Question

What weight of \(\mathrm{C}^{14}\) will have radioactivity one curie if \(\lambda\) (disintegration constant) is \(4.4 \times 10^{-12} \mathrm{sec}^{-1}\) ? (a) \(3.7 \times 10^{-6} \mathrm{~kg}\) (b) \(51 \times 10^{-3} \mathrm{~kg}\) (c) \(1.96 \times 10^{-4} \mathrm{~kg}\) (d) \(1.7 \times 10^{-6} \mathrm{~kg}\)

Short answer

The weight of C-14 for one curie is 1.96 x 10^-4 kg. Answer: (c).

Step-by-step solution

Understand the Relationship between Curie and Disintegration Rate

The Curie (Ci) is a unit of radioactivity, defined as 3.7 x 10^10 disintegrations per second. Thus, 1 Ci = 3.7 x 10^10 disintegrations per second.

Define Disintegration Rate using Mass, Avogadro's Number, and Disintegration Constant

The disintegration rate (R) of a sample is related to the number of radioactive nuclei (N) by the equation R = λN, where λ is the disintegration constant. N can be found with N = (mass / molar mass) x Avogadro's number (6.022 x 10^23 atoms/mol).

Formulate Equation with Given Values

Given λ = 4.4 x 10^-12 s^-1, equate 3.7 x 10^10 = λN: 3.7 x 10^10 = 4.4 x 10^-12 x N.

Solve for N (Number of Nuclei)

Rearrange the equation to solve for N: N = (3.7 x 10^10) / (4.4 x 10^-12) = 8.409 x 10^21 nuclei.

Calculate Mass using N, Molar Mass of C-14, and Avogadro's number

The molar mass of C-14 is approximately 14 g/mol or 14 x 10^-3 kg/mol. Using N = (mass / molar mass) x Avogadro's number: mass = (N x molar mass) / Avogadro's number. Substitute the known values: mass = (8.409 x 10^21 x 14 x 10^-3 kg/mol) / 6.022 x 10^23 atoms/mol = 1.96 x 10^-4 kg.

Key concepts

Disintegration Constant

The disintegration constant, often denoted as \( \lambda \), is a fundamental concept in nuclear physics that describes the decay process of radioactive substances. It is the probability per unit time that a given nucleus will decay. This constant provides insight into how quickly a radioactive substance undergoes disintegration. The smaller the disintegration constant, the longer it takes for half of the radioactive nuclei in a sample to decay.

Understanding the disintegration constant is essential for calculating radioactive decay rates. It helps us determine how active a substance is by relating to the disintegration rate or activity which refers to the number of decays per second. In mathematical terms, the relationship between the disintegration constant \( \lambda \) and the activity \( R \) is expressed by the equation:

• \( R = \lambda N \)

where \( N \) is the number of radioactive nuclei present. This relationship enables the calculation of how many atoms decay in a specific timeframe, assisting in understanding the material's radioactivity level.

Curie

The Curie, symbolized as Ci, is an older but still widely used unit of radioactivity. Named after the pioneering scientist Marie Curie, it quantifies the amount of radioactive decay that occurs in a given sample. One curie is defined as exactly 3.7 x 10^10 disintegrations per second, which reflects the decay rate of radioisotopes that are equivalent to about one gram of radium-226.

This unit of measurement allows scientists and industries to compare the radioactivity levels of different substances plainly. Its use can be seen in various fields, including medicine for radiotherapy, and nuclear physics for quantifying decay rates of unstable isotopes. The curie helps provide a tangible scale for expressing levels of radioactivity, contributing to safety standards and legal thresholds in handling materials.

Avogadro's Number

Avogadro's number, denoted by \( N_A \), is a fundamental constant in chemistry and physics. It represents the number of atoms, molecules, or particles in one mole of a substance. The value of Avogadro's number is approximately 6.022 x 10^23 entities per mole.

This number plays a crucial role in converting between atoms/molecules and moles, thus linking the microscopic world of atoms with the macroscopic world we can measure. In the context of radioactivity, Avogadro's number assists in determining the number of nuclei in a given mass. For example, while calculating the number of radioactive nuclei in a sample, we use:

• \( N = \frac{\text{mass}}{\text{molar mass}} \times N_A \)

This enables us to transition effectively between the microscopic properties and measurable mass in experiments or problem-solving scenarios.