Question

A radioactive material loses \(25 \%\) of its mass in 10 minutes. What is its half-life?

Short answer

Answer: The half-life of the radioactive material is approximately 23.93 minutes.

Step-by-step solution

Identify the given values

We're given that, - The radioactive material loses 25% of its mass in 10 minutes. - Our task is to find its half-life.

Use the radioactive decay formula

The formula for radioactive decay is given by: $$N_t = N_0 e^{-\lambda t}$$ Where, - \(N_t\) is the remaining mass after some time \(t\) - \(N_0\) is the initial mass - \(\lambda\) is the decay constant - \(t\) is the time elapsed After 10 minutes, 25% of the radioactive material has decayed, so the remaining mass is 75% or \(0.75N_0\). Therefore, we can write the equation as follows: $$0.75N_0 = N_0 e^{-\lambda \cdot 10}$$

Solve for the decay constant

Now, divide both sides of the equation by \(N_0\): $$0.75 = e^{-\lambda \cdot 10}$$ We'll then take the natural logarithm to isolate the lambda: $$\ln(0.75) = -\lambda \cdot 10$$ Now, divide both sides by -10 to find the decay constant: $$\lambda = \frac{\ln(0.75)}{-10}$$ Calculate the value of the decay constant: $$\lambda \approx 0.02899$$

Calculate the half-life

The half-life of a radioactive element is given by: $$t_{1/2} = \frac{\ln(2)}{\lambda}$$ Now, substitute the value of \(\lambda\) that we found in the previous step: $$t_{1/2} =\frac{\ln(2)}{0.02899}$$ Calculate the half-life of the radioactive material: $$t_{1/2} \approx 23.93 \text{ minutes}$$ The half-life of the radioactive material is approximately 23.93 minutes.

Key concepts

Radioactive Decay Formula

Understanding the process of radioactive decay is essential because it allows us to determine the rate at which an unstable atomic nucleus loses energy by emitting radiation. With the given exercise, we focused on how we can use the radioactive decay formula to determine the half-life of a substance.

The formula at the heart of this discussion is expressed as: \[N_t = N_0 e^{-\textbackslash\textlambda t}\]where \(N_t\) is the amount of substance that remains after time \(t\), \(N_0\) is the original amount of substance, \(e\) is the base of natural logarithms, \(\textbackslash\textlambda\) is the decay constant, and \(t\) is the time elapsed. In our exercise, we use this formula to find out how much material will be left after a certain period, from which we deduce the half-life of the material.

Decay Constant

The decay constant, represented by \(\textbackslash\textlambda\), is a vital term in the radioactive decay equation. It signifies the probability per unit time that a nucleus will decay. The value varies for different substances and directly affects the speed of the decay process.

To find this constant using the formula provided in the exercise, we solve for \(\textbackslash\textlambda\) from the equation \(0.75 = e^{-\textbackslash\textlambda \textbackslash\texttimes 10}\). Here, we interpret \(0.75N_0\) as the remaining mass after the decay, because 25% has been lost, leaving us with 75% of the original mass.

Half-Life Calculation

The half-life of a radioactive substance is the time it takes for half of the material to decay. This concept is critical in fields like archeology, medicine, and environmental science, as it helps to predict how long a substance remains active or dangerous. In the context of the exercise, once we've established the decay constant, we can calculate the half-life with the formula:\[t_{1/2} = \textbackslash\frac{\textbackslash\textln(2)}{\textbackslash\textlambda}\]By inserting the decay constant into this formula, we estimate the time in which half of the original material would remain. Our calculation results show how the decay constant directly influences the half-life—the smaller the decay constant, the longer the half-life and vice versa.

Through practical application, we can understand that the half-life provides a means to evaluate how quickly a substance will diminish to a point where it may no longer be effective or hazardous.

Natural Logarithm

The natural logarithm, often represented as '\(\textbackslash\textln\)', is a mathematical function that's the inverse of the function '\(e^x\)', where \(e\) is Euler's number (approximately 2.71828). Understanding this tool is crucial when dealing with exponential decay because it allows us to isolate variables, like the decay constant, from the equation involving \(e\).

In our solution, we used the natural logarithm to manipulate the decay equation and solve for \(\textbackslash\textlambda\). Specifically, by taking the natural logarithm of both sides of the equation \(0.75 = e^{-\textbackslash\textlambda \textbackslash\texttimes 10}\), we obtain \(\textbackslash\textln(0.75) = -\textbackslash\textlambda \textbackslash\texttimes 10\), which enables us to find the decay constant. It's an excellent example of logarithms making complex equations manageable.