Question

\({ }_{94}^{239} \mathrm{Pu}\) decays with a half-life of 24,100 yr via emission of a \(5.25-\mathrm{MeV}\) alpha particle. If you have a \(1.00 \mathrm{~kg}\) spherical sample of \({ }_{94}^{299} \mathrm{Pu},\) find the initial activity in becquerels.

Short answer

Answer: The initial activity of the sample is approximately \(1.03 \times 10^{11}\;\mathrm{Bq}\).

Step-by-step solution

Finding the decay constant

We can find the decay constant (\(\lambda\)) by using the half-life formula: \(\lambda = \frac{\ln{2}}{t_{1/2}}\) where \(t_{1/2}\) is the half-life. In this case, \(t_{1/2}=24,100\;\mathrm{yr}\). First, we should convert years to seconds to use the proper SI unit: \(1\;\mathrm{yr} = 365.25\;\mathrm{days} \cdot 24\;\mathrm{hr} \cdot 60\;\mathrm{min} \cdot 60\;\mathrm{s}\) Now, we can find the decay constant: \(\lambda = \frac{\ln{2}}{t_{1/2}} = \frac{\ln{2}}{24,100\;\mathrm{yr}\times 365.25\;\mathrm{days} \cdot 24\;\mathrm{hr} \cdot 60\;\mathrm{min} \cdot 60\;\mathrm{s}}\)

Calculating the number of atoms

The mass of \({ }_{94}^{239} \mathrm{Pu}\) is given as \(1\;\mathrm{kg}\). We will find the number of atoms (\(N\)) in the sample using Avogadro's number (\(N_A\)), and the molar mass (\(M\)) of plutonium-239: We know that: \(N = \frac{\text{mass of the sample}}{M} \times N_A\) The molar mass of plutonium-239 is approximately \(239\;\mathrm{g/mol}\). Next, we need to convert the mass of the sample to grams (\(1\;\mathrm{kg}=1000\;\mathrm{g}\)). Avogadro's number \((N_A)\) is \(6.022 \times 10^{23}\;\mathrm{atoms/mol}\). Now, we can calculate the number of atoms: \(N = \frac{1000\;\mathrm{g}}{{239\;\mathrm{g/mol}}} \times 6.022 \times 10^{23}\;\mathrm{atoms/mol}\)

Calculating the activity

Now that we have the decay constant \(\lambda\) and the number of atoms \(N\), we can find the activity in Bq. The formula for activity is: \(A = \lambda \times N\) Substitute the values we found from Steps 1 and 2: \(A = \left(\frac{\ln{2}}{24,100\;\mathrm{yr}\times 365.25\;\mathrm{days} \cdot 24\;\mathrm{hr} \cdot 60\;\mathrm{min} \cdot 60\;\mathrm{s}}\right) \times \left(\frac{1000\;\mathrm{g}}{{239\;\mathrm{g/mol}}} \times 6.022 \times 10^{23}\;\mathrm{atoms/mol}\right)\) Finally, compute the activity: \(A \approx 1.03 \times 10^{11}\;\mathrm{Bq}\) Thus, the initial activity of the sample is approximately \(1.03 \times 10^{11}\;\mathrm{Bq}\).

Key concepts

Decay Constant

The decay constant, denoted by the symbol \(\lambda\), is a critical factor that quantifies the probability of decay of a radioactive atom per unit time. It can be understood as the inverse of the average lifetime of a radioactive particle. The decay constant is specific to each radioactive isotope and is constant over time. To calculate it, we use the formula \(\lambda = \frac{\ln{2}}{t_{1/2}}\), where \(t_{1/2}\) represents the half-life, the time it takes for half of the radioactive atoms to decay. Using the respective half-life of a radioactive substance, we can find the decay constant which then can be used to determine the activity or the rate of decay.

For plutonium-239 with a given half-life of 24,100 years, the decay constant calculation would involve converting the half-life into seconds, since the decay constant's unit is inverse seconds (s^-1). The conversion ensures consistency with the SI unit system used in scientific calculations.

Half-Life Formula

The half-life, symbolized as \(t_{1/2}\), is a term describing the time required for half of the radioactive atoms in a sample to undergo decay. It's a commonly used measure to express the stability or the rate of decay of a radioactive element. The half-life formula is intimately connected with the decay constant and is instrumental in understanding the decay process. In practice, the half-life allows us to predict how long a radioactive sample will remain active or how quickly it will reduce to a safe level.

The relationship between the half-life and the decay constant is given by \(t_{1/2} = \frac{\ln{2}}{\lambda}\). For instance, we can determine the rate at which plutonium-239 will decay based on its half-life. Understanding this fundamental concept provides a window into the timing of radioactive decay processes.

Avogadro's Number

Avogadro's number is a constant named after the scientist Amedeo Avogadro, and it represents the number of atoms or molecules in one mole of a substance. The value is approximately \(6.022 \times 10^{23}\) entities per mole. This enormous number provides the link between the macroscopic scale of materials that we can manipulate and measure in the laboratory to the microscopic scale of individual atoms or molecules.

When dealing with radioactive materials, Avogadro's number allows us to convert between the mass of a substance and the number of its constituent radioactive atoms. This conversion is crucial for calculating characteristics such as activity, which depends on the number of radioactive atoms present in a sample.

Becquerel

The Becquerel (Bq) is the SI unit of radioactivity named after the French physicist Henri Becquerel. One Becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. It's a measure of the rate at which a sample of radioactive material is decaying and thus helps quantify the intensity of radiation.

When we calculate the initial activity of a radioactive sample, such as a kilogram of plutonium-239, the result is expressed in Becquerels. Large numbers of Becquerels correspond to high levels of radioactivity and hence a greater potential for radiation exposure.

Alpha Particle Emission

Alpha particle emission is one type of radioactive decay where an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons, essentially a helium-4 nucleus. This process decreases the atomic number by two and the mass number by four, leading to the formation of a new element.

For instance, when plutonium-239 undergoes alpha decay, it loses an alpha particle to become uranium-235. Alpha particles have relatively low penetration power and can be stopped by a sheet of paper or the skin. However, if alpha-emitting substances are ingested or inhaled, they can cause significant biological damage.

Plutonium-239 Decay

Plutonium-239 (Pu-239) is a radioactive isotope of plutonium that decays via the emission of an alpha particle and transforms into uranium-235. With a half-life of 24,100 years, its decay process is relatively slow, which contributes to its use in nuclear reactors and weapons.

The activity of a plutonium-239 sample is calculated by determining the decay constant from its half-life and the total number of plutonium atoms present in the sample. This activity illustrates the rate at which Pu-239 emits radiation in the form of alpha particles and is a key factor in the handling and storage of this radioactive substance.