Carbon dating using exponential growth

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).

They may ask, "What's the difference between an isotope and an atom?

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However, we generally refer to isotopes of a particular element (e.g., Rubidium-87 (Pb)).

The number associated with an isotope is its atomic mass (i.e., protons plus neutrons).

Radioactivity and radioactive decay are spontaneous processes.

Students often struggle with this concept; therefore, it should be stressed that it is impossible to know exactly when each of the radioactive elements in a rock will decay.

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.The atoms that are involved in radioactive decay are called isotopes.In reality, every atom is an isotope of one element or another.Isotopes of an element are atoms that all have the same atomic number (or number of protons in the nucleus) but have different atomic masses (hence different numbers of neutrons in the nucleus).For example, all atoms of oxygen have 8 protons in the nucleus and hence have an atomic number of 8.When an organism dies, the amount of 12C present remains unchanged, but the 14C decays at a rate proportional to the amount present with a half-life of approximately 5700 years.