Do all chlorine atoms have the same mass?

Mathematical induction is a mathematical proof technique. It is essentially used to prove that a property P(n) holds for every natural number n, i.e. for n = 1, 2, 3, and so on. The method of induction requires two cases to be proved:

1. The first case, called the base case, proves that the property holds for the number 1.
2. The second case, called the induction step, proves that, if the property holds for one natural number n, then it holds for the next natural number n + 1.

These two steps establish the property P(n) for every natural number n = 1, 2, 3, ...

Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. (via GIPHY)

Do all chlorine atoms have the same mass?

1. First we establish a base case for one chlorine atom (n=1). The case with just one chlorine atom is trivial. If there was only one chlorine atom in the universe, then clearly all chlorine atoms in that universe would have the same mass.
2. We then prove that if in any random set of n chlorine atoms every atom had the same mass, then in any random set of n+1 chlorine atoms every atom must also have the same mass. First, exclude the last chlorine atom and look only at the first n chlorine atoms; all these have the same mass since n chlorine atoms always have the same mass. Likewise, exclude the first chlorine atom and look only at the last n chlorine atoms. These too, must also have the same mass. Therefore, the first chlorine atom in the group has the same mass as the chlorine atoms in the middle, who in turn have the same mass as the last chlorine atom. Hence the first chlorine atom, middle chlorine atoms, and last chlorine atom have all the same mass, and we have proven that: If n chlorine atoms have the same mass, then n+1 chlorine atoms will also have the same mass.

We already saw in the base case that the rule ("all chlorine atoms have the same mass") was valid for n=1. The inductive step showed that since the rule is valid for n=1, it must also be valid for n=2, which in turn implies that the rule is valid for n=3 and so on. Thus in any group of chlorine atoms, all chlorine atoms must have the same mass. We have proof that in any universe, no matter how many chlorine atoms exist, all chlorine atoms must have the same mass.

Nevertheless, it is an experimental fact that it is false that all the atoms of the same element have the same mass. For instance, chlorine's atomic mass of 35.5 a.m.u. is an average of the masses of the different isotopes of chlorine. This is calculated by working out the relative abundance of each isotope. For example, in any sample of Chlorine 25% will be Cl-37 and 75% Cl-35, so there are chlorine atoms with different (35 a.m.u. and 37 a.m.u.) masses.

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook, Twitter or Instagram to give your answers.

Can an atom have a half-integer number of neutrons?

William Prout, by Henry Wyndham Phillips, 1820 - 1868 - From a miniature by Henry Wyndham Phillips, Public Domain, Link

Prout's hypothesis was an early 19th-century attempt to explain the existence of the various chemical elements through a hypothesis regarding the internal structure of the atom. In 1815 and 1816, the English chemist William Prout published two papers in which he observed that the atomic weights that had been measured for the elements known at that time appeared to be whole multiples of the atomic weight of hydrogen. He then hypothesized that the hydrogen atom was the only truly fundamental object, which he called "protyle", and that the atoms of other elements were actually groupings of various numbers of hydrogen atoms.

Prout's hypothesis was an influence on Ernest Rutherford, and that is the reason why he suggested in 1920 the name "protons" por the positive particles that live in the atomic nuclei. The name "proton" comes from the suffix "-on" for particles, added to the stem of Prout's word "protyle". Later, the English physicist Sir James Chadwick discovered the neutron. Both particles, proton and neutron, have almost the same mass (1 a.m.u.), which is much bigger that the electron mass, and that is why the elements known at Prout's time were measured to be whole multiples of the atomic mass of hydrogen, which is 1 a.m.u.

Nevertheless, nowadays we know that chlorine's atomic mass is 35.5 a.m.u. Since we know that each chlorine atom has 17 protons inside its nucleus, does it mean that chlorine atoms have a half-integer number of neutrons?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook, Twitter or Instagram to give your answers.

Why don't protons in a nucleus repel each other?

The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom.

A model of the atomic nucleus showing it as a compact bundle of the two types of nucleons:

protons (red) and neutrons (blue).

By Marekich - Own work (vector version of PNG image), CC BY-SA 3.0, Link

The atomic number Z of a chemical element is the number of protons found in the nucleus of an atom. Since neutrons are neutral while protons are positively charged with a charge that is, in absolute value, equal to the electron charge, the atomic number is identical to the charge number of the nucleus. In an uncharged atom, the atomic number is also equal to the number of electrons, and that is why the atomic number uniquely identifies a chemical element. For instance, Z=1 is called hydrogen and Z=6 carbon.

The world nuclide is referred to a 'species of nucleus', characterized by its number of protons Z, its number of neutrons N, and its nuclear energy state. Identical nuclei belong to one nuclide, for example each nucleus of the carbon-13 nuclide is composed of 6 protons and 7 neutrons. On the other hand, the members of the group of all the nuclides of the same elements are called isotopes. That is, the nuclides with equal atomic number, i.e., of the same chemical element but different neutron numbers, are called isotopes of that element.

Stable nuclides are nuclides that are not radioactive and so (unlike radionuclides) do not spontaneously undergo radioactive decay. It has been found that there are 80 elements with one or more stable isotope. For instance, carbon has 15 known isotopes, from carbon-8 to carbon-22, of which carbon-12 and carbon-13 are stable.

But, because of the fact all the protons have the same charge and are very closed to one another in the nucleus, one expects that they repel each other by a strong electric force, a force that must be much stronger than the force acting between the nucleus and the surrounding electrons. This force should make the nucleus explode. Therefore, no nuclide with more than one proton should be stable, that is, the only stable element should be hydrogen!

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers. 

Why does the death of a living being affect the decay of carbon-14?

Carbon-14 is a radioactive isotope of carbon with an atomic nucleus containing 6 protons and 8 neutrons. Carbon-14 decays into nitrogen-14 through beta decay:

By emitting a beta particle (an electron, e^-) and an electron antineutrino (ν_e), one of the neutrons in the carbon-14 nucleus changes to a proton and the carbon-14 nucleus becomes the stable (non-radioactive) isotope nitrogen-14.

The equation governing the decay of a radioactive isotope is

$$ N=N_0 e^{-\frac{t}{\tau}}$$

where N_o is the number of atoms of the isotope in the original sample (at time t = 0, when the organism from which the sample was taken died), and N is the number of atoms left after time t. On the other hand, the mean-life τ is the average or expected time a given atom will survive before undergoing radioactive decay.

Since the amount of carbon-14 inside a piece of wood or a fragment of bone decrease as the carbon-14 undergoes radioactive decay, measuring the amount of carbon-14 in a sample provides information that can be used to calculate when the animal or plant died. The mean-life of carbon-14 is 8267 years, so the equation above can be rewritten as:

Nevertheless, radioactive decay is a process that takes place inside the nucleus, so nor a change of temperature neither chemical reactions affect radioactive decay. Carbon-14 atoms inside a living being are decaying after and before the living being dies. So why is this method used efficiently to measure when the living being died? How do we know N_o, the amount of carbon-14 the living being had at the moment it died, if carbon-14 was also decaying when the plant or the animal was alive?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

If nuclear fusion is the reverse of fission, why is energy released in both processes?

Nuclear power is the use of nuclear reactions that release nuclear energy to generate heat. There are basically two ways to release energy from nuclei:

• nuclear fission, which is either a nuclear reaction or a radioactive decay process in which the nucleus of an atom splits into smaller parts (lighter nuclei).  This released energy is the one that is frequently used in steam turbines to produce electricity in nuclear power plants.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=486924

• nuclear fusion, which is a reaction in which two atomic nuclei fuse to form a heavier nucleus. Nuclear fusion reactors are not yet economically viable, but this technology is currently under research and it could become viable in a few decades.

By Wykis - Own work, based on w:File:D-t-fusion.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=2069575

If splitting a nucleus into two smaller nuclei releases energy, it seems that combining two smaller nuclei into one larger nucleus would require energy, not release it, because it is the inverse process. So, why can we obtain energy from both processes?

 Are you able to resolve this cognitive conflict?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.