Chapter 40: Problem 31
A certain radioactive isotope decays to one-eighth its original amount in \(5.00 \mathrm{~h} .\) How long would it take for \(10.0 \%\) of it to decay?
Short Answer
Expert verified
Answer: Approximately 1.37 hours.
Step by step solution
01
Decay Formula
The decay formula is given by:
\(N(t) = N_0 e^{-\lambda t} \)
Where \(N(t)\) is the amount of the radioactive isotope at time t, \(N_0\) is the initial amount of the isotope, \(\lambda\) is the decay constant, and \(t\) is the time in hours.
02
Determine the Decay Constant
Since the isotope decays to one-eighth its original amount in 5 hours, we can plug this information into the decay formula to determine the decay constant:
\(\frac{1}{8} N_0 = N_0 e^{-\lambda (5)} \)
We can get rid of \(N_0\) on both sides, as it cancels out:
\(\frac{1}{8} = e^{-5 \lambda} \)
Now, we need to find the value of \(\lambda\) by taking the natural logarithm of both sides:
\(ln(\frac{1}{8}) = -5 \lambda \)
\(\lambda = -\frac{1}{5} ln(\frac{1}{8}) \)
03
Find the Time for 10% Decay
We want to find the time it takes for 10% of the isotope to decay, which means there is 90% of the isotope left. We can set up the decay formula to find this time:
\(0.9 N_0 = N_0 e^{-\lambda t} \)
Again, the \(N_0\) values cancel out:
\(0.9 = e^{-\lambda t} \)
Now, we can plug in the value of \(\lambda\) that we found in the previous step:
\(0.9 = e^{-(-\frac{1}{5} ln(\frac{1}{8}))t} \)
04
Calculate the Time
Next, we take the natural logarithm of both sides to solve for time t:
\(ln(0.9) = -(-\frac{1}{5} ln(\frac{1}{8}))t \)
Now, we can isolate t:
\(t = \frac{ln(0.9)}{\frac{1}{5} ln(\frac{1}{8})} \)
05
Final Answer
By calculating this expression, we get the time:
\(t = \frac{ln(0.9)}{\frac{1}{5} ln(\frac{1}{8})} \approx 1.37 h\)
It would take approximately 1.37 hours for 10% of the radioactive isotope to decay.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Decay Formula
Understanding the decay formula is fundamental to studying radioactive isotope decay. Simply put, the decay formula models how the quantity of a radioactive substance diminishes over time. The general form of this formula is represented as:
\[\begin{equation}N(t) = N_0 e^{-\tau t}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag\end{equation}\]In this formula,
\[\begin{equation}N(t) = N_0 e^{-\tau t}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag{1}\tag\end{equation}\]In this formula,
- (t) represents the quantity of the substance at a specific time t,
- 0 is the initial quantity at time t=0,
- \tau is the decay constant uniquely associated with the substance, and e is the base of the natural logarithm.
Decay Constant
The decay constant, denoted by \tau, represents the probability per unit time that a given atom will decay. It's a fundamental property of each radioactive isotope and varies between different types of isotopes. The decay constant is related to the half-life of the substance, which is the time taken for half of the radioactive atoms to decay. This relation can be expressed as:\[\begin{equation}\tau = \frac{ln(2)}{t_{1/2}}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{2}\tag{\end{equation}\]Understanding \tau is essential for calculating the rate at which a radioactive isotope decays over time. By knowing the decay constant, one can predict how long it will take for a certain portion of the isotope to decay, which is particularly relevant in applications like radiometric dating, nuclear medicine, and radiation therapy.
Exponential Decay
Exponential decay is a process that decreases at a rate proportional to its current value. This decline isn't linear; instead, it follows a distinctive pattern where the rate of decay slows down over time. The radioactive decay formula we discussed earlier is a typical example of exponential decay.One can visually comprehend exponential decay by looking at a decay curve on a graph, which steadily 'flattens out' as time moves on. The basis of this behavior stems from the fact that the greater the quantity of the substance, the more particles there are that could potentially decay, increasing the absolute number of decays per time unit. As the quantity diminishes, there are fewer particles, and thus, the decay rate decreases, following the exponential pattern.
Natural Logarithm
The natural logarithm is a mathematical function that is the inverse of the exponential function, usually denoted as ln(x). It plays a pivotal role in calculating decay problems since the exponential decay equation involves the natural base e. To isolate and solve for variables such as the decay constant or time, we often take the natural logarithm of both sides of the equation. This process transforms an equation involving exponents into a linear form, which is generally simpler to manage and solve.For instance, if you wanted to solve for the decay constant in the equation arising from the amount of a radioactive isotope that decays to one-eighth of its original amount in a certain period, you would take the natural logarithm of both sides to proceed with the calculation, as was illustrated in the exercise above.
