Problem 33 - Chemistry Structure and Dynamics

Question

A 1.000 -gram sample of \({ }^{22} \mathrm{Na}\) decays to 0.20 gram in 6.04 years. Calculate the half-life for this decay, the rate constant, and the time it would take for this sample to decay to 0.075 gram.

Step-by-Step Solution

Answer

Half-life is 2.27 years, rate constant is 0.305 year^{-1}, decay to 0.075 gram in 8.89 years.

Step 1: Understanding the Problem

We have a sample of \(^{22} \mathrm{Na}\) that decays from 1.000 gram to 0.20 gram in 6.04 years. We need to calculate the half-life, the rate constant, and the time for it to decay to 0.075 gram.

Step 2: Formula for Radioactive Decay

The decay of a radioactive isotope is given by the formula: \[ N_t = N_0 e^{-kt} \]where \(N_t\) is the remaining quantity at time \(t\), \(N_0\) is the initial quantity, \(k\) is the rate constant, and \(t\) is the time.

Step 3: Calculate the Rate Constant

First, we find the rate constant \(k\). Using the decay formula:\[0.20 = 1.00 \cdot e^{-6.04k}\]Solve for \(k\):\[ e^{-6.04k} = \frac{0.20}{1.00} = 0.20 \]Thus,\[-6.04k = \ln(0.20)\]\[ k = -\frac{\ln(0.20)}{6.04} \approx 0.305 \text{ year}^{-1}\]

Step 4: Calculate the Half-Life

To find the half-life \(t_{1/2}\), we use the relationship:\[ t_{1/2} = \frac{\ln(2)}{k} \]Substitute the value of \(k\):\[ t_{1/2} = \frac{\ln(2)}{0.305} \approx 2.27 \text{ years}\]

Step 5: Calculate Time to Decay to 0.075 gram

We use the decay formula again to find the time it takes to reduce to 0.075 gram:\[0.075 = 1.00 \cdot e^{-kt}\]Substitute the value of \(k\):\[e^{-0.305t} = 0.075\]Thus,\[-0.305t = \ln(0.075)\]Solve for \(t\):\[t = \frac{\ln(0.075)}{-0.305} \approx 8.89 \text{ years}\]

Key Concepts

Half-life Calculation

The concept of half-life is key in understanding how radioactive isotopes decay over time. Half-life is the duration it takes for half of a radioactive substance to decay. In simpler terms, it's the time required for the substance's quantity to reduce to half its initial amount.

For the calculation, we use the formula:

• \[ t_{1/2} = \frac{\ln(2)}{k} \]

where:

• \( t_{1/2} \) is the half-life
• \( k \) is the rate constant

By substituting the calculated rate constant from our example (\( k \approx 0.305 \text{ year}^{-1} \)), we determine the half-life as approximately 2.27 years.

This means that every 2.27 years, the amount of \(^{22} \mathrm{Na} \) in our sample will reduce by half. This helps in predicting how long it will take for a substance to significantly decay, aiding in safety assessments and usage strategies.

Rate Constant Determination

The rate constant \( k \) is a crucial factor in radioactive decay, as it indicates the decay rate per unit time. It helps us understand how quickly a substance is losing its radioactive nature.

In our case, the formula for calculating \( k \) from the exponential decay equation is:

• \[ 0.20 = 1.00 \cdot e^{-6.04k} \]

Rearranging the equation gives:

• \[ k = -\frac{\ln(0.20)}{6.04} \]

Solving it, we find \( k \approx 0.305 \text{ year}^{-1} \).

With this rate constant, we quantify how the sample decays, which in this scenario is relatively quick. Knowing \( k \) allows one to model the decay over different time periods accurately, which is essential in many scientific and industrial applications.

Exponential Decay Formula

The process of radioactive decay is governed by the exponential decay formula:

• \[ N_t = N_0 e^{-kt} \]

Here:

• \( N_t \) is the remaining amount of the substance at time \( t \)
• \( N_0 \) is the initial amount at the start
• \( k \) is the rate constant
• \( t \) is the time elapsed

The formula shows how the substance's amount decreases exponentially with time.

In our example, we initially had 1.000 gram of \( ^{22} \mathrm{Na} \), which decays to 0.20 gram in 6.04 years. Using the formula allows us to calculate critical values, like the rate constant and predict future amounts. If we want to know when the quantity will reduce to 0.075 gram, we rearrange the formula:

• \[ e^{-0.305t} = 0.075 \]
• \[ t = \frac{\ln(0.075)}{-0.305} \]

This shows it'll take approximately 8.89 years to reach that amount.

Understanding the exponential decay formula is vital; it is extensively used in nuclear physics, medicine, and environmental science, making it a fundamental tool for calculations relating to time and decay.