Problem 18 - Precalculus

Question

We list some radioactive isotopes and their associated half-lives. Assume that each decays according to the formula \(A(t)=A_{0} e^{k t}\) where \(A_{0}\) is the initial amount of the material and \(k\) is the decay constant. For each isotope: \- Find the decay constant \(k\). Round your answer to four decimal places. \- Find a function which gives the amount of isotope \(A\) which remains after time \(t\). (Keep the units of \(A\) and \(t\) the same as the given data.) \- Determine how long it takes for \(90 \%\) of the material to decay. Round your answer to two decimal places. (HINT: If \(90 \%\) of the material decays, how much is left?) Uranium 235 , used for nuclear power, initial amount \(1 \mathrm{~kg}\) grams, half-life 704 million years.

Step-by-Step Solution

Answer

Decay constant \(k\) is \(-0.0009849\). The decay function is \(A(t) = e^{-0.0009849 t}\). 90% decay occurs in approximately 2345.86 million years.

Step 1: Find Decay Constant

To find the decay constant \(k\), we use the formula for half-life \(t_{1/2}\): \[ t_{1/2} = \frac{\ln(2)}{-k} \] Given the half-life of Uranium-235 is 704 million years, we solve for \(k\): \[ 704 = \frac{\ln(2)}{-k} \] Rearranging gives: \[ k = \frac{\ln(2)}{704} \] Using a calculator, \(k \approx -0.0009849\).

Step 2: Formulate Decay Function

Now that we have the decay constant \(k = -0.0009849\), we substitute it into the exponential decay formula: \[ A(t) = A_{0} e^{k t} \] For an initial amount \(A_{0} = 1 \text{ kg}\), the function becomes: \[ A(t) = 1 \times e^{-0.0009849 t} \] Thus, the function is \(A(t) = e^{-0.0009849 t}\).

Step 3: Calculate Time for 90% Decay

To determine how long it takes for 90% of the material to decay, we need to find the time \(t\) when 10% of uranium is left. Setting \(A(t) = 0.1A_{0}\) in the decay equation gives: \[ 0.1 = e^{-0.0009849 t} \] Taking the natural logarithm of both sides: \[ \ln(0.1) = -0.0009849 t \] Solving for \(t\): \[ t = \frac{\ln(0.1)}{-0.0009849} \] This calculation gives \(t \approx 2345.86\) million years.

Key Concepts

Exponential Decay Formula

A fundamental concept in understanding radioactive decay is the exponential decay formula. Imagine that you start with an initial amount of material, like a piece of radioactive Uranium-235 weighing 1 kg. As time goes by, this material gradually decays and the amount decreases. The exponential decay formula models this process mathematically. It's represented as \(A(t) = A_{0} e^{k t}\). Here, \(A(t)\) is the amount of the radioactive substance left after time \(t\), \(A_{0}\) is the initial amount, and \(k\) is the decay constant, which tells us how fast or slow the material decays.

A particular characteristic of this formula is that the decrease in quantity is not linear; instead, it happens at a rate proportional to the amount still present. This means that in equal time periods, the material loses the same fraction of itself. For example, if the material loses half its mass every 704 million years, it will keep losing half of what remains in each subsequent 704 million years. This makes the formula particularly useful for predicting how long it takes for a given portion of the material to decay.

Half-life Calculation

In the realm of radioactive decay, the half-life is an essential measure. It refers to the time required for half of the radioactive isotope to decay. To understand this in uranium's case, its half-life is 704 million years. This means if you have 1 kg of Uranium-235 at the beginning, only 0.5 kg would remain after 704 million years.

Calculating half-life involves a relationship with the decay constant \(k\). The formula to find \(k\) using half-life is \(t_{1/2} = \frac{\ln(2)}{-k}\). Given Uranium-235's half-life as 704 million years, you can find \(k\) by rearranging the formula to \(k = \frac{\ln(2)}{704}\). Using a calculator reveals \(k \approx -0.0009849\).

Knowing the half-life lets you predict how long a sample of a radioactive isotope will take to reach a certain level of decay. This is crucial in fields like nuclear medicine, archeology, and engineering, where precise timing is key.

Decay Constant

The decay constant \(k\) is a crucial parameter in exponential decay equations. It represents the rate of decay of the radioactive isotope and is inherently tied to the half-life. In essence, \(k\) signifies how quickly the material disintegrates.

To compute \(k\), you're essentially looking at how the material's amount changes in proportion to its current amount, using time as a factor. This is calculated through the formula derived from the half-life, \(k = \frac{\ln(2)}{t_{1/2}}\). For Uranium-235, with a half-life of 704 million years, this calculation gives a decay constant of \(k \approx -0.0009849\).

The negative sign of the decay constant indicates a decrease in the material's amount as time passes. It might seem like a small number, but even small decay constants result in significant changes over long periods, like millions of years, making this constant invaluable for predicting decay events in nuclear physics.