Chapter 6 - Electronic Structure of Atom

In this chapter, we're going to discuss the electronic structure of the ABS. By the end of this chapter, we'll be able to calculate the wavelength of electromagnetic radiation given its frequency or its frequency given its wavelength. We'll also be able to order the common kinds of radiation in the electromagnetic spectrum according to their wavelength or energy. You will be able to explain what photons are and be able to calculate their energies given their frequency or wavelength. Additionally, we'll be able to explain how line spectra relate to the idea of quantifying energy states of electrons in atoms. You will calculate the wavelength of a moving object, and finally, you will explain how the uncertainty principle limits how precisely we can specify the position and the momentum of subatomic particles such as electrons. This chapter is all about electronic structure and the arrangement and energy of the electrons. We're going to start by talking about waves. It may seem odd to start by talking about waves; however, extremely small particles have properties that can only be explained in this manner. So we're going to see the matter perspective but also the wave perspective of these structures. Let's talk about waves. Here we have a wave, and the distance between these crests and these troughs, is known as the wavelength. The distance from one crest to the next crest is the wavelength. To understand the electronic structure of atoms, one must understand the nature of electromagnetic radiation. The distance between corresponding points on adjacent waves is the wavelength, usually represented by the Greek letter lambda (λ). The number of waves passing a given point per unit of time is the frequency, represented by the Greek letter nu (ν). For waves, we have two types. We can see this one that has this form, and at the same time, this one that represents time. The black line represents time, and for this time, we can see that from this point to this point is a cycle. So we can go one and two cycles using this wavelength, which is shorter than this one. So let's see how many cycles we can have in the same time. We can go one, we can go two, we can go three, and we can go four cycles. At the same time, we can have four cycles at this wavelength, and for this wavelength, we can only have two cycles. So four waves are traveling at the same velocity. The longer the wavelength, the smaller the frequency. For the shorter wavelength, it was just two cycles, whereas for the longer wavelength, the frequency was higher. If the time associated with the lines to the left is one second, then the frequency would be two cycles per second for the shorter wavelength and four cycles per second for the longer wavelength. That will be the frequency for these examples. Most of the time, we talk about cycles per second, which is time. For the shorter wavelength, the frequency is two cycles per second, and for the longer one at the bottom, it's four cycles per second. Here we have the electromagnetic radiation, and this is the visible region. At this point, we also have other types of frequencies - gamma rays, x-rays, ultraviolet, infrared, microwaves, and radio frequency. We can see that the wavelength increases from left to right, while the frequency increases from right to left. Usually, the unit we use is a meter. So the wavelength increases from left to right, while the frequency increases from right to left. They are inversely proportional to each other. When one increases, the other one decreases. All electromagnetic radiation travels at the same velocity, which is the speed of light, denoted by c. The equation for the speed of light is c = λ (wavelength) x ν (frequency). This relationship is inverse because the speed of light always remains the same. If you increase the wavelength, the frequency must decrease, and if you decrease the wavelength, the frequency must increase. The yellow light emitted by a sodium vapor lamp used for public lighting has... A wavelength of 589 nanometers. What is the frequency of this radiation? So, we have the wavelength, and we need to find the frequency. We have this formula that relates the speed of light to frequency. We are given the wavelength of the radiation and asked to calculate the frequency. The relationship between wavelength and frequency is given by the equation: speed of light = frequency × wavelength. We can solve for frequency and use the values of lambda and the speed of light to obtain an answer. So, we have the speed of light, and the equation is c × frequency = λ. But we can rearrange this equation to obtain the frequency. Frequency is equal to the speed of light divided by the wavelength. Therefore, frequency = c / λ. We have the speed of light and the wavelength. Now, we can convert the units. In the given wavelength, we have units of nanometers, and in the speed of light, we have meters. We need to change from nanometers to meters to cancel out the units. We use the definition: 1 nanometer = 10^-9 meters. By doing this, we can cancel out the units and end up with the units of frequency in cycles per second. Once again, we have the equation: frequency = speed of light divided by lambda, as shown in the relationship between speed of light, frequency, and wavelength. We rearrange the equation to calculate the frequency, and the result is the speed of light divided by the wavelength. The speed of light is a constant value, and the wavelength for this specific yellow light is 589 nanometers. We need to cancel out the units of nanometer and meter using the relationship between meters and nanometers. After canceling out the units, we obtain the frequency in cycles per second. The high frequency is reasonable because of the short wavelength. The units are appropriate because frequency has the units of cycles per second. The wave nature of light does not explain how an object can glow when its temperature increases. This is basically due to energy. The energy is defined by Max Planck as coming in packets called quanta. A quantum is a package of energy that is needed to go from one step to another. Each step requires a specific amount of energy. This energy is known as a quantized energy. So, when you need to move from one step to another, you need a specific amount of energy that is called a quantum. The photoelectric effect was developed by Einstein. Einstein used quanta to explain this effect. Each metal has a different energy at which it ejects electrons. Electrons can be removed from metals by hitting them with a specific energy. It seems that different energies can remove or eject electrons. At lower energy, electrons are not emitted. Hence, Einstein concluded that energy is proportional to frequency. Energy equals Planck's constant (H) times frequency (ν). Energy is directly proportional to frequency. The higher the frequency, the higher the energy. This relationship is inversely proportional to wavelength. Higher energy corresponds to lower wavelength, and lower energy corresponds to higher or larger wavelength. Planck's constant (H) is the value of the constant. Now, let's calculate the energy of a photon of yellow light with a wavelength of 589 nanometers. We need to use the equation we saw before, but that equation was in terms of frequency. First, we need to calculate the frequency. Need to change from wavelength to frequency. We need to convert this wavefront to frequency using the same that we used before. Is any question? As the example six points like that, six point two. With it before, we obtained there the frequency. And then that frequency, we can use this formula and we can multiply it by the Planck constant. So, ten, the energy. As we saw in example six point six point two, this is the frequency, okay, of this yellow light at that wavelength. And now, this frequency, we can put it here in this equation. So, we can multiply it by the Planck constant. We have the frequency obtained by this calculation. And now, this will be the energy of this wavelength. Let's talk about the atomic emissions. Another mystery in the early 20th century involved emission spectra observed from energy emitted by atoms and molecules. We have, for example, for me in this kind of emission, it does kind of light. And for hydrogen, these other colors, okay? So, we have basically different colors for, in this case, different atoms. And also, it has been found that we have continuos and it's line spectrum. So, for definition, basically, this one is a continuous spectrum. For atoms and molecules, one does not observe a continuous spectrum as one gets from white light. So, the white light produces a continuous spectrum. You don't see any interruption between the colors. It goes from purple to basically red, orange and red. Okay, all the way without interruptions. Now, only a line spectrum of this discrete wavelength is observed when we have elements. Each element has a unique line spectrum. This, instead of this, is basically the same range from here, okay? But, as you can see, instead of seeing a continuous spectrum, you can just see line spectrums. This wave line, this one, and this one. Just, you can see just four lines for hydrogen, for me. And you're going to see a much larger number of lines. But, it's the same concept, the same characteristic. You can't see a continuous spectrum. You're going to see line spectrum for the elements. And the question is, why? So, let's try to explain. Explain first the most simple, that is the hydrogen. The hydrogen spectrum. A summation will have just four lines. Johan Balmer in 1885 discovered a simple formula related to four lines, two integers. Integers. And Johan’s a rad berg advanced this also formula. And they established that the equation is going to be this one over the wavelength is going to be equal to the Rydberg constant times 1 over n square at 1 square minus n 2 square. Niels Bohr explain why this mathematical relationship works. Ok, so in this case, the n represent the energy levels. We're going to see that in a little bit later. But, this basically represents the states, the energy with where the electrons are. Because the emission of the light is going to be due to the transition of the position of the electrons. When they jump from one energy to another energy states, from one level to another one. I'm going to talk a little bit of that in a few slides. So, let's talk about the Bohr model. Niels Bohr adopted the Planck's assumption and explain this phenomena in this way. Level 1 electrons in an atom can only occupy certain orbits. These orbits represent energy. So, they are in specific orbits around the atom. Electron in permitted orbits have specific allowed energies. These energies will not be regulated from the atom. So, if you want to be in a specific orbit, that orbit has specific energy. And you need to have this energy to be at that level. It's the same as when we talk about doing the steps. If you have enough step to get to the next one, you're going to be in the next or in the next step. So, it's the same with the electrons. They reside in specific orbitals. They need to have that energy to be in that specific orbit. Energy is only absorbed or emitted in such a way as to move an electron from one allowed energy state to another. The energy is defined as e equals to the constant Planck and the frequency. So, this means that if you want to go from one orbit to another one, if it's in the eyes if it has a higher energy, you need to gain enough absorb enough energy to be in that next orbit. If you want to go another bit lower than where you are, you need to release or emit that specific amount of energy where you can have the energy associated with that level. You can't be in a lower level with more energy that it requires to be in. That orbit. So, as you can see here, we basically have one electron. We can have, for example, one electron in the ground level, one N equals to one, and we want to reach N equals two. So, this is the energy required for the electron to absorb in order to reach the second orbit. Also, it can reach the third orbit if it gains or absorbs this amount of energy. From this point to this point, it can reach here. But if it has enough energy to reach this point, it can't reach either this one or this one. The same thing happens when we want to go from orbit number two to orbit one. We need to release all this amount of energy in order to reach this point. You can also go from orbit four to two. If one electron wants to release enough energy to reach number two, you need to release or emit this amount of energy from this point to this point. Because this represents energy here on the left side of the axis. So, the Bohr model shows all the different kinds of emissions or movements of the electron from orbit one to two, from two to three, from two to four. You can see that this represents basically the energy that needs to be absorbed if you want to go up, or released if you want to go down. The amount of energy is constant between orbits. So, the energy absorbed or emitted from the process of electrons' promotion or demotion can be calculated by the following equation: the change in delta E is equal to minus H, which is Planck's constant, times C, which is the speed of light, times R, E, H, which is the Rydberg constant, times in parentheses 1 over N final squared minus 1 over N initial squared. N represents the initial orbit, and S represents the final orbit. This model has some limitations because it only works for hydrogen. In classical physics, an electron would fall into the positively charged nucleus, but the Bohr model assumed it would not. Circular motion is not observed in nature. But there are very important ideas from this model that have been incorporated into the current atomic model. These ideas include the following: 1. That electrons exist only in certain discrete energy levels. 2. That energy is involved in the transition of an electron from one level to another. Now let's talk about the wave nature of matter. Louis de Broglie theorized that if light can have wave properties, matter should exhibit wave properties as well. He demonstrated that the relationship between mass and wavelength had this equation: wavelength = Planck's constant / mass. This equation incorporates the concept of matter having mass. So now, instead of wavelength, we are talking about the wave nature of matter. Because we are incorporating the concept of mass. This wave nature of matter is used to produce electron micrographs, as shown in this basic picture of metal in an electron micrograph technique. Now let's talk about Heisenberg's uncertainty principle. Heisenberg showed that the more precisely the momentum of a particle is known, the less precisely its position is known. So in other words, the position of an electron is really hard to establish. It is basically unknown. We can have an idea, but we can't precisely say, "The electron is right here at this point in time." To explain how we know where the electron is, we need to invoke what is known as quantum mechanics. This was developed by a mathematician known as Erwin Schrödinger. He developed a mathematical treatment in which both the wave and particle nature of matter could be incorporated. This is known as quantum mechanics. Essentially, instead of knowing the specific location of the electron, we are talking about probabilities. Where it is more probable to find the electron. So, we don't know the specific location, but we know the specific area. Where you are going to have a higher probability of finding that electron is determined by the solution of the Schrodinger equation. The wave equation is assigned by a lower Greek letter, psi. Basically, the wave equation represents the electron density or probability of where an electron is likely to be found at any given time. So the result of the wave equation allows us to know where there is a higher probability of finding the electron. In the intense blue regions, there is the highest probability of finding the electron, while in the regions with low density, there is a lower probability. Solving the wave equation gives us a set of wave functions, also called orbitals, and their corresponding energies. Each orbital obtained from the wave equation has a specific energy and describes a spatial distribution of electron density. An orbital is described by a set of three quantum numbers. The first quantum number is the principal quantum number, represented by N. It describes the energy level on which the orbital resides. The values of N are integers, starting from 1 and going higher. This corresponds to the values in the Bohr model, where N represents the different energy levels. The second quantum number is the angular momentum quantum number, represented by the letter L. This quantum number defines the shape of the orbital. The allowed values of L are integers ranging from 0 to (N-1). Each value of L corresponds to a specific shape and type of orbital. The letter designation is used to communicate the different values of L and the corresponding shapes and types of orbitals. The third quantum number is the magnetic quantum number, represented by Ml. It describes the three-dimensional orientation of the orbital. The allowed values of Ml are integers ranging from (-L) to L, including zero. So on any given energy level, there can be up to one s orbital, three p orbitals, five d orbitals, seven f orbitals, and so on. Orbitals with the same value of N form an electron shell, and different types of orbitals within a shell are called subshells. Here is a table illustrating this concept: (Please refer to the original text for the table) In the first shell, where the principal quantum number (N) is 1, the possible angular momentum (L) is 0. Therefore, the magnetic quantum number (Ml) is also 0. The letter representation for this subshell is "s", and there is only one orbital in this subshell and a total of one orbital in the shell. Moving on to the second shell... Have 2 possible values of n. Remember that they may be mL. Okay, basically the L is going to go from 0 to n-1. So starting at 0, n-1 is 2-1=1. So it goes from 0 to 1. That's why we have a possible mL of 0 and 1. What about 0? Ok, now let's determine b mL. The magnetic quantum number goes from 0 to -L to L. But in this case, since L is 0, we are going to have a value of 0, okay? And that's the same as we saw before. Now, when we are in L=1, the subshell is going to be 2p. Because now the L value of 1 is represented by p, and we are in the second orbital shell (2s). So we are 2s, and the mL is going to be 1, 0, -1. Why? Because remember that mL goes from L to -L, going through 0, and L is 1. So it's going to be 1, 0, -1. So from this, we are going to have 3 orbitals in the 2p subshell, and 1 in the 2s subshell. So, in the end, we are going to have 4 types of orbitals in the second orbital shell: 3 from the 2p subshell, and 1 from the 2s subshell. If we go, for example, to level 3, the possible values of L go from 0 to n-1. So n-1 is 3-1=2. So it goes from 0, 1, 2. And this is going to be the same, but now we are in the third orbital shell (3s). So it's going to be 3s, 3p, and 3d. And the 3s has an mL value of just 0. And the p is going to have mL values from -L to L, so it goes from 1, 0, -1. And the d (3d) has an L value of 2, so mL goes from -2 to 2. So it goes to 2, 1, 0, -1, -2. So we have 5 orbitals in this subshell. So in the end, this third orbital shell is going to have 9 orbitals: 1 from the 3s, 3 from the 3p, and 5 from the 3d. And the same thing applies to the principal number of 4. Now let's talk about the orbitals. The s orbital (n = 4) has an L value of 0. And the shape of this orbital is spherical. It's basically like a circle, even though it's a sphere specifically. And the radius of the base of this sphere is going to increase with the value of n. The higher the principal quantum number, the larger this sphere of the s orbital is going to be. So when you have n=1, the 1s orbital is here, n=2, you have the 2s orbital here, n=3, you have the 3s orbital here. That means that the probability also increases. As you go from this part of the sphere to this one here, the area represents the probability for the 1s orbital to find the electron. When you have a 2s, you can also have the 1s, because you need to fill further down the smaller spheres. So you have the 1s, the 2s, and here the 3s. These are the probabilities. This represents the 1s, 2s, and 3s. For an ns orbital, the number of peaks is equal to n, and the number of nodes is equal to n-1. So here we have an n=1, so we don't have any nodes here. But we are in n=2. So the number of nodes is going to be 2-1=1. So we have 1 node between this area and this area. This node represents that you have a 0 probability of finding the electron. So here we have this case, that's it. That is the 2s. Then for the 2p, for example, 2-1 is going to be 1. So we have 1 node, so this is where we have a node. And then for the 3s, 3-1 is going to be 2. So we are supposed to have 2 nodes. One is going to be here and one is going to be here. These two areas or positions have a probability of 0 to find the electron. As n increases, the electron density is more spread out and there is a greater probability of finding an electron further from the nucleus. So as we can see here, this is for the 1s. This is for the 2s. The white areas represent the nodes. So we have a very intense blue in the middle, which is the 1s. Then a white area, which is the node. Then another intense blue outer sphere, which is the 2s. Here we have the 3s. And for the 3s, we have two possible nodes. We have one node here, which is this white area, and also another one here, which is another white area. So we have the 1s in the middle, the 2s around here, and the 3s as the outermost area of this orbital. Sphere, okay. So the p orbitals: the value of L for a p orbital is one, and they have two lobes with a node between them. Basically, these are the representation of a pure "p". It looks like a Dumbo, okay? So, you have this as the representation of a pure "p". We can also describe it as having a p orbital in each of the axes of a three-dimensional representation. So, this represents the higher probability, and also this one. This point here, between the two lobes, is known as a node. We have a node here, and the probability to find the electron there is zero. Whenever we talk about nodes, that means the probability to find the electron at that specific point is zero. So, we have a lobe in the z-axis, the y-axis, and the x-axis. Okay, then we have the d orbitals. The value of L for a d orbital is two, and we basically have five different shapes. Four of the five d orbitals have four lobes. The other resembles a p orbital with a donut around the center. So, these are the representations of the p orbitals, five different shapes. Okay, these are basically the planes of the z, using also the y. This is for z and x, x and y, and this one also fits the x on the y, but using the z-axis. And this is between the axes. So, there are four representations with four different lobes, and this one has basically the donut shape of the p and also a donut around the center. So, these are four representations mathematically. That's the way that they observed this equation of form. Okay, these are just mathematical equations. Remember that these are just probabilities to find the electrons. So, these are the five different shapes of the d orbitals. For the f orbitals, the shapes are really complicated, so we're not going to show that in the text. But they have seven equivalent orbitals in the outer shell, and L is equal to three. Now, let's talk about the energy of orbitals, and let's use the one that we saw before, which is hydrogen. For a one-electron hydrogen atom, orbitals on the same energy level have the same energy. Chemists call them degenerate orbitals. So here we have one orbital that has this energy, and here we have a cluster of different subshells that we also have different energy. But we have found that as the number of electrons increases, so does the repulsion between them. So they specifically have different kinds of energies. Each of them, these are the ones that are called degenerate orbitals because they have the same energy. These three orbitals, for example, here we have five degenerate subshells because they have the same energy. And also we have three here. Therefore, in atoms with more than one electron, not all orbitals on the same energy levels are degenerate. Orbitals sets in the same sublevel are still degenerate. Energy levels start to overlap in energies. For example, the 4s is lower than the 3d. This is not like a physical perspective. We can see this as a physical perspective. It's just with an energy perspective. So if you're going to start to add electrons, remember that you start with the lowest level of energy. So you start putting electrons at this point, at the 1s, then go to the 2s, then go to the 2p, and then go to 3s, 3p, and tend to go to the 3d. That is the next one with the lower energy. The 4s has a lower energy than the 3d. So that's why you fill the 4s and then the 3d, and then the 4p. There is another quantum number that is known as the spin quantum number. In the 1920s, it was discovered that two electrons in the same orbital don't have exactly the same energy. The spin of an electron describes its magnetic field, which affects its energy. So even though they're in the same orbital and it was suspected to have basically the same energy, they have different energies due to the spin. This led to the spin quantum number, Ms. Electrons can have a minus one-half or a positive one-half as the value of spin, Ms. So if positive one-half or minus one-half, those are the two possibilities for the Ms. So the principle is, in principle, no two electrons in the same atom can have exactly the same energy. Therefore, two electrons in the same atom cannot have identical sets of Quantum numbers. This means that every electron in an atom must have at least one of the four quantum number values. So, we have the n, the L, the ml, and the Ms. Each electron is going to have four specific numbers, and it can't have the same numbers as another electron. They need to be different in at least one way. Let's talk about electron configuration. The way electrons are distributed in atoms is called the electron configuration. The most stable organization is the lowest possible energy, called the ground state. Each component consists of a number denoting the energy level. The letter denotes the type of orbital, with P being a type of orbital, and a superscript denoting the number of electrons in those orbitals. For example, we have five electrons in the p orbital. Now, let's discuss the orbital diagrams. Each box in the diagram represents one orbital, and a half arrow represents an electron. The direction of the arrow represents the relative spin of the electron. For an ml value of positive 1/2, we represent it by pointing the arrow upward. And for a -1/2 value, we represent the electron with an arrow pointing downward. Each box can have a maximum of two electrons. Here is the orbital diagram for lithium. We have three electrons, with two in the 1s orbital and one in the 2s orbital. For beryllium, which has an atomic number of 4, we have two electrons in the 1s orbital and two electrons in the 2s orbital. So, the electron configuration for beryllium is 1s2 2s2. Now, let's consider boron. Boron has an atomic number of 5, so it has 5 electrons. We have two in the 1s orbital, two in the 2s orbital, and one in the 2p orbital. Therefore, the electron configuration for boron is 1s2 2s2 2p1. Next, let's discuss carbon. For carbon, we add 2 electrons in the 1s orbital, 2 electrons in the 2s orbital, and instead of adding two electrons in the same 2p orbital, we split them. This is because electrons are particles with a negative charge, and putting them together in the same box would result in repulsion. By separating them, we avoid this repulsion, and they still occupy orbitals with the same energy. So, for carbon, we have one electron in each 2p orbital. The electron configuration for carbon is 1s2 2s2 2p2. Next, let's move on to nitrogen. Nitrogen has an atomic number of 7, so we have 2 electrons in the 1s orbital, 2 electrons in the 2s orbital, and one in the 2p orbital. We need to add 3 more electrons in the 2p orbitals, so we add one electron to each. The electron configuration for nitrogen is 1s2 2s2 2p3. For oxygen, which has 8 electrons, we put 2 in the 1s orbital, 2 in the 2s orbital, and then we don't have any more spaces to put the electron. So, we pair it up in the 2p orbital. The electron configuration for oxygen is 1s2 2s2 2p4. Similarly, for neon, which has 10 electrons, we put 2 in the 1s orbital, 2 in the 2s orbital, and then 6 in the 2p orbital. So, the electron configuration for neon is 1s2 2s2 2p6. This pattern continues for other elements, with electrons being filled up in the orbitals following specific rules.