Question

232 Ra has a half-life of 11.4 d. How long would it take for the radioactivity associated with a sample of \(^{232} \mathrm{Ra}\) to decrease to \(1 \%\) of its current value?

Short answer

The radioactivity associated with a sample of \(^{232}Ra\) would decrease to 1% of its current value in approximately 76.29 days.

Step-by-step solution

Understand what are the known and unknown variables

Here, we know the half-life \(T = 11.4\) days, the initial and final amounts of \(^{232}Ra\), \(N_0\) and \(N(t)\). We know that \(N(t) = 0.01 * N_0\), because we want the radioactivity to decrease to 1% of its initial value. We want to find out time 't'.

Apply the formula of substance decay

Substitute the given values into the equation: \(0.01 * N_0 = N_0 * (0.5)^{t/11.4}\). Because both sides of the equation involve \(N_0\), we can divide each side by \(N_0\) making it cancel out.

Solve for time 't'

After simplifying, the equation becomes \(0.01 = (0.5)^{t/11.4}\). To solve for 't', we can use the natural logarithm and properties of logarithm: \(ln(0.01) = ln((0.5)^{t/11.4}) = \frac {t}{11.4} * ln(0.5)\). Now, solving for 't' gives \(t = \frac {11.4 * ln(0.01)} {ln(0.5)}\).

Calculate the value of 't'

't' is about 76.29 days after calculation.

Key concepts

Understanding Half-Life

The concept of half-life is central to the understanding of radioactive decay. Half-life, represented by the symbol T, refers to the time required for half of a radioactive substance to decay. In the context of the exercise with radium-232 (^{232} Ra), the half-life is given as 11.4 days. This means that every 11.4 days, the amount of ^{232} Ra will reduce to half of its original quantity. Understanding the half-life of a substance allows us to predict how long it will take for it to reduce to a certain fraction of its initial amount.

For instance, if we start with 100g of a radioactive substance with a half-life of 11.4 days, after 11.4 days, only 50g would remain. After another 11.4 days (a total of 22.8 days), only 25g would be left, and this pattern of reduction would continue. The half-life concept is not only vital in radioactive decay calculations but also in fields like medicine, archaeology, and environmental science.

Exponential Decay

Exponential decay describes the process through which a quantity diminishes at a rate proportional to its current value. This process is modeled by an exponential function, which in the realm of radioactive decay is represented by N(t) =N_0 * (0.5)^{t/T} where N(t) is the quantity at time t , N_0 is the initial quantity, T is the half-life, and the exponent t/T signifies the number of half-lives that have elapsed.

Visualizing Exponential Decay

Imagine plotting the amount of a radioisotope against time on a graph. The curve would start at N_0 and gradually bend downwards, approaching zero but never quite reaching it. This reflects the intrinsic nature of exponential decay: the reduction is rapid at first but slows down over time. In practical situations, this model helps to determine the remaining quantity of a substance after a given period, which is applicable to various scientific and industrial processes.

Natural Logarithm in Decay Calculations

The natural logarithm, denoted as ln, is a mathematical function that is pivotal in solving equations involving exponential decay. It is the inverse function of the exponential function e^x, where e is Euler's number (~2.71828). In our example with Radium-232, to isolate the variable t, we use the natural logarithm to linearize the exponential decay equation. This process allows us to find the time it would take for ^{232} Ra to decrease to a certain percentage of its initial quantity.

When the equation is expressed as 0.01 = (0.5)^(t/T), taking the natural logarithm of both sides gives us ln(0.01) = (t/T) * ln(0.5). The natural logarithm simplifies the process of solving for t since it converts the process of dealing with an exponent to a simple algebraic manipulative process. A solid understanding of logarithms is thus essential when dealing with exponential decay problems.

Radioisotope Dating

Radioisotope dating is a technique used to determine the age of objects based on the known half-lives of the isotopes they contain. By measuring the current amount of the isotope and comparing it to the expected original amount, scientists can calculate the time that has passed since the isotope started decaying. This technique is commonly used in fields such as geology, archaeology, and paleontology.

For example, Carbon-14 dating is a widely known method for estimating the age of organic materials by measuring the level of radioactivity from the Carbon-14 isotope. Similarly, using our ^{232}Ra example from the exercise, if we knew the initial quantity of the isotope in a sample, and measured its present quantity, we could apply the decay formula to estimate how long ago the sample was formed. Radioisotope dating relies on precise measurements and a good understanding of half-life and decay processes to arrive at accurate age estimates.