Question

A certain radioactive isotope decays to one-eighth of its original amount in \(5.00 \mathrm{~h} .\) How long would it take for \(10.0 \%\) of it to decay?

Short answer

Answer: It takes approximately 1.45 hours for 10.0% of the radioactive isotope to decay.

Step-by-step solution

Understanding exponential decay

Exponential decay can be represented by the equation: \(N(t) = N_0e^{-\lambda t}\) where \(N(t)\) is the remaining mass at time t, \(N_0\) is the initial mass, \(\lambda\) is the decay constant, and \(t\) is the elapsed time.

Identify the given information and set up the equation

In this case, we know that after t = 5.00 hours, the isotope decayed to one-eighth of its original amount. Thus, we can set up the equation as: \(\frac{1}{8}N_0 = N_0e^{-\lambda (5.00)}\) We need to find the decay constant (\(\lambda\)), which will allow us to answer the question of how long it will take for 10.0% of the isotope to decay.

Find the decay constant (\(\lambda\))

Divide both sides by \(N_0\) and take the natural logarithm of both sides: \(\frac{1}{8} = e^{-\lambda (5.00)}\) \(-\ln(8) = - 5.00 \lambda\) \(\lambda = \frac{\ln(8)}{5.00}\)

Set up the equation for the 10.0% decay

Now we need to find the time it would take for 10.0% (\(0.10 N_0\)) of the original isotope to decay. We can set up the equation as: \(0.90 N_0 = N_0e^{-\lambda t}\) Here, \(0.90N_0\) represents 90.0% remaining mass because 10% has decayed.

Solve for the elapsed time (\(t\))

Divide both sides by \(N_0\): \(0.90 = e^{-\lambda t}\) Take the natural logarithm of both sides: \(-\ln(0.90) = -\lambda t\) Substitute the value of \(\lambda\) that we found in Step 3: \(-\ln(0.90) = -\frac{\ln(8)}{5.00}t\) Now, isolate \(t\): \(t = \frac{-5.00 \ln(0.90)}{\ln(8)}\)

Calculate the elapsed time (\(t\))

Plug the numbers into the equation and calculate the time t: \(t \approx 1.45 \mathrm{~h}\) So, it would take approximately 1.45 hours for 10.0% of the radioactive isotope to decay.

Key concepts

Radioactive Isotope Decay

The process of radioactive isotope decay is a transformation that some elements undergo as unstable nuclei lose energy by emitting radiation in the form of particles or electromagnetic waves. This decay occurs naturally and spontaneously, but it follows a predictable pattern described by the exponential decay law. Understanding this concept is key in fields like nuclear physics, radiometric dating, and medicine, as it helps predict how long it takes for a quantity of radioactive material to diminish to a certain level.

Exponential decay can be mathematically represented in the form of an equation: \[\[\begin{align*} N(t) = N_0e^{-\frac{t}{\tau}} \
\end{align*}\]\] Here, \(N(t)\) represents the number of undecayed nuclei at time \(t\), \(N_0\) is the initial quantity of nuclei, \(e\) is the base of the natural logarithm (approximately equal to 2.71828), \(\tau\) is the time it takes for the number of undecayed nuclei to be reduced by half known as half-life, and \(t\) is the elapsed time. The rate at which this decay happens is consistent for a given isotope, reflecting in its unique half-life value.

Decay Constant

The decay constant, denoted by \(\lambda\), is a probability rate that characterizes how quickly a radioactive isotope decays. It is inversely related to the half-life (\(t_{1/2}\)) of the substance, meaning that isotopes with a large decay constant decay more rapidly and thus have a shorter half-life.

The relationship between the decay constant and the half-life is given by the formula: \[\[\begin{align*} \lambda = \frac{\ln(2)}{t_{1/2}} \
\end{align*}\]\] When solving problems in radioactive decay, determining the decay constant is a crucial step, as it allows you to forecast the rate of decay over time accurately. Once you have calculated \(\lambda\), you can use it in the exponential decay equation to predict the remaining undecayed quantity of the isotope at any given time.

Natural Logarithm

The natural logarithm is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. In the context of exponential decay, the natural logarithm, denoted as \(\ln\), is used to solve for variables in the decay expression when dealing with ratios of initial and remaining quantities of a radioactive isotope.

Taking the natural logarithm of both sides of an equation is a valuable step in isolating the decay constant or time variable. For example, if we have an equation of the form \(A = Be^{-\lambda t}\), taking the natural logarithm of both sides gives us \(\ln(A) = \ln(B) - \lambda t\), which can be rearranged to solve for \(t\). This is precisely what we do when analyzing radioactive decay with known initial amounts and remaining fractions.