All ordinary matter is made up of combinations of chemical elements, each with its own atomic number, indicating the number of protons in the atomic nucleus.

Additionally, elements may exist in different isotopes, with each isotope of an element differing in the number of neutrons in the nucleus.

The method of carbon dating makes use of the fact that all living organisms contain two isotopes of carbon, carbon-12, denoted 12C (a stable isotope), and carbon-14, denoted 14C (a radioactive isotope).

The ratio of the amount of 14C to the amount of 12C is essentially constant (approximately 1/10,000).

Then after time equals one half-life, we'd have 50% of our substance.

At time is equal to two half-lives, we'd have 25% of our substance, and so on and so forth.

A fossil found in an archaeological dig was found to contain 20% of the original amount of 14C. I do not get the $-0.693$ value, but perhaps my answer will help anyway.

If we assume Carbon-14 decays continuously, then $$C(t) = C_0e^,$$ where $C_0$ is the initial size of the sample. Since it takes 5,700 years for a sample to decay to half its size, we know $$\frac C_0 = C_0e^,$$ which means $$\frac = e^,$$ so the value of $C_0$ is irrelevant.and is now the principal source of information about the absolute age of rocks and other geological features, including the age of fossilized life forms or the age of the Earth itself, and can also be used to date a wide range of natural and man-made materials.Together with stratigraphic principles, radiometric dating methods are used in geochronology to establish the geological time scale.We have the negative change in the concentration of A, over the change in time, and that's equal to the rate constant k times the concentration of A squared. So instead of expressing this as an average rate, change in A over change in time, we can think about the instantaneous rate.So the rate of change of the concentration of A with respect to time. All right, and we can solve our differential equation and get a function, and the first thing that we do to solve a differential equation is to separate our variables. So we're going to integrate the left here, and we're going to integrate the right.- Let's say that our reaction here is second order in A. And when time is equal to zero, we're starting with our initial concentration of A.

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