In nature a decay chain starts with \(\mathrm{Th}^{232}\) and finally terminates at \(\mathrm{Pb}^{208}\). A thorium ore sample was found to contain \(6.72 \times 10^{-5} \mathrm{ml}\) of \(\mathrm{He}\) (at \(273 \mathrm{~K}\) and \(\left.1 \mathrm{~atm}\right)\) and \(4.64 \times 10^{-7} \mathrm{~g}\) of \(\mathrm{Th}^{232}\). Find the age of the sample assuming that source of He to be only due to decay of \(\mathrm{Th}^{232}\). Also assume complete retention of He within the ore. \(\left(t_{1 / 2}\right.\) of \(\mathrm{Th}^{232}=1.38 \times 10^{10}\) years, \(\log 2=0.3)\) (a) \(2.3 \times 10^{10}\) years (b) \(2.3 \times 10^{9}\) years (c) \(4.6 \times 10^{9}\) years (d) \(9.2 \times 10^{9}\) years

Radioactive decay is a natural process by which an unstable atomic nucleus loses energy by emitting radiation. This transformation results in the creation of different elements or isotopes. There are several types of decay, including alpha, beta, and gamma decay. In the context of our exercise, Thorium-232 undergoes alpha decay, producing Helium and other decay products over time.

In alpha decay, which is relevant to Thorium-232, the nucleus ejects an alpha particle (which is essentially a Helium nucleus) and transforms into a different element. The rate of this decay is described by the half-life, which indicates the time it takes for half of a sample of radioactive substance to decay. The concept of half-life is crucial in determining the age of geological samples, like the thorium ore in our original problem, by measuring the ratio of the remaining radioactive material to the decay products (in this case, Helium).

The ideal gas law is a crucial equation in chemistry and physics that relates the pressure, volume, temperature, and amount (in moles) of an ideal gas. It is often expressed as PV=nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature in Kelvin.

Determining the moles of a gas, like Helium in the exercise, can be done using this law, assuming standard temperature and pressure conditions (STP, which are 273 K and 1 atm). At STP, 1 mole of an ideal gas occupies 22.4 liters. Thus, by knowing the volume of gas at these conditions, as given in the exercise, we can calculate the amount of gas in moles. This calculation is a vital step in linking the physical measurement of the gas to the amount of Thorium decayed.

Half-life is defined as the time required for half of the radioactive nuclei in a sample to undergo decay. This constant value is unique for each radioactive isotope. For example, Thorium-232 has a half-life of about 14 billion years, meaning that after this period, only half of the original sample would remain undecayed.

The half-life concept allows scientists to understand the rate of decay and to estimate the length of time a radioactive substance has been decaying. In the context of our exercise, by comparing the amount of decay product (Helium) to the remaining Thorium-232, and knowing the half-life of Thorium-232, we can estimate the age of the sample. It's a profound tool in the fields of geology, archaeology, and even cosmology to date objects and events up to billions of years old.

The decay formula is a mathematical representation of the radioactive decay process. It connects the initial number of radioactive nuclei, the remaining amount after a certain period, and the half-life of the substance. It can be expressed as N = N_0 * (1/2)^(t/T), where N is the current number of atoms, N_0 is the initial number of atoms, t is the time elapsed, and T is the half-life of the radioactive isotope.

By rearranging this formula, it's possible to solve for the elapsed time if the half-life and the ratio of the remaining to the initial nuclei are known. In our exercise, we manipulate the formula to find the age of the thorium sample, using the logarithms of the ratios and the known half-life. It's essential to understand and apply this formula correctly, as it forms the basis of the calculations needed for radioactive dating techniques, such as the one used to date the thorium ore.