Strontium-90 is one of the products of the fission of uranium-235. This strontium isotope is radioactive, with a half-life of 28.1 yr. Calculate how long (in yr) it will take for \(1.00 \mathrm{~g}\) of the isotope to be reduced to \(0.200 \mathrm{~g}\) by decay.

The concept of half-life is crucial in the study of radioactive decay and is a key factor in understanding how radioactive substances diminish over time. A half-life is the time required for one-half of the radioactive nuclei in a sample to undergo decay. It is a constant that is characteristic of each radioactive isotope.

For example, the half-life of Strontium-90, as mentioned in the original exercise, is 28.1 years. This means that every 28.1 years, half of a given amount of Strontium-90 will have decayed into another element or isotope. After another 28.1 years, half of the remaining amount will decay, and so on.

The importance of the half-life in radioactive decay lies in its ability to allow scientists to predict the amount of radioactive material that will remain after a certain period. It is also critical in various applications, such as dating archaeological finds or managing nuclear waste.

An exponential decay function is a mathematical model used to describe the process of radioactive decay. The formula representing exponential decay is generally expressed as:
\(N = N_0 . (1/2)^{t/t_{1/2}}\)

In this formula, \(N_0\) represents the initial quantity of the substance, while \(N\) represents the remaining quantity after time \(t\). The exponent \(t/t_{1/2}\) indicates how many half-life periods have elapsed. Radioactive decay follows an exponential pattern because the rate at which the material decays is proportional to its current amount.

This function shows a rapid decrease initially, which gradually levels off as time passes. This pattern occurs because each half-life reduces the remaining amount by half, so although the percentage reduction is constant, the actual amount of decay decreases with each half-life.

Solving exponential equations is a critical skill in many scientific fields, especially when dealing with processes like radioactive decay. These equations often require the use of logarithms to find an unknown variable when the equation involves an exponent. Here’s how you solve an exponential decay equation, as seen in the step by step solution provided:

First, we need to isolate the exponential expression. This allows us to take the logarithm of both sides. For instance, in the original problem, we have:\[(1/2)^{t/28.1} = 0.200\]
Taking the natural logarithm of both sides gives us:\[\ln((1/2)^{t/28.1}) = \ln(0.200)\]
Applying the laws of logarithms, which state that \(\ln(a^b) = b \ln(a)\), we can bring down the exponent as a multiplier:\[(t/28.1) \ln(1/2) = \ln(0.200)\]
Finally, solving for \(t\) involves rearranging the equation and solving for the unknown value, which gives us the time it takes for the sample to decay from its initial mass to the given remaining mass.

Although these steps can be mathematically challenging, understanding how to manipulate and solve exponential equations is fundamental to handling problems involving radioactive decay.