## Sunday, February 12, 2023

### Isotopes: UNIZOR.COM - Physics4Teens - Atoms - Nucleus and Electrons

Notes to a video lecture on http://www.unizor.com

Isotopes

Isotopes are atoms that have the same number of protons (atomic number Z) but different number of neutrons (N).
Since the number of electrons and their distribution among shells and subshells are the same for different isotopes of atoms with the same atomic number, these isotopes have practically the same chemical properties.

At the same time, since the number of neutrons of different isotopes of atoms with the same atomic number is different, certain physical properties of these atoms might be quite different.

Identification of different isotopes involves specifying their atomic mass.
For example, carbon has 6 protons and 6 electrons, but the number of neutrons can be 6, 7 or 8 in its nucleus with atomic masses of these isotopes, correspondingly,
A=Z+N=6+6=12,
A=Z+N=6+7=13 and
A=Z+N=6+8=14.
These isotopes are named, according to their atomic mass, carbon-12, carbon-13 and carbon-14.

Another example: uranium-235 (symbol 92235U) with 92 protons, 92 electrons (atomic number is Z=92) and 143 neutrons (N=143), which has atomic mass A=235, and uranium-238 (symbol 92238U) with the same number of protons, electrons and the same atomic number Z=92, but with 146 neutrons (N=146) and atomic mass A=238.

In most cases one particular isotope of an atom is sufficiently stable and can be found in nature, while other isotopes of the same atom might be more or less radioactive, that is their atoms break with time into other (smaller) atoms - a process called radioactive decay.

The role of neutrons in a nucleus is to stabilize it. Since protons are positively charged and repel each other, neutrons serve as a buffer between them and, therefore, their number is usually equal or greater than the number of protons.
Obviously, so-called strong forces inside a nucleus that bind together all particles (protons to protons, protons to neutrons, neutrons to neutrons) are the main forces that hold a nucleus as a whole.

Interestingly, most nuclei with even atomic number Z and even number of neutrons N demonstrate more stability than those with either Z or N or both are odd.

As an example of the usage of isotopes, let's describe the determination of age of certain samples found by archeologists using radiocarbon dating.

The most stable and abundant isotope of carbon is the one with 6 protons and 6 neutrons in a nucleus - carbon-12 612C.
This carbon isotope plays extremely important role in organisms, as carbon is one of the main building blocks of all living beings on Earth.

Firstly, let's examine where carbon-14 comes from.
High energy cosmic rays bombard the Earth and break atoms of the upper layer of atmosphere, releasing some elementary particles, electrons, protons and neutrons.

Next, some neutrons (01n) hit the atoms of nitrogen (714N) in the air causing production of carbon-14 (614C) and hydrogen (11H) in the following nuclear reaction:
714N + 01n → 614C + 11H
The reaction above describes the process of a neutron hitting an atom of nitrogen, kicking out a proton and an electron that connect to each other in the form of an atom of hydrogen, while replacing a missed proton with itself, thereby reducing the atomic number by one, while retaining the atomic mass, which transforms it into an isotope carbon-14.

Now carbon-14 is formed in the air in some small quantity and participates in the cycle of life. Any living organism absorbs it the same way it absorbs regular carbon-12 during its life time and, therefore, it's present in some quantity inside this organism in proportion to regular carbon-12 similar to its proportion that exists in nature.

As we know, carbon-14 is not a stable element, it decays with a half-life about 5730 years.

The process of decaying is rather complex, but can be described in an oversimplified form as follows.
A neutron in the nucleus of unstable carbon-14 transforms into a pair proton+electron. This does not change the electric neutrality of the atom, does not change the atomic mass, it remains 14, but atomic number increases by 1, thereby creating an atom of stable nitrogen.

This nuclear reaction can be described (in an oversimplified form) as
614C → 714N + e + ?
where ? signifies additional participants in this transformation that we cannot discuss at this point because it requires knowledge of other elementary particles beyond the main ones (electron, proton and neutron).

While an organism lives, the relative amount of carbon-14 in it is maintained on the same level. Whatever is decayed is replenished from the outside world as part of the organism's existence.
As soon as life stops, absorption of carbon-14 stops and, whatever is present in the organism is no longer renewed, but decays according to the laws of half life.

If the amount of carbon-14 in the dead tree is 1/2 of whatever is normal, the tree died approximately 5730·1=5730 years ago.
If the amount of carbon-14 in the dead tree is 1/4 of whatever is normal, the tree died approximately 5730·2=11460 years ago.
If the amount of carbon-14 in the dead tree is 1/8 of whatever is normal, the tree died approximately 5730·3=17190 years ago.
etc.
In general, if the amount of carbon-14 in the dead tree is 1/N of whatever is normal, the tree died approximately 5730·log2N years ago.