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 do nonmetals have both positive and negative oxidation numbers?

The chemical elements can be broadly divided into metals and nonmetals according to their tendency to loose or gain electrons:

• Atoms that belong to metallic elements tend to loose electrons. When they loose electrons, they become cations, positive ions with a charge that equals the number of electrons they have lost. That number is given by the oxidation number. For instance, sodium's oxidation number is +1, while calcium's oxidation number is +2.
• On the other hand, atoms that belong to nonmetallic elements tend to gain electrons, so they become anions, ions with a negative charge that equals the number of electrons they have gain. For instance, fluorine tends to gain one electron and becomes F^-. That is why it has oxidation number -1. 

But, as we can see in the following periodic table, most nonmetals have both positive and negative oxidation number:

Why do nonmetals have both positive and negative oxidation number if they always tend to gain electrons?

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 do metals seem colder although they have the same temperature?

By Anonimski - Own work, CC BY-SA 3.0, Link

All the objects that have been inside your room for more than one hour are at room temperature. That is because heat flows from hotter to colder objects, so if you put a cold object inside the room, heat will flow to it until it reaches room temperature.

If we touch a piece of metal that is inside the room it feels cold. But when we touch the other objects of the room they don't feel as cold. Why is that? Why do metals seem colder than the other objects in the room although they have the same temperature?

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.

What is the difference between a mixture and a compound?

According to most textbooks:

• a compound is an entity consisting of two or more atoms, commonly from different chemical elements, which associate via chemical bonds.
• On the other hand, a mixture is a material made up of two or more different substances which are mixed but are not combined chemically.

So the difference is that a mixture refers a physical combination of substances, whereas a compound refers to a chemical combination.

By Chris 73 / Wikimedia Commons, CC BY-SA 3.0, Link

But these definitions do not say anything unless we establish the difference between a chemical and a physical combination and, according to the same textbooks:

• a chemical process is a method or means of somehow changing one or more compounds,
• whereas physical changes are changes affecting a substance, but not its chemical composition, because they do not change chemical bonding.

As you probably have already realized, the definitions are circular! We put two substances together. If the process is not chemical, what we obtain is a mixture, not a compound. But we defined a non-chemical process as the one where the compounds are still the same compounds, but mixed. Who is Alice? She is Bob's cousin. And who is Bob? Alice's cousin. We still do not know who is Alice!

Are you able to give a definition of compounds and mixtures that is not circular? How can we define chemical process without saying that a chemical process is different from a physical one in that compounds change?

Notice the distinction cannot come from the physical properties if the substance, because the physical properties of a mixture may differ from those of the components. In addition, evolved or absorbed heat cannot be the solution because, both in chemical reactions and in mixtures, heat is either evolved (an exothermic process) or absorbed (an endothermic process).

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 do ice cubes melt faster in fresh water than in salt water?

The melting point of a solid is the temperature at which it changes state from solid to liquid at atmospheric pressure. When considered as the temperature of the reverse change from liquid to solid, it is referred to as the freezing point.

The freezing point of a solvent is depressed when another compound is added, meaning that a solution has a lower freezing point than a pure solvent. This phenomenon is used in technical applications to avoid freezing, for instance by adding salt or ethylene glycol to water. If you live in a place that has lots of snow and ice in the winter, then you have probably seen the highway department spreading salt on the road to melt the ice.

Now, let us consider the following experiment:

1. Make two almost identical ice cubes.
2. Mix 1 teaspoon of salt in an 8 oz. cup of water. This will be our salt water cup.
3. Fill a 8 oz. cup with water, but with no salt added. This will be our fresh water cup
4. Place one ice cube into each cup simultaneously. Which ice cube do you predict would melt the fastest?

Naively, one would think that, according to the previous information, since salt lowers the freezing/melting point of water, the ice cube in the salt water cup should melt the fastest.

Nevertheless, if you carry out the experiment, it leaves no doubt. The ice cube in the fresh water cup melts faster!

Why do ice cubes melt slower in salt water? 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.

I will give you a clue: repeat the experiment, but this time, after you place the ice cubes in the cups, wait 30 seconds and add a couple of drops of food coloring to each cup without disturbing the water in the cups.

Is hydrogen atom in bifluoride connected by two covalent bonds?

In chemistry, a valence electron is an electron that is associated with an atom, and that can participate in the formation of a chemical bond. In a single covalent bond, both atoms in the bond contribute one valence electron in order to form a shared pair. For instance, hydrogen atoms have one valence electron, while oxygen atoms have two. That is why a water molecule contains one oxygen and two hydrogen atoms that are connected by singles covalent bonds:

By Dbc334 (first version); Jynto (second version) - File:Water-3D-vdW.png, Public Domain, https://commons.wikimedia.org/w/index.php?curid=1332739

So the Lewis structure of water is H-O-H.

But let us consider a more complex example: bifluoride. Bifluoride is an inorganic anion with the chemical formula [HF₂]⁻.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=1496287

It is not a strange anion. Some [HF₂]^- salts are common, examples include potassium bifluoride (KHF2) and ammonium bifluoride ([NH4][HF2]).

As shown in the figure above, the structure of the anion is symmetric, with the hydrogen situated in the mid-point of the F-F distance. In addition, the H-F contacts in this ion are very short, like in covalent bonds, and the corresponding H.F interactions are also strong enough to be classified as covalent bonds. So the corresponding Lewis structure of the anion should be [F-H-F]^-. But, how is this possible? Hydrogen atoms have one valence electron. They cannot be connected by two covalent bond!

Are you able to give an explanation? Accept the challenge!

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.