(a) What is the activity, in millicuries, of a 0.500 -g sample of \({ }_{93}^{237} \mathrm{~Np}\) ? (This isotope decays by \(\alpha\) -particle emission and has a half-life of \(2.20 \times 10^{6}\) yr.) (b) Write a balanced nuclear equation for the decay of \({ }^{237} \mathrm{~Np}\)

Understanding the activity of radioactive isotopes is essential for comprehending nuclear decay processes. Activity is a measure of the rate at which radioactive atoms decay, typically expressed in becquerels (Bq) where one Bq is one decay per second. In our example, the activity is measured in millicuries, an older unit where one curie is equivalent to 3.7 x 10¹⁰ Bq.

To calculate the activity of a sample, you need to determine the number of decaying atoms and the decay constant. The decay constant, related to the half-life of the isotope, represents the probability per unit time that a given nucleus will decay. The activity, A, can then be given by the equation:
\[ A = \lambda \times N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive atoms present in the sample. In practice, this translates to monitoring how often a radioactive material releases particles as it transforms into a more stable element.

Nuclear chemistry involves studying the properties and reactions of atomic nuclei. It encompasses various phenomena, including radioactivity, fission, and fusion. In the context of our problem, nuclear chemistry principles are applied to understand the decay of the isotope \( ^{237}_{93}Np \) through alpha radiation.

Alpha radiation occurs when an unstable nucleus emits an alpha particle (a helium nucleus consisting of 2 protons and 2 neutrons), which results in the formation of a new element with a lower atomic number. Balancing nuclear equations, like the one provided for neptunium decay, is crucial for maintaining the conservation of both atomic number and mass number in nuclear reactions.

Avogadro's number plays a central role in chemistry, linking the macroscopic world with the atomic scale. It is defined as the number of constituent particles, usually atoms or molecules, per mole of substance. Avogadro's number is approximately 6.022 x 10²³ particles per mole.

Using Avogadro's number, we can calculate the number of atoms in any given mass of a substance by the formula:
\[ N = \frac{\text{Weight}}{\text{Molar mass}} \times \text{Avogadro's number} \]
This conversion is vital when dealing with nuclear decay problems since we need the number of atoms present to determine the activity of our sample of neptunium-237.

The half-life of a radioactive isotope is the time required for half of the radioactive nuclei in a sample to undergo decay. Calculating the half-life is fundamental to understanding the longevity and stability of isotopes.

When provided with the half-life, we can determine the decay constant with the formula:
\[ \lambda = \frac{\ln 2}{T_{1/2}} \]
The decay constant obtained allows us to calculate the activity of the sample, as it informs us of the rate at which decay processes are occurring. For instance, the half-life of neptunium-237 is 2.20 x 10⁶ years, which tells us that over a very long period, the quantity of neptunium-237 will decrease to half of its original amount due to radioactive decay. This understanding of half-lives is crucial for applications ranging from carbon dating to nuclear medicine.