In this problem, you will verify the statement (in Section...

In this problem, you will verify the statement (in Section 29.4) that the $^{14} \mathrm{C}$ activity in a living sample is $0.25 \mathrm{Bq}$ per gram of carbon. (a) What is the decay constant $\lambda$ for $^{14} \mathrm{C} ?$ (b) How many $^{14} \mathrm{C}$ atoms are in $1.00 \mathrm{g}$ of carbon? One mole of carbon atoms has a mass of $12.011 \mathrm{g},$ and the relative abundance of $^{14} \mathrm{C}$ is $1.3 \times 10^{-12} .$ (c) Using your results from parts (a) and (b), calculate the $^{14} \mathrm{C}$ activity per gram of carbon in a living sample.

Key Concepts

Radioactive Decay

Radioactive decay is the spontaneous process by which unstable atomic nuclei lose energy by emitting radiation. This decay follows an exponential law, where the number of radioactive atoms decreases over time according to the decay constant. The process is probabilistic, meaning that the decay of individual atoms cannot be predicted exactly, but the overall behavior of a large ensemble of atoms is well-described by this exponential relationship.

Decay Constant

The decay constant is a fundamental parameter in the description of radioactive decay, representing the probability per unit time that a single nucleus will decay. It is inversely related to the half-life of the isotope, with a larger decay constant implying a faster decay process. The decay constant is used in mathematical models to predict how the number of undecayed nuclei decreases over time.

Radioactive Activity

Radioactive activity refers to the rate at which nuclear decays occur in a given sample, usually measured in units of becquerels (Bq), where one becquerel equals one decay per second. This concept links the microscopic properties of individual nuclei, governed by the decay constant, to a macroscopic observable quantity. It is central to applications like radiocarbon dating, where the measured activity provides insights into the age of materials.

Mole Concept and Avogadro's Number

The mole concept is a fundamental principle in chemistry that provides a bridge between the microscopic scale of atoms and the macroscopic scale of grams. In essence, one mole represents a specific number of atoms, defined by Avogadro's number. This concept is critical when calculating the total number of atoms in a given mass of a substance, enabling the transition from mass measurements to a count of individual atomic constituents.

Isotopic Abundance

Isotopic abundance describes the proportion of a particular isotope relative to the total amount of the element found in nature. It is expressed as a fraction or percentage and is crucial in calculations that require knowing the exact number of atoms of a specific isotope in a material. Understanding isotopic abundance is key when applying formulas related to radioactive decay and activity, especially in contexts like radiocarbon dating.

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