of exponential equations such as radioactive decay.