Problem 33 - College Algebra

Question

For the following exercises, use this scenario: A tumor is injected with 0.5 grams of Iodine-125, which has a decay rate of \(1.15 \%\) per day. A scientist begins with 250 grams of a radioactive substance. After 250 minutes, the sample has decayed to 32 grams. Rounding to five signifi ant digits, write an exponential equation representing this situation. To the nearest minute, what is the half-life of this substance?

Step-by-Step Solution

Answer

The half-life of the substance is 62 minutes.

Step 1: Identify Key Information

We are given that 250 grams of a radioactive substance decays to 32 grams in 250 minutes. We need to find its half-life using an exponential decay equation. The decay rate is expressed as a percentage of decay per time unit.

Step 2: Write the Exponential Decay Equation

The general form of an exponential decay equation is given by \[ N(t) = N_0 e^{-kt} \]where \(N(t)\) is the quantity at time \(t\), \(N_0\) is the initial quantity, \(k\) is the decay constant, and \(t\) is time. Substitute \(N_0 = 250\) grams, \(N(t) = 32\) grams, and \(t = 250\) minutes.

Step 3: Solve for the Decay Constant

Plug the values into the equation:\[ 32 = 250 e^{-250k} \]This can be rearranged to:\[ e^{-250k} = \frac{32}{250} \]Take natural logarithm of both sides to solve for \(k\):\[ -250k = \ln\left(\frac{32}{250}\right) \]\[ k = -\frac{1}{250} \ln\left(\frac{32}{250}\right) \]

Step 4: Calculate the Half-Life

The half-life \(T_{1/2}\) is calculated using the formula:\[ T_{1/2} = \frac{\ln 2}{k} \]Substitute the expression for \(k\) into this formula:\[ T_{1/2} = \frac{\ln 2}{-\frac{1}{250} \ln\left(\frac{32}{250}\right)} \]Solve the value of \(T_{1/2}\) and round to the nearest minute.

Step 5: Perform Calculation

First, calculate \(k\) using a calculator: \( k \approx -\frac{\ln\left(\frac{32}{250}\right)}{250} \approx 0.01116 \) per minute.Then, calculate the half-life: \[ T_{1/2} \approx \frac{\ln 2}{0.01116} \approx 62 \] minutes.Round the result to get the half-life.

Key Concepts

half-life calculation

The concept of half-life is crucial in understanding radioactive decay. It refers to the time it takes for half of a sample of a radioactive substance to decay. This decay leads to a reduction in its quantity by half. Calculating the half-life involves determining how long it takes for half of the original amount of the substance to remain.

To find the half-life, we use the decay constant, which is a fixed value related to the speed of decay. The formula for half-life calculation is given by:

• \( T_{1/2} = \frac{\ln 2}{k} \)

where \( T_{1/2} \) is the half-life and \( k \) is the decay constant. The natural logarithm \( \ln 2 \) is approximately 0.693.

By plugging the decay constant into the formula, you can determine the time it will take for a substance to reduce to half its original mass, as performed in the exercise to find the half-life of the substance in minutes.

radioactive decay

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a specific rate, which is generally stated as a percentage indicating how much of the substance decays over a set period of time.

In the exercise, the scientist observes how a radioactive substance decays from 250 grams to 32 grams over 250 minutes. This illustrates the concept of radioactive decay as it reflects the transformation of the substance at a predictable rate. The substance loses its mass due to the emission of radiation.

The exponential decay equation can model this process:

• \( N(t) = N_0 e^{-kt} \)

where \( N(t) \) is the remaining amount at time \( t \), \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is the time lapsed. This clear relationship between the initial amount, the decay constant, and the remaining quantity makes it possible to predict how much of a substance will remain after a given time.

decay constant

The decay constant is a specific value that characterizes the rate at which a radioactive substance decays. It denotes how quickly atoms within a substance undergo radioactive decay, usually expressed per unit of time such as per second, per minute, or per day.

In this exercise, determining the decay constant involves using the initial and final amounts of the substance, and the time period over which decay occurs. The calculation follows this rearranged formula from the exponential decay equation:

• \( e^{-250k} = \frac{32}{250} \)
• Taking the natural logarithm leads to \( -250k = \ln\left(\frac{32}{250}\right) \)

Solving for \( k \),

• \( k = -\frac{1}{250} \ln\left(\frac{32}{250}\right) \),

where \( k \) is approximately 0.01116 per minute. This decay constant tells us that with each passing minute, a fixed percentage of the substance decays. Understanding the decay constant is crucial for predicting how fast a substance will decay and for calculating the half-life.