The half-life of 238U is 4.5 * 109 yr. A sample of rock of mass 1.6 g produces 29 dis/s. Assuming all the radioactivity is due to 238U, find the percent by mass of 238U in the rock.

Half-life is a fundamental concept in the field of nuclear chemistry and physics, which describes the time it takes for half of the atoms in a radioactive isotopic sample to decay. It is a measure of the rate at which a radioactive substance undergoes radioactive decay, and it is a key factor in understanding processes such as carbon dating, medical imaging, and nuclear power generation.
Understanding the half-life of an isotope allows scientists and engineers to predict how long a radioactive material will remain active or hazardous. The half-life of an isotope is a fixed value and does not depend on the amount of the substance or its chemical form. This makes it a valuable tool for a variety of applications, including geological dating and medical treatments that use radioactive isotopes.

The decay constant, represented by the Greek letter \(\lambda\), is a probability factor that describes the likelihood of a single atom's decay over a unit of time. It is inherently connected to an isotope's half-life through the relationship \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(\ln(2)\) is the natural logarithm of 2, and \(t_{1/2}\) is the half-life. The decay constant is a critical parameter in the equations modeling radioactive decay processes.
A high decay constant indicates that the substance decays more rapidly; this means a shorter half-life. Conversely, a small decay constant points to a slower decay process and a longer half-life. In calculations, understanding the decay constant is essential for accurately determining activity levels, dosages in medical treatments, and the longevity of radioactive materials.

Avogadro's number, denoted as \(N_A\), is a fundamental constant in chemistry known for defining the number of units in one mole of any substance. It is approximately equal to \(6.022 \times 10^{23}\) entities per mole. This number is crucial when dealing with chemical reactions and stoichiometry, as it allows scientists to convert between atoms, molecules, or ions and moles, providing a link between the microscopic scale of atoms and the macroscopic world we can measure in the laboratory.
The concept of Avogadro's number is applied extensively in radioactive decay calculations to establish the relationship between the decay rate or activity and the number of radioactive atoms present. It enables the translation of observable decay events, measured in disintegrations per second, into a quantity of matter in moles, thereby bridging the subatomic processes and measurable quantities.

The concept of percent by mass (also known as mass percent) is crucial in chemistry for expressing the concentration of a component in a mixture or a chemical compound. It is defined as the mass of the solute divided by the total mass of the solution or mixture, multiplied by 100% to convert the ratio into a percentage. The formula to calculate percent by mass is \(percent\;by\;mass = \left( \frac{mass\;of\;solute}{total\;mass\;of\;solution} \right) \times 100%\).
In radioactive decay problems, such as assessing the purity of a radioisotope in a sample, calculating the percent by mass allows us to determine how much of the sample's total mass is made up of the radioactive isotope in question. This percentage enables comparisons of samples or the determination of isotope concentration, which is critical in fields such as geology, pharmacology, and environmental science.