Calculate the activity \(R,\) in curies of \(1.00 \mathrm{g}\) of \(^{226} \mathrm{Ra}\). (b) Explain why your answer is not exactly 1.00 Ci, given that the curie was originally supposed to be exactly the activity of a gram of radium.

The decay constant, symbolized as \(\lambda\), is a crucial parameter in radioactivity that quantifies how quickly radioactive atoms undergo nuclear decay. It is uniquely defined for each radioactive isotope and is inversely proportional to its half-life. This means that a higher decay constant indicates a faster decay.
To calculate the decay constant, you use the half-life of the radioactive isotope, which is the time taken for half of the radioactive atoms in a sample to decay. The formula to find the decay constant using half-life is \(\lambda = \frac{\ln(2)}{t_{1/2}}\), where \(\ln(2)\) is the natural logarithm of 2 and \(t_{1/2}\) is the half-life of the isotope. This calculation helps scientists predict how a radioactive isotope will behave over time in terms of its radioactivity.

Avogadro's number, denoted \(N_A\), is a fundamental constant in chemistry and physics, representing the number of constituent particles, usually atoms or molecules, in one mole of a substance. The value of Avogadro's number is approximately \(6.022 \times 10^{23}\) particles per mole.
When dealing with radioactive substances like Radium-226, Avogadro's number allows us to connect the macroscopic mass of a sample with the microscopic number of atoms it contains. This is essential for calculating the activity of a radioactive sample, as it provides the total number of decaying particles. To find the number of atoms in any given sample, you divide its mass (in grams) by its molar mass and multiply the result by Avogadro's number. Rationalizing the very large scale of Avogadro's number in atomic scales versus the practical lab scales ensures that students grasp the extremes of scales involved in this concept.

Half-life is a term commonly used in nuclear physics to describe the time required for half of the radioactive nuclei in a given sample to undergo decay. It provides an average measure of the stability of a radioactive isotope. Each radioactive element has a characteristic half-life that can range from fractions of a second to millions of years.
In the context of Radium-226, the half-life is a significant figure as it directs the rate of decay, and therefore, the activity of the isotope. By understanding the half-life, scientists and engineers can make predictions about the behavior of the material over time and make safety calculations for its handling and disposal. Half-life also plays a crucial role in the fields of archaeology and geology for age estimation using dating techniques.

Nuclear decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This process can include the release of alpha particles, beta particles, gamma rays, and other subatomic particles. It's an intrinsic natural process that can't be influenced by external conditions such as temperature or pressure.
Understanding nuclear decay allows us to calculate the activity of a radioactive sample, which is the number of decays occurring per second. The activity of a sample is directly proportional to the number of unstable nuclei and the decay constant of the isotope. By analyzing the trends in nuclear decay, physicists and chemists can gauge the potent energy encapsulated within radioactive materials, facilitating uses in medicine, industry, and electricity generation, while also respecting the need for meticulous safety protocols.