Exercises: 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?

The principal quantum number $n$ labels the energy levels of the atom.

1. a) What values can $n$ take?
2. b) What happens to the energy of the level and the average size of the electron cloud when $n$ increases?

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?

In $1927$, Heisenberg stated his uncertainty principle. Explain what this principle states about the position and velocity of the electron, and why, because of it, physicists depict the electron as a "probability cloud" rather than a small ball following a trajectory.

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?

Since the 1960s, we have known that protons and neutrons are not elementary particles: they are made up of quarks.

1. a) How many quarks make up a proton? And a neutron?
2. b) What difference is there, in this respect, between those two particles and the electron?

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?

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.

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 outermost occupied shell?

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 ${}^{A}_{Z}\text{X}$.

Tooth enamel contains calcium and many toothpastes contain fluoride.

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?

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 a whole number if each chlorine atom has a whole number of particles in its nucleus.

In a PET scan, the patient receives glucose labeled with fluorine-18 ($Z = 9$), an isotope that emits positrons. When a positron meets an electron, they annihilate each other and two gamma photons are emitted in opposite directions, which are detected by the ring of the scanner. The diagram of the scanner is shown in the figure.

1. a) Calculate the number of neutrons in fluorine-18 and fluorine-19, the stable isotope of fluorine.
2. b) Why, by detecting the two photons at the same time, does the equipment know that the annihilation occurred along the line connecting the two detectors?
3. c) Tumors consume more glucose than healthy tissue: explain why PET allows them to be located.

Quarks have fractional electric charges: the up quark has $+\frac{2}{3}$ of the charge of the proton and the down quark, $-\frac{1}{3}$. The proton consists of two up quarks and one down quark (uud), and the neutron, of one up and two down (udd). The figure summarizes this composition.

1. a) Verify, by adding up the charges of its quarks, that the proton has a charge of $+1$ and the neutron, a charge of $0$.
2. b) Which combination of three up or down quarks would give a particle with a charge of $-1$?
3. c) And one with a charge of $+2$? Justify both answers by adding fractions.

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 part.

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.

Naturally occurring magnesium has three isotopes: magnesium-24 ($78.7\ \%$), magnesium-25 ($10.1\ \%$) and magnesium-26 ($11.2\ \%$).

1. a) Calculate the average atomic mass of magnesium as a weighted average.
2. b) Explain why the result is much closer to $24$ than to $26$.
3. c) Could the average mass of an element be equal to an integer? Explain in which case this would occur.

Consider these three species: $\text{F}^{-}$ ($Z = 9$), $\text{Ne}$ ($Z = 10$) and $\text{Na}^{+}$ ($Z = 11$).

1. a) Calculate the number of electrons of each one.
2. b) Are they the same element? Are they isotopes of each other? Justify both answers.
3. c) Write the common electron configuration of all three species in the order $1s$, $2s$, $2p$ and explain why they are said to be isoelectronic.

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?

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.

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, $E = m\times c^{2}$.

The copper in electrical cables is a mixture of two isotopes, copper-63 and copper-65, with an average mass of $63.5\ \text{u}$.

1. a) Without doing any calculations, state which of the two isotopes is more abundant and explain your reasoning.
2. b) Set up the weighted average equation and calculate the percent abundances of both isotopes.
3. c) In nuclear medicine, artificially produced copper-64 is sometimes used. Is it still copper? Explain what determines whether an atom is copper and not another element.

Unit challenge. How do we know what is inside matter if we cannot see an atom? Investigate the atomic scale and the transition from Thomson's model to the quantum model through a home experiment.

1. Experimental measurement: Cut 5 aluminum foil squares of increasing side lengths ($L = 5.0\text{ cm}$; $8.0\text{ cm}$; $10.0\text{ cm}$; $12.0\text{ cm}$; $15.0\text{ cm}$), with an uncertainty of $\pm 0.1\text{ cm}$. Measure the mass of each one on a precision scale ($\pm 0.01\text{ g}$).

2. Data processing: Construct a data table with area ($A = L^2$), mass ($m$), and associated uncertainties. Plot the graph of $m$ versus $A$. From the slope and the density of aluminum ($2.70\text{ g/cm}^3$), calculate the thickness of the foil and estimate how many stacked atoms it contains ($d \approx 0.28\text{ nm}$).

3. Reasoned proposal: Write a brief report: explain why Rutherford's $\alpha$ particles almost all passed through similar foils with minimal deflection while a few were strongly deflected or bounced back, and why these large deflections refuted Thomson's model, and justify how the current quantum model describes that atomic volume using orbitals and electron probability.

Evaluation criteria:

• Rigor in taking measurements, data table with uncertainties, and linear graph.

• Accuracy in calculating the thickness and estimating atomic layers.

• Scientific consistency when contrasting classical models with the quantum model.