Electronic structure and interaction with electromagnetic radiation

Passing the light from a hydrogen discharge lamp through a prism produces a few colored lines on a black background, whereas light from a filament bulb breaks down into a continuous rainbow. The figure compares the two spectra:

1. a) What is the difference between a continuous spectrum and a line spectrum?
2. b) Which of the two is an atomic emission spectrum? What information does it provide about the element producing it?

Electrons in an atom can only occupy certain energy levels, which are different in each element, as illustrated in the figure:

Explain, based on this idea:

1. a) Why each element emits light of characteristic colors and not others.
2. b) Why the set of these lines serves to identify an element, just as a fingerprint identifies a person.

Fireworks contain strontium salts (red light), barium salts (green light) and sodium salts (yellow light).

1. a) What happens to the electrons in these atoms when the combustion of gunpowder heats the salts?
2. b) Where does the colored light we see come from? Explain the full process using the concept of transitions between energy levels.

An electron in an atom is in level $n = 1$.

1. a) What needs to happen for it to move to level $n = 3$? Does the atom absorb or release energy in this process?
2. b) When the electron falls back to level $n = 1$, what does the atom emit? Draw the two processes in your notebook using arrows between the levels.

You can do it on paper and send a photo with your answer.

In $1814$, Joseph Fraunhofer observed that the spectrum of the Sun is crossed by hundreds of dark lines: they appear at exactly the same wavelengths that elements emit in the lab, but in absorption, because the gases in the solar atmosphere absorb that light. The diagram compares the two spectra:

1. a) Explain, using the concept of energy levels, why an element absorbs exactly the same wavelengths that it is capable of emitting.
2. b) Two of the dark lines of the Sun coincide with the yellow doublet of sodium: what conclusion can be drawn about the solar atmosphere?
3. c) Explain how it is possible to know the chemical composition of a distant star without ever retrieving a sample of it.

Many old street lamps used sodium vapor lamps, which give a very intense yellow light.

1. a) Why is their light practically monochromatic (single-colored), unlike the white light of a filament bulb?
2. b) One of these street lamps illuminates a road and everything appears in shades of yellow and gray: explain, using the concept of a line spectrum, why car colors are harder to distinguish than under white light.

The characteristic yellow line of sodium has a wavelength of $\lambda = 589\ \text{nm}$. Light travels in a vacuum at $c = 3\times10^{8}\ \text{m/s}$.

1. a) Express the wavelength in meters.
2. b) Calculate the frequency of this light using $c = \lambda\times\nu$.

The energy of a photon is $E = h\times\nu$, with Planck's constant $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$.

1. a) Calculate the energy of the sodium yellow photon, of frequency $\nu = 5.1\times10^{14}\ \text{Hz}$.
2. b) An ultraviolet photon has a higher frequency than visible light: is its energy greater or smaller? Justify your answer using the formula.

Germicidal lamps in operating rooms emit ultraviolet light of $\lambda = 254\ \text{nm}$. Data: $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$ and $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the energy of one of these photons using $E = \frac{h\times c}{\lambda}$.
2. b) UV photons damage the DNA of microorganisms while visible light photons do not: relate this fact to the energy calculated in the previous part.

A fireworks display simultaneously emits red ($\lambda \approx 700\ \text{nm}$), green ($\lambda \approx 530\ \text{nm}$) and blue ($\lambda \approx 450\ \text{nm}$) light. The figure positions the three lines in the visible spectrum:

1. a) Arrange the three colors in order of increasing frequency, reasoning from $c = \lambda\times\nu$.
2. b) Then arrange them in order of increasing photon energy, using $E = h\times\nu$: which color corresponds to the most energetic electron transitions?

The yellow light of sodium is actually a doublet: two very close lines of $\lambda_1 = 589.0\ \text{nm}$ and $\lambda_2 = 589.6\ \text{nm}$. Data: $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$ and $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the energy of each of the two photons using $E = \frac{h\times c}{\lambda}$.
2. b) Calculate the energy difference between them.
3. c) This difference corresponds to the energy gap between two subshells of the sodium atom: what does the existence of the doublet reveal about the structure of the atom's energy levels?

Red LEDs ($\lambda = 660\ \text{nm}$) have existed since the 1960s, but the blue LED ($\lambda = 470\ \text{nm}$) was only achieved in the 1990s, an advance awarded the Nobel Prize in Physics in $2014$. Data: $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$ and $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the energy of the red photon and that of the blue photon.
2. b) How many times more energetic is the blue photon than the red one?
3. c) An LED emits light when electrons jump a characteristic energy "step" of the semiconductor: using the previous results, explain why the blue LED was technologically much harder to manufacture than the red one.

The first three energy levels of hydrogen are $E_1 = -13.6\ \text{eV}$, $E_2 = -3.4\ \text{eV}$ and $E_3 = -1.5\ \text{eV}$. When an electron drops from one level to a lower one, it emits a photon whose energy is exactly the difference between the two levels. The diagram shows the three levels and the two transitions:

Data: $1\ \text{eV} = 1.602\times10^{-19}\ \text{J}$, $h = 6.63\times10^{-34}\ \text{J}\times\text{s}$, $c = 3\times10^{8}\ \text{m/s}$.

1. a) Calculate the energy emitted in transitions $E_3 \rightarrow E_2$ and $E_2 \rightarrow E_1$, in $\text{eV}$ and in joules.
2. b) Calculate the wavelength of each photon in $\text{nm}$ and state which one falls in the visible range ($390$–$750\ \text{nm}$) and which one in the ultraviolet. Transition $E_3 \rightarrow E_2$ is the famous red H-alpha line of the Balmer series.
3. c) Using your results, explain why the spectrum of hydrogen consists of discrete lines and is not continuous.

A highlighter glows brightly under the ultraviolet light of a nightclub: its molecules absorb invisible UV photons and emit brightly colored visible light.

1. a) Which photon is more energetic, the absorbed one (UV) or the emitted one (visible)? Justify your answer using $E = \frac{h\times c}{\lambda}$.
2. b) Propose an explanation using the concept of transitions between energy levels: the electron goes up in a single transition, but comes down in two smaller transitions, losing energy along the way.
3. c) Could a material emit blue light when illuminated with red light? Reason your answer in terms of energy.

The order of filling of subshells is $1s$, $2s$, $2p$, $3s$, $3p$, with maximum capacities of $2$ electrons in each s subshell and $6$ in each p subshell, as shown in the diagram:

1. a) Write the electron configuration of nitrogen ($Z = 7$).
2. b) Write the electron configuration of magnesium ($Z = 12$).

The number of the highest energy level occupied by electrons corresponds to the period of the element in the periodic table.

1. a) Which period is an atom in if its configuration ends in $3s^1$? Which element is it?
2. b) And which period is an atom in if its configuration ends in $2p^5$? Identify it using the periodic table.

The electrons in the outermost level, called valence electrons, determine the group of the element.

1. a) How many valence electrons does an atom whose configuration ends in $3s^2 3p^4$ have? Which group does it belong to?
2. b) And one whose configuration ends in $3s^2 3p^3$? Name both elements.

After the $3p$ subshell, $4s$ is filled before $3d$, as shown in the energy level diagram:

Maximum capacities: s, $2$ electrons; p, $6$; d, $10$.

1. a) Write the electron configurations of potassium ($Z = 19$) and calcium ($Z = 20$).
2. b) Write the electron configuration of iron ($Z = 26$), taking into account that after $4s$ is filled, $3d$ starts to fill.

1. a) Write the electron configuration of the ions $\text{Na}^{+}$ ($Z = 11$) and $\text{Cl}^{-}$ ($Z = 17$) and verify that each matches that of a noble gas: which noble gases are they?
2. b) Argon, the ion $\text{K}^{+}$ ($Z = 19$) and the ion $\text{Ca}^{2+}$ ($Z = 20$) have the same number of electrons: how many? What are species that have the same number of electrons called?

An element has the electron configuration $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^3$.

1. a) What is its atomic number? Which element is it?
2. b) In which period and in which group of the periodic table is it located? Justify both answers using its configuration.

The shorthand electron configuration of an element is $[\text{Ar}]\ 4s^2 3d^{10} 4p^5$.

1. a) Add up the electrons to find its atomic number and identify the element using the periodic table.
2. b) How many valence electrons does it have? What ion does it tend to form? Justify your answer by considering the configuration of the next noble gas.

An element has atomic number $Z = 37$.

1. a) Write its electron configuration (filling order: $1s$, $2s$, $2p$, $3s$, $3p$, $4s$, $3d$, $4p$, $5s$).
2. b) Deduce its period and group, and identify the element using the periodic table.
3. c) Predict, justifying your answer from the configuration: how many valence electrons it has, which ion it forms most easily, and whether it will react violently with water.

Iron ($Z = 26$) has the electron configuration $[\text{Ar}]\ 4s^2 3d^6$.

1. a) Atoms lose the electrons from the highest shell first: write the electron configuration of the $\text{Fe}^{2+}$ ion and that of the $\text{Fe}^{3+}$ ion.
2. b) A half-filled subshell is particularly stable: what configuration does $\text{Fe}^{3+}$ reach in its $3d$ subshell? Use this fact to explain why $\text{Fe}^{3+}$ is so abundant in nature (rust and many iron minerals contain it).

According to the usual filling order, chromium ($Z = 24$) should have the configuration $[\text{Ar}]\ 4s^2 3d^4$ and copper ($Z = 29$), $[\text{Ar}]\ 4s^2 3d^9$. In reality, their configurations are $[\text{Ar}]\ 4s^1 3d^5$ and $[\text{Ar}]\ 4s^1 3d^{10}$, respectively. The orbital box diagram shows the case of chromium:

1. a) Compare the "expected" and "actual" configurations of each element: how many electrons are in the $3d$ subshell in each case?
2. b) Half-filled ($d^5$) and fully filled ($d^{10}$) subshells are particularly stable: use this information to explain the two actual configurations.
3. c) What does this example teach us about the difference between a simple filling rule and the actual electronic structure of atoms?