Question

The half-life for the radioactive decay of U-238 is 4.5 billion years and is independent of initial concentration. How long will it take for \(10 \%\) of the U- 238 atoms in a sample of U- 238 to decay? If a sample of U-238 initially contained \(1.5 \times 10^{18}\) atoms when the universe was formed 13.8 billion years ago, how many U- 238 atoms does it contain today?

Short answer

The time it will take for 10% of the U-238 atoms to decay is approximately 0.65 billion years. The number of U-238 atoms remaining today is approximately \(3.20 \times 10^{17}\) atoms.

Step-by-step solution

- Understand the Concept of Half-Life

Half-life is the time taken for half of the radioactive nuclei in a sample to decay. The decay process is logarithmic, which means that in each half-life, half of the remaining radioactive nuclei will decay. The formula to calculate the remaining quantity after a certain time is given by: \( N(t) = N_0(1/2)^{t/T} \), where \( N(t) \) is the remaining quantity at time \( t \), \( N_0 \) is the initial quantity, and \( T \) is the half-life.

- Calculate Time for 10% Decay

We want to find the time \( t \) when 90% of the U-238 remains (since 10% has decayed). We will set up the following equation using the half-life formula: \( 0.9N_0 = N_0(1/2)^{t/T} \ 0.9 = (1/2)^{t/T} \). Taking the natural logarithm on both sides we get: \( \ ln(0.9) = ln((1/2)^{t/T}) \), which simplifies to \( ln(0.9) = (t/T) ln(1/2) \ ln(0.9) / ln(1/2) = t/T \). After substituting the half-life value for \( T \), we can solve for \( t \).

- Determine Time for 10% Decay

By substituting the half-life \( T = 4.5 \) billion years into the previous equation, we can calculate the time \( t \): \( t = T(ln(0.9) / ln(1/2)) \). Use a calculator to find the value of \( t \).

- Calculate the Remaining Atoms Today

Using the time elapsed since the formation of the universe (13.8 billion years) and the half-life of U-238 (4.5 billion years), we apply the half-life formula to find the remaining number of U-238 atoms today: \( N(13.8) = 1.5 \times 10^{18}(1/2)^{13.8/4.5} \). Compute this using a calculator to obtain the number of remaining atoms.

Key concepts

Half-Life Calculation

Understanding the concept of half-life is crucial when studying radioactive decay. It refers to the time required for half of the radioactive nuclei—or isotopes—in a given sample to undergo decay into other elements or isotopes. This property is important because it is constant and unique for each radioactive substance, independent of the initial quantity present.

In the context of our U-238 decay problem, the half-life is given as 4.5 billion years. This means that after 4.5 billion years, any given amount of U-238 will be reduced to half of its original quantity due to radioactive decay. To calculate the time it takes for a certain percentage of U-238 to decay, we rearrange the half-life formula: \[N(t) = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T}}\]where \(N(t)\) represents the remaining atoms at time \(t\), \(N_0\) is the initial number of atoms, and \(T\) is the half-life. For instance, if we want to calculate the time required for 10% of the U-238 to decay—meaning 90% remains—we can modify the equation to reflect this: \[0.9N_0 = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T}}\]By isolating \(t\), we can solve for the specific time that correlates with the 10% decay threshold. This approach simplifies complex decay processes into manageable calculations that reveal the timeframes over which radioactive materials change.

Logarithmic Decay Process

The decay of radioactive substances is inherently a logarithmic process, meaning that the rate at which these substances transform is proportional to their current quantity. As more material decays, the number of radioactive nuclei available for decay decreases, thereby slowing the rate of the overall decay process over time. This is a fundamentally different behavior than linear processes where the rate of change is constant irrespective of the amount present.

This process is best understood through logarithmic functions, which are the inverse of exponential functions. The key to solving a logarithmic decay problem, like that of U-238, is to use logarithms to rearrange the decay formula so that we can isolate the variable of interest—usually time. In our example, taking the natural logarithm of both sides of the half-life equation and solving for time \(t\) gave us a direct way to compute the period required for a given percentage of decay: \[ln(0.9) = t \times \frac{ln(1/2)}{T}\]Once you use the properties of logarithms to manipulate the decay equation, the solution becomes more apparent and tangible. It is a prime example of how understanding logarithms can unlock the mystery behind the timing of radioactive decay.

Exponential Decay Formula

Exponential decay is a fundamental concept underlying the process of radioactive decay. The decay rate is proportional to the remaining quantity of the substance, which results in an exponential decrease over time. The standard form of the exponential decay formula for any radioactive substance is: \[N(t) = N_0 \times e^{-\frac{t}{\tau}}\]Here, \(N(t)\) denotes the quantity of the substance after time \(t\), \(N_0\) the initial quantity, \(e\) is the base of the natural logarithm, and \(\tau\) is the mean lifetime of the substance. For half-life calculations, the formula is slightly modified because rather than using the mean lifetime, we work with the half-life \(T\), resulting in: \[N(t) = N_0 \times \bigg(\frac{1}{2}\bigg)^{\frac{t}{T}}\]It is this formula that you will often encounter in homework exercises and practical problems involving radioactive decay. By using the exponential decay model, we can predict the number of atoms remaining in a sample after any given period, as was done in the example of the U-238 decay over 13.8 billion years since the formation of the universe. Applying the formula with the appropriate variables provides a precise and reliable prediction of the number of nuclei remaining at any point in time.