The Quantum Model and Subatomic Particles

In the current quantum model, we no longer speak of orbits, as Bohr did, but of orbitals. Explain the difference between an orbit and an orbital: what information does each provide about the electron, and which of the two concepts describes a definite trajectory?

Within each energy level, there are subshells. In this topic, we will study the s and p subshells: the s orbital is spherical, and each p orbital is shaped like two symmetrical lobes. Their shapes are shown in the figure.

1. a) What is the maximum number of electrons that can fit in an s orbital?
2. b) A p subshell has $3$ orbitals: how many electrons in total can fit in a p subshell?

A single orbital can hold a maximum of $2$ electrons and, according to the Pauli exclusion principle, they must differ in their spin: one with spin $+\frac{1}{2}$ and the other with spin $-\frac{1}{2}$ (you can imagine this as two senses of rotation about their own axis).

1. a) Using this rule, how many electrons can a p subshell, which has $3$ orbitals, hold at most?
2. b) How many electrons can fit in total in level $n = 2$, which has an s subshell and a p subshell?
3. c) A student writes: "The $2p$ orbital is the closed trajectory followed by the electrons in the second level." Correct the sentence using what you know about the quantum model.

State the electric charge (positive, negative or zero) and location within the atom (nucleus or electron shell) of the proton, neutron and electron. Which of the three particles have almost the same mass? Which is about $2000$ times lighter than the other two?

Hospital PET scanners work thanks to positrons.

1. a) What is a positron? How is it similar to, and different from, an electron?
2. b) What happens when a positron meets an electron?

Cosmic rays arriving at the upper atmosphere produce positrons, the antiparticle of the electron.

1. a) When a positron and an electron almost at rest annihilate each other, two photons are emitted in opposite directions. Explain, using the concept of conservation of momentum (if the system was almost at rest, its total momentum was zero), why a single photon cannot be emitted.
2. b) Explain why we do not find stable antimatter in our everyday environment.
3. c) The Large Hadron Collider (LHC) produces antiparticles and other new particles in high-energy collisions: why is so much energy needed to create massive particles? Think about the mass–energy equivalence, $E = m\times c^{2}$.

Nuclear power plants use uranium-235 as fuel, whereas the most abundant uranium in nature is uranium-238; both have atomic number $Z = 92$. The figure represents the two nuclei.

1. a) Explain what isotopes of an element are.
2. b) Calculate how many neutrons each of these two uranium isotopes has in its nucleus.

Iodine is an essential trace element added to table salt; its most common isotope has $Z = 53$ and $A = 127$.

1. a) Calculate the number of protons, neutrons and electrons in a neutral iodine atom.
2. b) Write its symbol using the notation ${}^{A}_{Z}\text{X}$.

Natural chlorine is a mixture of two isotopes: $75\ \%$ of chlorine-35 and $25\ \%$ of chlorine-37.

1. a) Calculate the average atomic mass of chlorine as a weighted average of the masses of its isotopes.
2. b) The periodic table gives a mass of $35.5\ \text{u}$ for chlorine: explain why this value is not an integer if each chlorine atom has an integer number of particles in its nucleus.

The average atomic mass of boron is $10.8\ \text{u}$ and its two natural isotopes are boron-10 and boron-11.

1. a) Call $x$ the fraction of boron-10 and set up the equation satisfied by the weighted average: $10\times x + 11\times(1 - x) = 10.8$.
2. b) Solve it and express the abundances of each isotope as percentages.
3. c) Check the result by calculating the average mass again using these abundances.

Tooth enamel contains calcium and many toothpastes contain fluorine.

1. a) The calcium atom ($Z = 20$) forms the $\text{Ca}^{2+}$ ion: how many electrons does this ion have?
2. b) The fluorine atom ($Z = 9$) forms the $\text{F}^{-}$ ion: how many electrons does this ion have?
3. c) Explain, considering their electron configurations, why atoms tend to gain or lose electrons until they achieve a full outer shell.

In cancer radiotherapy, cobalt-60 is used as a radioactive source with $Z = 27$.

1. a) Write the symbol for cobalt-60 using the notation ${}^{A}_{Z}\text{X}$.
2. b) Calculate its protons, neutrons and electrons, assuming a neutral atom.
3. c) What is the difference between cobalt-60 and cobalt-59, the stable isotope of the same element?

Calculate the number of protons, neutrons and electrons for each of these species and then answer:

1. a) ${}^{40}_{20}\text{Ca}^{2+}$
2. b) ${}^{37}_{17}\text{Cl}^{-}$
3. c) ${}^{35}_{17}\text{Cl}$

1. d) Which of them are isotopes of each other? Why?
2. e) Which ones have the same number of electrons?

Electrons occupy the subshells in this order: $1s$, $2s$, $2p$, $3s$, $3p$. Each s subshell can hold $2$ electrons and each p subshell, $6$.

1. a) Write the electron configuration of sodium ($Z = 11$) and of chlorine ($Z = 17$).
2. b) How many electrons does each of the two atoms have in its outer occupied shell?

Level $n = 1$ has a single s subshell; level $n = 2$ has an s subshell and a p subshell.

1. a) How many electrons fit at most in level $n = 1$? And in level $n = 2$?
2. b) The first row of the periodic table has $2$ elements and the second, $8$: relate these numbers to the results in the previous section.

The order of filling of the subshells is $1s$, $2s$, $2p$, $3s$, $3p$, $4s$ (note: after $3p$, $4s$ is filled). Each s subshell holds up to $2$ electrons and each p subshell, $6$.

1. a) Write the electron configuration of potassium ($Z = 19$).
2. b) Write the configuration of the sulfide ion, $\text{S}^{2-}$ ($Z = 16$), and check that it matches that of a noble gas: which one is it?
3. c) Potassium is a metal that reacts violently with water: explain, based on its configuration, why it easily loses an electron.