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 \(\mathrm{Na}^{131} \mathrm{I}\) 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 \(26.57\) days for the amount of Iodine-131 to decrease to 1/100 of its original amount.

Step-by-step solution

Find the decay constant \(\lambda\)

We can find the decay constant \(\lambda\) using the following formula: $$T_{1/2} = \frac{ln(2)}{\lambda}$$ Where: - \(T_{1/2}\) is the half-life of the radioactive substance - \(\lambda\) is the decay constant We are given \(T_{1/2} = 8.0\) days. Plug this value into the formula to find \(\lambda\): $$\lambda = \frac{ln(2)}{T_{1/2}} = \frac{ln(2)}{8.0}$$

Set up the decay equation

Now that we have the decay constant \(\lambda\), we can set up the decay equation: \(N(t) = N_0 e^{-\lambda t}\) We are given the initial amount of Iodine-131 consumed by the patient: \(N_0 = 10 \mu g\). We are also given that the amount of Iodine-131 decreases to 1/100 of its initial amount. Therefore, \(N(t) = \frac{N_0}{100}\). Substitute these values into the decay equation: $$\frac{N_0}{100} = N_0 e^{-\lambda t}$$

Solve for the time \(t\)

Now, we need to solve this equation for \(t\). First, divide both sides by \(N_0\): $$\frac{1}{100} = e^{-\lambda t}$$ Next, take the natural logarithm of both sides: $$ln\left(\frac{1}{100}\right) = -\lambda t$$ Finally, divide by \(-\lambda\): $$t = \frac{ln\left(\frac{1}{100}\right)}{-\lambda}$$ Substitute the value of \(\lambda\) that we found earlier: $$t = \frac{ln\left(\frac{1}{100}\right)}{-\frac{ln(2)}{8.0}}$$

Calculate the time \(t\)

Now, calculate the time \(t\) using the expression we derived: $$t = \frac{ln\left(\frac{1}{100}\right)}{-\frac{ln(2)}{8.0}} \approx 26.57 \; \text{days}$$ So, it will take approximately 26.57 days for the amount of Iodine-131 to decrease to 1/100 of its original amount.

Key concepts

Half-Life Calculation

The half-life of a substance is a key concept in understanding radioactive decay. It represents the time it takes for half of the radioactive isotopes in a sample to decay, reducing their quantity by half. In practical terms, if you start with 10 micrograms of a substance, a single half-life is the time it will take for only 5 micrograms to remain.
For Iodine-131, the half-life is given as 8.0 days. This means every 8 days, the amount of Iodine-131 will reduce by half. To find the decay constant (a crucial part in predicting decay over time), we use the known relation between half-life and decay constant: \[ T_{1/2} = \frac{ln(2)}{\lambda} \]where \( \lambda \) is the decay constant and \( T_{1/2} \) is the half-life. Plug in the half-life to find \( \lambda \).

Decay Constant

The decay constant, represented as \( \lambda \), is pivotal when discussing how quickly a radioactive isotope decays over time. It's derived from the relationship with half-life discussed earlier: \[ \lambda = \frac{ln(2)}{T_{1/2}} \]For Iodine-131 with a half-life of 8.0 days, calculate the decay constant:- Substitute 8.0 days into the formula: \( \lambda = \frac{ln(2)}{8.0} \).The decay constant offers insight into the rate of decay, allowing us to not only compute the remaining quantity at any given time but also reliably predict future quantities less than the initial amount due to decay.
The larger the decay constant, the quicker the decay process, indicating a faster decrease in the radioactive isotopes within a given timeframe.

Radioactive Isotopes

Radioactive isotopes, or radioisotopes, are variants of chemical elements that possess unstable nuclei and emit radiation during decay, transforming into isotopes of different elements. Such radioactive behavior is a natural phenomenon in numerous elements, including Iodine-131, commonly used in medicine. Key aspects of radioactive isotopes include: - **Instability**: Their nuclei tend to be unstable, which results in decay over time and the emission of particles. - **Applications**: Utilized extensively in fields like medicine (for diagnostic imaging or treatment), archaeology (carbon dating), and energy (nuclear power). - **Decay Types**: Can undergo alpha, beta, and gamma decay, each involving the release of specific types of particles or energy. Understanding these isotopes is essential for applications requiring precise management of the decay process, like in medical treatments where accurate dosage over time is crucial.

Exponential Decay Equation

The exponential decay equation is fundamental in predicting how much of a radioactive isotope remains after a certain time period. The equation:\[ N(t) = N_0 e^{-\lambda t} \]Where:- \( N(t) \) is the amount remaining at time \( t \).- \( N_0 \) is the original amount.- \( \lambda \) is the decay constant.- \( t \) is time.This formula captures the essence of how substances undergo exponential decay, showing a decreasing curve over time. For practical application, it’s used to determine how long it will take for a certain quantity of a substance to decay to a specific level. In our example, with initial Iodine-131 being 10.0 micrograms reducing to 0.1 micrograms (\(1/100\) of the original), plugging in these numbers and solving allows us to predict that roughly 26.57 days is required to reach this reduced level.
This predictable pattern of decay makes exponential functions powerful tools in radioactivity studies, allowing scientists and professionals to plan and track material usage or exposure over time.