Question

After 30 days of radioatctive decay, \(100 \mathrm{mg}\) of a radioactive substance was observed to remain. After 120 days, only \(30 \mathrm{mg}\) of this substance was left. (a) How much of the substance was initially present? (b) What is the half-life of this radicactive substance? (c) How long will it take before only 18 of the original amount remains?

Short answer

For this exercise, we have calculated the following values: a) The initial amount of the radioactive substance present was approximately \(267.7\mathrm{mg}\). b) The half-life of the radioactive substance is approximately \(45\mathrm{days}\). c) It will take approximately \(162.1\mathrm{days}\) before only 18 times the original amount of the radioactive substance remains.

Step-by-step solution

Understand the exponential decay equation

The exponential decay equation is given by the formula: \(N(t) = N_0 e^{-\lambda t}\), where \(N(t)\) is the amount of the substance remaining at time t, \(N_0\) is the initial amount of the substance, \(\lambda\) is the decay constant, and \(t\) is the time elapsed in the given units (days, in our case).

Solve for the initial amount (a)

First, we will use the given information, \(N(30) = 100\mathrm{mg}\) and \(N(120) = 30\mathrm{mg}\), to create two equations using the exponential decay formula: 1. \(100 = N_0 e^{-\lambda \times 30}\) 2. \(30 = N_0 e^{-\lambda \times 120}\) We can solve these equations simultaneously by dividing Equation 2 by Equation 1: \(\frac{30}{100} = \frac{N_0 e^{-\lambda \times 120}}{N_0 e^{-\lambda \times 30}}\) We can simplify this equation to: \(0.3 = e^{-\lambda \cdot 90}\) Now, we can take the natural logarithm of both sides and solve for \(\lambda\): \(-\lambda \cdot 90 = \ln(0.3)\) \(\lambda = \frac{\ln(0.3)}{-90}\) Now, we can substitute \(\lambda\) back into Equation 1 and solve for \(N_0\): \(100 = N_0 e^{\frac{\ln(0.3)}{-90} \cdot 30}\) \(N_0 = \frac{100}{e^{\frac{\ln(0.3)}{-90} \cdot 30}}\) Calculating this yields: \(N_0 \approx 267.7 \mathrm{mg}\) So, there were initially approximately \(267.7\mathrm{mg}\) of the radioactive substance present. This is the answer to part (a).

Calculate the half-life (b)

Half-life is related to the decay constant by the equation: \(T_{1/2}=\frac{\ln(2)}{\lambda}\). We have already calculated \(\lambda\) in Step 2. Now, we just need to substitute the value of \(\lambda\) to find \(T_{1/2}\): \(T_{1/2}=\frac{\ln(2)}{\frac{\ln(0.3)}{-90}}\) Calculating this yields: \(T_{1/2} \approx 45 \mathrm{days}\) So, the half-life of the radioactive substance is approximately \(45\mathrm{days}\). This is the answer to part (b).

Calculate the time for the desired amount to remain (c)

We are asked to find how long it will take before only 18 times the original amount of \(N_0\) remains. Therefore, we want to solve the decay equation for the time at which the substance has decayed to a value of \(\frac{1}{18}N_0\): \(\frac{1}{18}N_0=N_0 e^{-\lambda t}\) We can cancel the term \(N_0\) on both sides: \(\frac{1}{18}=e^{-\lambda t}\) Taking the natural logarithm of both sides: \(\ln\left(\frac{1}{18}\right)=-\lambda t\) Now, substitute the value for \(\lambda\): \(\ln\left(\frac{1}{18}\right)=-\left(\frac{\ln(0.3)}{-90}\right) t\) Solving for \(t\): \(t=\frac{\ln\left(\frac{1}{18}\right)}{\frac{\ln(0.3)}{-90}}\) Calculating this yields: \(t \approx 162.1 \mathrm{days}\) So, it will take approximately \(162.1\mathrm{days}\) before only 18 times the original amount of the radioactive substance remains. This is the answer to part (c).

Key concepts

Exponential Decay Equation

Understanding the exponential decay equation is essential for solving various problems involving radioactive decay. This equation describes how a quantity decreases over time at a rate proportional to its current value. The general form of the exponential decay equation is:

\[ N(t) = N_0 e^{-\text{λ}t} \]
Where:

• \( N(t) \) is the amount of the substance remaining after time \( t \).
• \( N_0 \) is the initial amount of the substance.
• \( \text{λ} \) is the decay constant, which indicates the rate of decay.
• \( t \) is the time that has elapsed.

By using this equation, one can determine how the quantity changes as time progresses. In practice, you'll need two sets of data points—\( N \) at two different times—to establish the decay constant and then solve for any unknowns.

Half-Life Calculation

The concept of half-life is crucial in the realm of radioactive decay. It represents the time it takes for half of the radioactive substance to decay. The half-life is calculated using the following relationship:

\[ T_{\frac{1}{2}} = \frac{\ln(2)}{\text{λ}} \]
Once the decay constant \( \text{λ} \) is known from the exponential decay equation, one can easily find the half-life of a substance by plugging the constant into the above equation. A shorter half-life means the substance decays more quickly, while a longer half-life indicates slower decay. This concept helps in understanding the longevity and decay behavior of different radioactive substances.

Initial Amount Determination

To solve problems involving radioactive decay, it's often necessary to determine the initial amount of the substance, labeled as \( N_0 \). The initial amount can be found by using the exponential decay equation along with certain known values. By substituting the known amounts of the substance at different time intervals into the equation and solving for \( N_0 \), you can back-calculate what the original amount was before decay began. This information is crucial for various practical applications, such as determining doses of medical isotopes, or understanding natural processes in geology and archaeology.

Decay Constant

The decay constant \( \text{λ} \) is foundational to understanding exponential decay. It is inversely related to the half-life and directly determines the rate at which a substance will decay. A higher decay constant means the substance will decay more rapidly. It is unique for each radioactive isotope and can be derived from experimental data by measuring the amount of substance remaining at two or more different times. Using logarithmic manipulation, as shown in the step-by-step solution, the decay constant can be isolated and subsequently used to find both the initial amount of the substance and its half-life. Knowledge of the decay constant is imperative for accurate dating in fields like archaeology, geology, and environmental science.