Problem 43 - Chemistry and Chemical Reactivity

Question

Sodium-23 (in a sample of NaCl) is subjected to neutron bombardment in a nuclear reactor to produce \(^{24}\) Na. When removed from the reactor, the sample is radioactive, with \(\beta\) activity of \(2.54 \times 10^{4} \mathrm{dpm} .\) The decrease in radioactivity over time was studied, producing the following data: $$\begin{array}{lc}\hline \text { Activity }(\mathrm{dpm}) & \text { Time }(\mathrm{h}) \\\\\hline 2.54 \times 10^{4} & 0 \\ 2.42 \times 10^{4} & 1 \\\2.31 \times 10^{4} & 2 \\\2.00 \times 10^{4} & 5 \\\1.60 \times 10^{4} & 10 \\\1.01 \times 10^{4} & 20 \\\\\hline\end{array}$$ (a) Write equations for the neutron capture reaction and for the reaction in which the product of this reaction decays by \(\beta\) emission. (b) Determine the half-life of sodium- 24.

Step-by-Step Solution

Answer

(a) Neutron capture: \(^{23}_{11}Na + \: ^{1}_{0}n \rightarrow ^{24}_{11}Na\); Beta decay: \(^{24}_{11}Na \rightarrow ^{24}_{12}Mg + \beta^- + \overline{\nu}_e\). (b) Half-life is approximately 15 hours.

Step 1: Neutron Capture Reaction

In this step, we need to write the equation for the neutron capture reaction. The target nucleus is sodium-23, which captures a neutron to form sodium-24: \[ \text{\(^{23}_{11}Na + \: ^{1}_{0}n \rightarrow \: ^{24}_{11}Na\)} \]

Step 2: Beta Decay Reaction

Next, we write the reaction for beta decay. Sodium-24 undergoes \(\beta^-\) decay to form magnesium-24, emitting a beta particle and an antineutrino: \[ \text{\(^{24}_{11}Na \rightarrow \: ^{24}_{12}Mg + \beta^- + \overline{u}_e\)} \]

Step 3: Understanding Half-Life

The half-life is the time taken for the radioactivity of a substance to decrease to half its initial value. We will use the activity data provided to determine the half-life of sodium-24.

Step 4: Initial and Final Activities

The initial activity (\(A_0\)) of the sample is \(2.54 \times 10^4\,\mathrm{dpm}\) at time \(t = 0\). We will analyze the activities to see when it reduces to half, which is \(1.27 \times 10^4\,\mathrm{dpm}\).

Step 5: Calculating Half-Life Using Data

Examining the data, the activity decreases to \(1.60 \times 10^4\,\mathrm{dpm}\) at \(t = 10 \text{ hours}\), and \(1.01 \times 10^4\,\mathrm{dpm}\) at \(t = 20 \text{ hours}\). Interpolating between these points, the half-life can be estimated slightly above \(10 \text{ hours}\).

Step 6: Exact Half-Life Calculation

Using the decay constant formula and activities: \[ A_t = A_0 \times e^{-\lambda t} \] Re-arranging to find \(t_{1/2}\): \[ \lambda = \frac{\ln(2)}{t_{1/2}} \] Calculating \(t_{1/2}\) using given activities, we set up a linear equation using logarithmic values to solve for an exact value. The calculations yield a half-life close to \(15 \text{ hours}\).

Key Concepts

Neutron Capture

In nuclear chemistry, neutron capture plays a critical role in nuclear reactions. It occurs when a nucleus collides with a neutron and absorbs it, forming a heavier nucleus. This process alters the atomic mass but not the atomic charge, hence the element remains the same but in a different isotope.

For instance, in the example provided with sodium-23 (Na), neutron capture results in the formation of sodium-24 (Na). The nuclear reaction can be written as:

• Initial nucleus: ext{Na-23}
• Neutron captured: ext{n}
• Resulting nucleus: ext{Na-24}

This process is essential in creating unstable isotopes which then undergo other nuclear processes, including beta decay, to achieve stability.

Beta Decay

Beta decay is one of the pivotal types of radioactive decay and occurs when an unstable nucleus transforms into a more stable one by emitting a beta particle. A beta particle is a high-energy, high-speed electron (or positron) that is ejected from the nucleus.

In the specific case of sodium-24, it undergoes beta-minus ( ext{β}^- ) decay to become magnesium-24 ( ext{Mg-24} ). During beta decay, a neutron is transformed into a proton, releasing an electron and an antineutrino:

• Initial Nucleus: ext{Na-24}
• Emitted Beta Particle: ext{β}^-
• New Nucleus: ext{Mg-24}

This process increases the atomic number by one but conserves the atomic mass, leading to a new element. Understanding beta decay is crucial in predicting the behavior and stability of nuclear substances.

Half-Life

The half-life of a radioactive isotope is a fundamental concept in nuclear chemistry, representing the time required for half of a sample's radioactive nuclei to decay. Knowing an element's half-life allows scientists to understand the behavior of radioactive materials over time.

To determine the half-life of sodium-24, we can use the activity data provided. Here, the initial activity is given as ext{2.54 × 10^4 } dpm (disintegrations per minute). The goal is to see when this value halves to ext{1.27 × 10^4 } dpm.

By examining the activity values over time, and using the decay constant ext{λ}, we apply the formula A_t = A_0 imes e^{-λt} , where A_0 is the initial activity and A_t is the activity at time t. Solving this gives us an accurate determination of the half-life, which is approximately 15 hours. This principle also helps in dating materials and understanding the longevity of different isotopes.

Radioactivity Measurement

Measuring radioactivity is crucial for studying nuclear reactions and radiation safety. The rate of radioactive decay is typically measured in terms of activity, which is the number of disintegrations per unit time. The unit of measurement is often in disintegrations per minute (dpm) or becquerels (Bq).

For example, when sodium-24 is removed from the reactor, its initial activity is recorded as 2.54 imes 10^4 dpm. Over time, as the sodium-24 decays, these measurements decline, providing data on the sample's radioactivity at various time intervals.

To ensure precise and reliable radioactivity measurements:

• Use accurate detectors that are sensitive to the type of radiation emitted.
• Regularly calibrate equipment to maintain measurement accuracy.
• Consider background radiation and subtract it from recorded data to find the true activity.
• Make multiple measurements for accuracy and precision.

These measurements play a vital role in assessing radiation exposure safety, developing nuclear technology, and performing scientific research.