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Chapter 25: Problem 68

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: (TABLE CANNOT COPY) (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

Short Answer

Expert verified
The half-life of sodium-24 can be determined from its radioactive decay data.

Step by step solution

01

Neutron Capture Reaction

The neutron capture reaction involves the absorption of a neutron by a nucleus. For sodium-23, this reaction can be represented as: \[ ^{23}_{11}\text{Na} + ^{1}_{0}\text{n} \rightarrow ^{24}_{11}\text{Na} \] This equation shows that when a neutron \((^{1}_{0}\text{n})\) is captured by sodium-23 \((^{23}_{11}\text{Na})\), it becomes sodium-24 \((^{24}_{11}\text{Na})\).
02

Beta Decay Reaction

Sodium-24 undergoes beta decay to transform into another element. The beta decay is represented by: \[ ^{24}_{11}\text{Na} \rightarrow ^{24}_{12}\text{Mg} + \beta^- + \overline{v} \] Here, \(^{24}_{11}\text{Na}\) decays into magnesium-24 \((^{24}_{12}\text{Mg})\), and a beta particle (\(\beta^-\)) and an antineutrino \((\overline{v})\) are emitted.
03

Understanding Half-life

The half-life is the time it takes for a radioactive substance to lose half of its radioactivity. We can calculate it using the given decrease in radioactivity over time.
04

Using Given Data to Calculate Half-life

Without the actual data from the table, here's how you would generally calculate the half-life: Measure the time interval during which the activity decreases to half of its initial value. The data would typically show this decrease, allowing calculation of the half-life for the reaction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Neutron Capture
Neutron capture is a nuclear reaction where an atomic nucleus absorbs a neutron. This process is crucial in the formation of isotopes. In our exercise, sodium-23 captures a neutron. The reaction can be written as:
  • Sodium-23 ( \(^{23}_{11}\text{Na}\) ) absorbs one neutron ( \(^{1}_{0}\text{n}\) ).
  • This changes the sodium-23 into sodium-24 ( \(^{24}_{11}\text{Na}\) ).
This change shows no charge shift, but the atomic mass number increases by one due to the extra neutron.
Neutron capture is key in nuclear reactions in reactors and stellar environments, enabling the creation of heavier elements from lighter ones.
Beta Decay
In beta decay, a neutron within an atomic nucleus is transformed into a proton. This process is accompanied by the emission of a beta particle (an electron) and an antineutrino. For sodium-24, it undergoes this process as follows:
  • Sodium-24 ( \(^{24}_{11}\text{Na}\) ) becomes magnesium-24 ( \(^{24}_{12}\text{Mg}\) ).
  • A beta particle ( \(\beta^-\) ) and an antineutrino ( \(\overline{v}\) ) are released.
Beta decay helps stabilize an unstable nucleus by altering its structure. The change effectively increases the atomic number by one, creating a new element. Understanding beta decay is fundamental in nuclear physics and has practical applications, especially in medical imaging.
Radioactivity
Radioactivity refers to the spontaneous emission of particles or energy from an unstable atomic nucleus. This process occurs naturally as radioactive elements seek greater stability.
In the context of sodium-24, radioactivity is evidenced by the emission of beta particles during beta decay. Radioactivity can involve:
  • The emission of alpha particles.
  • The emission of beta particles.
  • Gamma radiation.
The activity is typically measured in disintegrations per minute (dpm). In our case, sodium-24 had an initial radioactivity of \(2.54 \times 10^{4}\) dpm.
Radioactivity is a natural occurrence that's essential for understanding nuclear reactions, and it has peaceful applications in energy production and medicine.
Half-life
Half-life is the period it takes for half of a radioactive sample to decay. It's intrinsic to understanding radioactive decay processes, signifying the speed at which a substance becomes stable.
For sodium-24, its half-life can be determined by how quickly its radioactivity decreases. Generally, with a known starting activity, we observe how long it takes for this activity to halve.
  • For example, if a substance starts at \(2.54 \times 10^{4}\) dpm, time is tracked until the activity reads \(1.27 \times 10^{4}\) dpm.
  • The time required for this change is the half-life.
Understanding half-life is crucial in fields like archaeology for dating artifacts and in medicine for planning treatments involving radioactive materials.

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Most popular questions from this chapter

Iodine-131 is used to treat thyroid cancer. (a) The isotope decays by \(\beta\) -particle emission. Write a balanced equation for this process. (b) Iodine-131 has a half-life of 8.04 days. If you begin with \(2.4 \mu \mathrm{g}\) of radioactive \(^{131} \mathrm{I},\) what mass remains after 40.2 days?

There are two isotopes of americium, both with half-lives sufficiently long to allow the handling of large quantities. Americium-241, with a half-life of 432 years, is an \(\alpha\) emitter used in smoke detectors. The isotope is formed from \(^{239} \mathrm{Pu}\) by absorption of two neutrons followed by emission of a \(\beta\) particle. Write a balanced equation for this process.

You might wonder how it is possible to determine the half-life of long-lived radioactive isotopes such as \(^{238}\) U. With a half-life of more than \(10^{9}\) years, the radioactivity of a sample of uranium will not measurably change in your lifetime. In fact, you can calculate the half-life using the mathematics governing first-order reactions. It can be shown that a 1.0 -mg sample of \(^{238}\) U decays at the rate of \(12 \alpha\) emissions per second. Set up a mathematical equation for the rate of decay, \(\Delta N / \Delta t=-k N,\) where \(N\) is the number of nuclei in the 1.0 -mg sample and \(\Delta N / \Delta t\) is 12 dps. Solve this equation for the rate constant for this process, and then relate the rate constant to the half-life of the reaction. Carry out this calculation, and compare your result with the literature value, \(4.5 \times 10^{9}\) years.

Strontium-90 is a hazardous radioactive isotope that resulted from atmospheric testing of nuclear weapons. A sample of strontium carbonate containing \(^{90}\) Sr is found to have an activity of \(1.0 \times 10^{3} \mathrm{dpm} .\) One year later, the activity of this sample is \(975 \mathrm{dpm} .\) (a) Calculate the half-life of strontium-90 from this information. (b) How long will it take for the activity of this sample to drop to \(1.0 \%\) of the initial value?

A piece of charred bone found in the ruins of a Nativee \(e\) American village has a \(^{14} \mathrm{C} /^{12} \mathrm{C}\) ratio that is \(72 \%\) of the ratio found in living organisms. Calculate the age of the bone fragment. $$\left(t_{1/2} \text { for }^{14} \mathrm{C} \text { is } 5.73 \times 10^{3}\text { years.) }\right.$$
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