Problem 10 - Essential Calculus Early Transcendentals

Question

A sample of tritium-3 decayed to 94.5\(\%\) of its original amount after a year. (a) What is the half-life of tritium-3? (b) How long would it take the sample to decay to 20\(\%\) of its original amount?

Step-by-Step Solution

Answer

Half-life is about 12.36 years. It takes approximately 28.71 years to decay to 20%.

Step 1: Understand Exponential Decay

The decay of a radioactive substance is modeled by the exponential decay function, given by \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the amount of substance remaining at time \( t \), \( N_0 \) is the original amount, and \( k \) is the decay constant.

Step 2: Calculate the Decay Constant

We are given that 94.5\(\%\) of tritium-3 remains after one year. First, express this as an equation: \( 0.945N_0 = N_0 e^{-k(1)} \). Canceling \( N_0 \) from both sides, we have \( 0.945 = e^{-k} \). Solving for \( k \), we take the natural logarithm of both sides: \( k = -\ln(0.945) \approx 0.0561 \).

Step 3: Determine the Half-Life

The half-life \( t_{1/2} \) is the time it takes for the substance to reduce to half its initial amount, given by the formula \( t_{1/2} = \frac{\ln(2)}{k} \). Substituting \( k \approx 0.0561 \) into the formula, we get \( t_{1/2} = \frac{\ln(2)}{0.0561} \approx 12.36 \text{ years} \).

Step 4: Calculate Time for 20% Decay

To find the time \( t \) required for the sample to decay to 20\(\%\) of its original amount, use the equation \( 0.20N_0 = N_0 e^{-kt} \). Simplifying, we have \( 0.20 = e^{-kt} \). Taking the natural logarithm, \( -kt = \ln(0.20) \). Solving for \( t \) using \( k = 0.0561 \), \( t = \frac{\ln(0.20)}{-0.0561} \approx 28.71 \text{ years} \).

Key Concepts

Exponential Decay

Exponential decay is a fundamental concept used to describe the process of a quantity decreasing exponentially over time. This concept is very important in fields like physics, chemistry, and biology, where it often describes how substances transform or break down.
The formula for exponential decay is expressed as:

• \( N(t) = N_0 e^{-kt} \)

Here, \( N(t) \) represents the remaining quantity of the substance at time \( t \), \( N_0 \) is the initial amount, and \( k \) is the decay constant that determines the rate of decay. In an exponential decay process, the decrease is not linear but accelerates over time. If you were to graph this, you'd see a steep curve that falls sharply at first and then gradually slows down. The constant \( e \) (approximately 2.718) is the base of natural logarithms, which makes the equation non-linear. Exponential decay captures several real-world processes, from cooling in thermodynamics to interest decay in finance.

Radioactive Decay

Radioactive decay is a specific type of exponential decay where the number of atoms of a radioactive isotope decreases over time. This is due to unstable atomic nuclei losing energy by emitting radiation in various forms, such as alpha or beta particles or gamma rays.
The key characteristic of radioactive decay is that it is random and spontaneous, yet predictable on a macro scale. This means while we cannot predict when a specific atom will decay, we can calculate the average decay behavior for a large number of atoms. Radioactive decay is modeled with the exponential decay equation, utilizing the decay constant \( k \), which is unique to each radioactive isotope. The half-life of a radioactive substance is particularly important, as it helps estimate the time it takes for half of a sample to decay. Understanding this process is essential in areas such as nuclear power generation, medical imaging, and geologic dating.

Decay Constant Determination

The decay constant \( k \) is a crucial parameter in understanding exponential and radioactive decay. It describes the speed at which a substance undergoes exponential decay. To determine the decay constant, we apply the mathematical form of exponential decay and available data about the substance's behavior over time.
For example, if you know that a substance decays to 94.5\(\%\) of its original amount in one year, you can use the following steps:

• Set up the equation: \(0.945N_0 = N_0 \, e^{-k}\times(1)\)
• Simplify it to: \(0.945 = e^{-k}\)
• Take the natural logarithm of both sides to solve for \(k\): \(k = -\ln(0.945)\)
• You find \(k\approx 0.0561\).

The decay constant allows us to predict how long it'll take for a substance to reach a certain level, such as calculating its half-life, which is given by \(t_{1/2} = \frac{\ln(2)}{k}\). This knowledge is widely applied in scientific fields to infer decay processes and timetables accurately.