Problem 6 - Chemistry The Central Science

Question

The accompanying graph illustrates the decay of \(_{42}^{88} \mathrm{Mo}\) which decays via positron emission. (a) What is the half-life of the decay? (b) What is the rate constant for the decay?(c) What fraction of the original sample of \(_{42}^{88} \mathrm{Mo}\) remains after 12 \(\mathrm{min}\) ? (d) What is the product of the decay process? [Section 21.4\(]\)

Step-by-Step Solution

Answer

In summary, for the decay of \(_{42}^{88} \mathrm{Mo}\) via positron emission, we find that the half-life (T_1/2) comes from the accompanying graph, the rate constant (k) is calculated using the formula \(k = \frac{\ln 2}{T_{1/2}}\), the remaining fraction of the original sample after 12 minutes is determined using the formula \(\frac{N(t)}{N_0} = e^{-kt}\), and the product of the decay process is \(_{41}^{88} \mathrm{P}\).

Step 1: Determine the half-life of the decay

Based on the graph, we locate the point where half of the sample has decayed. The half-life can be read from the time-axis. We do not have the actual graph, but let's call the half-life T_1/2, which is a variable for now.

Step 2: Calculate the rate constant for the decay

The rate constant for the decay is related to the half-life by the following formula: \[k = \frac{\ln 2}{T_{1/2}}\] Where \(k\) is the rate constant, \(T_{1/2}\) is the half-life, and \(\ln 2\) comes from the nature of exponential decay. Plug the half-life T_1/2 calculated in step 1 into the formula above to find the rate constant.

Step 3: Find the remaining fraction of the original sample after 12 minutes

To find the remaining fraction of the original sample after 12 minutes, we'll use the decay formula: \[N(t) = N_0 e^{-kt}\] Where \(N(t)\) is the amount of the sample remaining at time \(t\), \(N_0\) is the initial amount of the sample, \(k\) is the decay constant, and \(t\) is the time. We can rearrange the equation to find the remaining fraction after 12 minutes: \[\frac{N(t)}{N_0} = e^{-kt}\] Plug in \(t = 12\,\mathrm{min}\) and the decay constant (\(k\)) found in step 2 to find the remaining fraction of \(_{42}^{88} \mathrm{Mo}\).

Step 4: Identify the product of the decay process

Since \(_{42}^{88} \mathrm{Mo}\) decays via positron emission, the decay process is as follows: \[_{42}^{88} \mathrm{Mo} \rightarrow \, _{41}^{88} \mathrm{P} + \,_{1}^{0}\beta^+\] The product of the decay process is \(_{41}^{88} \mathrm{P}\). Solution: In this exercise, we learned that 1. The half-life of the decay must be determined from the given graph (T_1/2). 2. The rate constant for the decay can be calculated using the half-life value: \(k = \frac{\ln 2}{T_{1/2}}\). 3. The remaining fraction of the original sample after 12 minutes can be found using the decay formula: \(\frac{N(t)}{N_0} = e^{-kt}\). 4. The product of the decay process for \(_{42}^{88} \mathrm{Mo}\) decaying via positron emission is \(_{41}^{88} \mathrm{P}\).

Key Concepts

Understanding Nuclear Decay

Nuclear decay is a fundamental concept in both chemistry and physics that describes the process by which an unstable atomic nucleus loses energy by emitting radiation. This process can result in the nucleus transforming into a different element or a different isotope of the same element. There are several types of nuclear decay, with positron emission being one form.

During positron emission decay, an unstable nucleus emits a positron, which is the antimatter counterpart of the electron. This process occurs when a proton is transformed into a neutron, and a positron and a neutrino are released. As a result, the atomic number of the element decreases by one, producing a different element with the same mass number. For example, in the decay of \( _{42}^{88} \mathrm{Mo} \) via positron emission, the product is \( _{41}^{88} \mathrm{P} \) along with the emitted positron.

How to Calculate Half-Life

The half-life of a radioactive substance is a measure of the time it takes for half the nuclei in a sample to undergo decay. It is a crucial aspect of understanding nuclear decay because it helps determine the stability of a nuclide and its potential applications or hazards. To calculate the half-life of a radionuclide, one would typically observe its decay over time, plotting the activity on a graph and identifying the point where activity decreases to half of its initial value.

This observed value can be used to predict the behavior of the substance over time. For instance, in our example with \( _{42}^{88} \mathrm{Mo} \) if the half-life is found to be, say, 6 minutes, it means that every 6 minutes, the quantity of \( _{42}^{88} \mathrm{Mo} \) will drop to half of the previous value. Understanding and calculating half-life are vital for a range of applications from dating archaeological finds to medical treatments.

Exponential Decay Formula

The exponential decay formula is a mathematical expression that describes how the quantity of a decaying substance decreases over time. The formula reflects the fact that the decay process is random and follows an exponential pattern. In mathematics, the formula is represented as \( N(t) = N_0 e^{-kt} \), where:

• \( N(t) \) is the amount of the substance that remains at time \( t \).
• \( N_0 \) is the initial amount of the substance.
• \( k \) is the decay constant, which is related to the half-life \( T_{1/2} \) via \( k = \frac{\ln 2}{T_{1/2}} \).
• \( t \) is the time elapsed.

The exponential decay formula is key in understanding the rate at which a radioactive material decreases. By knowing the initial amount and the half-life, one can calculate the decay constant \( k \) and predict how a sample will change over any given time. It illustrates the power of exponential functions in modeling real-world phenomena in nuclear chemistry.