The periodic table arose primarily from the work of Dmitri Mendeleev. He noticed correlations in both the chemical and physical properties of elements. In fact, he was able to predict the existence and approximate properties of Ga and Ge before these elements were discovered. Furthermore, even though the atomic weight of Te is greater than that of I, he reversed them in his table. This was really the start of the predominance of atomic number over atomic weight but, at that time, the distinction was not yet clear (for instance, it would be many years before people thought of nuclei with Z protons as determining the atomic number). In any event, we now have some more modern insights into periodic properties. For instance, note the following figure of atomic radius vs. atomic number.For now note that there are bumps, hills, and valleys in these numbers. The why of this will be discussed later in this and later chapters!
Light and the Electromagnetic
Spectrum:
Much of what we know about atoms (and molecules, too) comes from spectroscopy. (At this point, you may wish to view the movie on flames with some of the 1A metals.) Visible light is just one form of electromagnetic radiation. This radiation has this name since, if a electric field is changing, there must be a magnetic field of equal energy formed at right angles to it (and vice versa).Electromagnetic radiation is best described in terms of either its wavelength, l, or its frequency, n. (These are the Greek letters, "lambda" and "nu," respectively.) For electromagnetic radiation in physical and chemical processes, the waves propagate at the speed of light, c. Wavelengths in SI units are given in units of meters and frequencies in Hz (1Hz = 1s-1). We can express the relationship among these quantities as
c = ln.
The speed of light is 2.99792458 x 108 m/s and, thus, you should have some grasp of l and n. For instance, radiation with l = 1.00000 m would have a frequency of ca. 299.79 MHz. Or, to put this another way, radiation with a frequency of 89.7MHz (the frequency of WUSF) would have a wavelength of ca. 3.342m.
Keeping these things in mind, let's look at the following picture:
Here you see the entire EM (electromagnetic) spectrum. Below the main part we see frequencies and, in the top line, wavelengths along with objects of about the size of those wavelengths. At the very bottom we see the visible spectrum (what our eyes can see). This goes from violet light (at 380 nm or so) to red light (at 780 nm or so). We see g rays, x rays, and uv (ultraviolet) rays at shorter wavelengths and IR (infrared) rays, microwaves, and radio waves at longer wavelengths. Note that the shortest wavelengths have the highest frequencies!
We generally think of the EM waves we encounter in spectroscopy as propagating with simple sinusoidal motion. For instance look at the following simple examples:
Here we see the waves expressed in terms of their wavelengths. Also, the height of a wave could be expressed in terms of its amplitude, A. Note that you would see a similar picture if we had plotted vs. time instead of length. In that case, you would have had the same curve in terms of frequency. Indeed, if we were to express the above waves as plotted in terms of the distance travelled, x, we would come out with an equation that looks something like this:
You math guys and gals will recognize 2p as just 360o. x is then just graduated into units of l to get things to come out right in terms of wavelengths.
If we put time (t) on the horizontal axis, then we come out with the following equation (again this is just for the math folks):
Note that the sine wave has the same amplitude and the same basic shape--it's just plotted vs. a different axis!
Given this "artillery" you should have a pretty good understanding of wave motion by now. At this point, it should suffice to be able to use and write the following three equations (being careful with units, of course):
Electronic Radiation and Atomic Spectra:
Many of the keys to the "quantum revolution" of the early 20th century were found first in spectroscopy. What was found was that each element has a discrete spectrum which is unique to it. This is, for example, how we know that there is sodium in the sun as well as many other elements. (In fact helium was discovered in the sun before it was discovered on earth!) We look at a line spectrum now, namely that of atomic hydrogen.
Here we see four discrete lines. The interesting thing about these is that an arithmetic teacher named Balmer found that these exactly fit the equation,
where the constant, "Rydberg's constant," is given as
In the book we use just "R" for this; the infinity subscript is there for reasons we do not need to discuss but which are quite valid.
Also, note that the reciprocal of the wavelength (sometimes called a "wave number") is used above. This is a common unit of frequency in certain branches of spectroscopy (mostly IR). In any event, if you prefer to use frequency units, the above equation can be rewritten as
It was later found that other lines in the spectrum of the H-atom (in the uv and IR mainly) can also be expressed by a similar equation. In this case the wave numbers take the form
Other elements also have discrete spectra but they do not obey relationships quite this simple. Reasons for this became apparent later.
Particlelike Properties of Electromagnetic Radiation: The Planck Equation
As if some of the things shown by spectrscopy were not hint enough of some underlying fundamental theory of matter and radiation, the work of Max Planck showed a major breakthrough in the pivotal year 1900. In essence, he came up with an equation which explained the concept of black body radiation. Black body radiation (a black body itself is an idealized object you learn about in physics courses) is of interest since all matter, at a given temperature, has the same distribution of radiation frequencies. For instance, things which are "red-hot" are at about the same temperature; "white-hot" things are at much higher temperatures--in fact, the filament of an ordinary W lightbulb is hotter than the surface of the sun!
What is interesting about this work of Planck is shown in the next figure.
The "expected curves" come from classical theory and give the absurd conclusion that all matter is radiating infinite amounts of energy at infinitely high frequencies (this was dubbed the "ultraviolet catastrophe")! What Planck did was to find an equation which fit the data (and it does so exactly). What he found was that energy must be "quantized" as discrete "lumps" as is the case with matter. This "lumping" or "quantizing" was found to obey the equation,
(Note the slight historical inaccuracy in the textbook!) As we shall see momentarily, h is a fundamental physical constant (called Planck's constant) and has the value of 6.626069 x 10-34J-s. In Planck's work, it was actually an adjustable parameter to make the equation give a best fit to the data--it's fundamental importance was not recognized until a little later.
The next breakthrough came with Einstein's work on the photoelectric effect in 1905. (You may wish to view the photoelectric effect movie at this point!) In the photoelectric effect, electrons are expelled by light rays above a certain frequency (called the threshhold frequency, n0). According to classical theory, the energy of electrons should be proportional to the square of the light intensity; in actuality, only the number of electrons expelled is proportional to this whereas the energy of the expelled electrons is given by
The interesting thing here is that h is here again in a new role!
Light is, of course, a wave, but Einstein's work showed that it also has a particle aspect. In essence, this is an even deeper revelation of the nature of radiation than was Planck's insight. In fact, it led Einstein to propose that light quanta exist as discrete (but massless) particles with an energy,
These particles are called PHOTONS.
So, light exists as both waves and particles and both concepts are needed to fully explain its properties. What about matter? We discuss this in the next section!
Wavelike Properties of Matter: The de Broglie Equation.
Except for the work of Rutherford and Bohr, among others, no really fundamental progress was made until the work of Prince Louis de Broglie in 1924. While what he did seems to be just a simple formula, the results of his work resulted in a major leap forward in quantum theory (which was realized in 1926 in the work of Schrödiner and Heisenberg).
The derivation of de Broglie's equation in the book is in error since photons do not have mass. The derivation I give now is a little more accurate; however, it is really reasoning by analogy rather than a rigorous derivation. First, let us start with Einstein's famous equation from special relativity:
This is true for matter in general. For photons, howver, we need to express the energy a different way. Photons do not have mass but do possess momentum; the symbol used in physics for momentum is p and, for a photon,
Now, let us combine this with the equation for the energy of a photon in terms of its frequency. If we do this, we get (also using the fact that n = c/l)
Rearranging and solving for the wavelength gives
So far, we have not done anything new or different. But, now, de Broglie took a "leap in the dark" and said, essentially that we can apply this to matter. For matter,
and, hence, we can say that particles have a wavelength given by their mass and velocity and corresponding to the equation,
This is an astonishing conclusion! However, experimental confirmation came almost immediately when the wavelengths of rapidly moving electrons were discovered. This is the principle behind electron diffraction and also the underlying reason behind the functioning of electron microscopes.
Two Consequences of the de Broglie Wave Relation:
De Broglie showed that matter cannot be considered as just being particles--it's wavelike nature is also of importance. For matter at the macroscopic level (i.e., for objects the size of people or planets) the wavelike nature is unobservable (see for instance the problem involving the baseball in the assigned problems section). However, for things the size of electrons, protons, and neutrons, these effects cannot be ignored. The wave-particle aspects of matter exactly parallel those of photons. With very small particles, both the particle and wave nature of matter must be considered. We now look at two consequences of this.
The Heisenberg Uncertainty Principle:Werner Heisenberg showed that because of the wave nature of matter, there is a limit of the ultimate accuracy with which we can describe a particle. Since matter exists as waves, it is impossible to precisely fix a particle's position without causing an error in measuring its momentum (and vice versa). He put these ideas into the following expression:The Schrödinger Wave Equation:
(The latter quantity, called "h-bar" is a useful shortcut for h/2p, a quantity which occurs enough to have its own special symbol.)
What this principle states is that there is a fundamental limit on how well we can know both position and momentum! We can know either one as accurately as we wish, but always at the expense of less certainty about the other. The above expression of the products of the "uncertainties" of these two quantities, position and momentum, thus shows an ultimate limitation on the accuracy of experiments. In other words, we move from an exact view to a statistical view of things!
In 1926, Erwin Schrödinger in a series of papers showed the power of the wave concept of matter. He derived an equation which proved to be very powerful. Niels Bohr had, years earlier, derived an equation which explained the spectrum of the hydrogen atom and found the value of the Rydberg constant in terms of more fundamental constants (including Planck's constant, of course). However, his model could not be extended to atoms of more than one electron and, more importantly, could not be extended to molecules. On the other hand Schrödinger's equation was immediately seen to be much more general and, along with the work of Heisenberg, constituted the heart and soul of modern quantum mechanics.The solution of the Schrödinger equation for a given system gives that system's wave function, y. The square of this function corresponds to the probability of finding a particle in a particular place. Note that, again, we use the term probability and that our interpretation of the motion of small particles such as electrons must be statistical. The Schrödinger equation is a second-order differential equation in space and a first-order differential equation in time. We shall not bother to write it here; if you are interested, look at my quantum mechanical notes for CHM4411!
Wave Functions and Quantum
Numbers:
Details regarding the probability interpretations of wave functions are given in the supplementary notes to this course. What is most important here is an examination of Schrödinger's solution of the hydrogen atom and its extension to more complex atoms.What Schrödinger's equation accomplishes is a formulation of the concept of orbitals for electrons in an atom (and, as we shall see later, in molecules). Orbitals, in turn, are labelled quite effective by what are called quantum numbers--these arise quite naturally from the solution of Schrödinger's equation for the hydrogen atom and, with just a few modifications, also come out of solutions for more complicated atoms. We now look at these.
The principal quantum number, n:
The principle quantum number is related to the energy of an orbital. It can take on the following values:The azimuthal (or angular-momentum) quantum number: ln = 1, 2, 3, 4, ...
In other words, n can be any positive integer. Sometimes we use the term shell to designate electrons and orbitals with a given value of n.
The quantum number, l, is related to the shapes of the atomic orbitals. It also is an indication of their angular momenta (but you have to be careful here--the solution of the Schrödinger equation involves complex functions which are not necessarily the ones you see discussed in the book). l can have any integer value (including zero), up to n - 1. For instance, if n = 3, l can take on the values 0, 1, and 2.The magnetic quantum number: mll-values are often used to refer to subshells. Subshells are important enough to warrent their own notation as we now show.
l-value: 0 1 2 3 4 Subshell notation: s p d f g When discussing a particular orbital we give the n-value and the subshell letter. For instance, the following terms are quite common: 1s, 2p, 3d, 8g, etc.
This quantum in essence describes the direction of an orbital (except, of course for s-orbitals which are spherical in shape and can have no distinct orientation). ml can be either a positive or negative integer or zero. Its allowed values are determined by l (which, in turn, is determined by n). For instance, if l = 3, ml can take on the values -3, -2, -1, 0, +1, +2, and +3 (or 7 values overall). How these values come together and how we name orbitals is shown in the following table.
The electron-spin quantum number: msWe now come to the last, and surely the strangest of the quantum numbers, the spin quantum number, ms. You know that the rows of the periodic table are of length, 2, 8, 18, and 32. Yet, in the table above, we see just 1, 4, 9, and 16 for the number of orbitals in a shell. The Schrödinger equation predicts the latter to be the number of elements in a period. What went wrong? Actually, nothing. The Schrödinger equation in its usual form neglects the effects of relativity; if you put these in, the last quantum number ms develops naturally (c.f. the work of the British physicist P. A. M. Dirac).ms is, nevertheless, a little strange. It takes on just two values:
This property is called "spin" because electrons have a definite, measurable, magnetic moment. From our usual experience, such a moment can only arise if there is movement of electrical charge (which, for a single particle would come, presumably from rotation, or "spin"). Whether or not it is actually spinning in a physical way understandable to us is a moot point. For instance, the spin axis is what is sometimes called an "improper rotation axis"--from whence the +1/2 arises--and an electron, in order to completely return to its starting point needs to rotate 720o! (Most objects we are familiar with need only 360o.)
Be this as it may, this last quantum number now makes the rows of the periodic table 2, 8, 18, and 32 elements in length and all is well!
Orbital Energy Levels:The Schrödinger equation correctly predicts the energies of orbitals in atoms. In the case of the simple H-atom, it can be solved exactly. With atoms possessing two or more electrons, numerical methods must be used. With hydrogen, all energies are the same for a given value of n. However, when we have more electrons, the energies become different for different l-values (but stay the same for a given set of ml-values). This is shown in the next figure.
To use some of the jargon of quantum mechanics, the energies for a given n-value of hydrogen are said to be degenerate. When extra electrons are added and the energies change for the various possible values of l for a given n, the degeneracy is said to be lifted (i.e., removed). In atoms, each subshell is 2l + 1 degenerate. That is, the degeneracy is 1 for l = 0, 3 for l = 1, 5 for l = 2, and so on. The above diagram is not completely clear except in terms of energies. Note that each p-level is actually 3 orbitals, each d-level is actually 5 orbitals, and each f-level is actually 7 orbitals. And, of course, each orbital can hold two electrons (maximum!) since ms can take on only two values.
Shapes of orbitals and their orientations are set by their values for n, l, and ml. Check the supplementary notes for more complete discussions of these and their names.
Before concluding this section, we note one thing: Orbitals are actually standing-wave patterns. Standing waves are what are responsible for the pitch of a piano string, violin string, or organ pipe. In the case of these things, we have so-called linear harmonics. The standing waves in atoms are three-dimensional and correspond to what are called spherical harmonics. These are somewhat more complicated than linear harmonics but, nevertheless, follow the same basic laws (but in three dimensions instead of one dimension).
The Aufbau Principle and
the Electronic Configurations of Multielectron Atoms:
If things weren't "pHun" before, they are definitely going to become so now! Elements, as we move through the periodic table, become more and more complex. Exactly how electrons enter into orbitals in atoms are dictated by various principles. These could also be called "rules." In any event, the way in which electrons enter into atoms of successively higher atomic number (Z ) can be stated in what is the "Aufbau principle." This idea was first stated by Niels Bohr and the details became clearer as the principles of quantum mechanics were better developed. We now state the relevant rules.
Rules of the Aufbau principle:
- Lower energy orbitals fill before higher energy orbitals. We showed this in a figure which we now repeat:
You see that multielectron atoms have energies which are different for a given l along with the given n.
We are now ready to look at elements in the next row of the periodic table. These fill in much the same way as did the elements in the second row. To show this, we just look at the electron configurations:- An orbital can hold only two electrons, which must have opposite spins. In fact, no two electrons can have the same 4 quantum numbers. This is a statement of the Pauli exclusion principle. This principle is one of the mainstays of atomic and molecular quantum mechanics. Stated another way it is just this: No two electrons may be in exactly the same state. (States for atoms are defined by the four quantum numbers; in molecules it is the symmetry of the various wave functions which rules.)
- Electrons enter into degenerate orbitals (in a subshell) one at a time with their spins parallel until all orbitals are half-filled. The electrons after these then continue to fill that subshell.
- Half-filled subshells are especially stable.
- Filled subshells are especially stable.
- A noble gas configuration is the stablest situation of all!
Following these rules, we construct the electron configurations of the first ten elements. Note that we do this in two ways:
- Orbital diagrams
- Electron configurations
Hydrogen:
Helium:
From H to He we fill the 1s orbital with one electron for H and two electrons (of opposite spin) for He.
We now look at the rest of the elements in the first 10 (H- Ne) of the periodic table.
Lithium:
Beryllium:
(Please note that the energies are given in the proper order but that the scale is only schematic; that is, it only reflects proper order--not magnitude! However, subshells at the same level do have the same energy.)
Boron: (Here, p orbitals are filled for the first time!)
(The next element, carbon, takes its next electron in the next p subshell. This is the first example of Hund's rule at work!)
Carbon:
Nitrogen (first example of a half-filled subshell):
Oxygen (now, the electrons in the subshell begin pairing):
Fluorine (at this point, just one unpaired electron):
Neon: At this point, we attain a stable (noble-gas) configuration:
For the rest of the elements in this row, we shall start with the "neon core, " [Ne] and just add the other electrons:
We have reached the next inert gas core at the end of this row, namely, [Ar]. The electrons fill the orbitals in the same way as they did in the previous row. For instance, the electron configuration of phosphorus (P) can be shown as
Electrons continue to fill orbitals according to the Aufbau principle in a fairly regular fashion (except for a few so-called "anomalous" configurations which we shall discuss shortly). In essence, the chemistry of elements is determined by which block of the periodic table (s, p, d, or f) is being filled at a particular time. How this is envisioned is shown in the next figure.
The particular configurations for all the elements known are shown next. Notice that things get complicated rather fast!
Except for a few "glitches," these seem to go quite well!
The "Glitches":
Some "Anomalous Electron Configurations
We now look at some configurations which are, on occasion, termed "anomalous." Actually, what occurs in nature is not anomalous; rather, these so-called anomalies are just exceptions to our rules, in this case of the Aufbau principle! We look at the first two which occur. Are they really anomalous? Let's re-answer this question after doing the examination!The first case of interest is chromium (Cr). This is element #24 and has 6 more electrons than its nearest lower rare gas, Ar (#18). So, we would expect the following electron configuration:
[Ar] 4s2 3d 4.
Instead, we get the following:
[Ar] 4s1 3d 5.
What has happened here? The answer is clearer if one shows the individual electrons:
In this case, one of the 4s electrons has transferred to the 3d subshell. What has happened is that the half-filled subshell is particularly stable and, thus, the transfer has occurred! Note, also, that the 4s subshell is also half-filled in this case. This is a corollary of Hund's rule: half-filled subshells are particularly stable!
The next thing we note is that fully filled subshells are particularly stable! We see this with copper (Cu). For this element, we would expect the configuration
[Ar] 4s2 3d 9.
Instead, we get
[Ar] 4s1 3d 10.
This is easily understood if you look that the relevant diagram for Cu:
Here, the rule is just as stated above: fully filled subshells are particularly stable. We note that Ag and Au (in the same family as Cu) behave in the same way. If we go back to Cr, we see that the element immediately below it, Mo has a similar "anomalous" configuration; however, the next element under it, W, does not! (Is this an exception to an exception?)
If you look at other elements in the figure above which gives electronic configurations, you will see a few more "anomalies." Can you find them? We leave all this pHun to you!
Assigned Problems:
Before starting on these, we shall write the values of some of the physical constants used in this chapter. If you have a pocket calculator with a good memory, you may wish to store these permanently.
Some other numbers such as conversion factors will probably also be entering later and you may wish to store them too. (It's up to you!)
Problem 5.28 (p. 195):
Which of the following three spheres represents a Ca atom, which an Sr atom, and which a Br atom?This is an example of periodic trends: Ca and Sr are group 2A elements. Br is in the same row as Ca. Thus, Sr is bigger than Ca which, in turn, is bigger than Br. Thus, the gray sphere is Sr, the wine-colored one is Ca and the yellow one is Br.
- Size increases as you go from top to bottom in a column and
- Size decreases as you go from left to right in a row.
Problem 5.30 (p. 196):
Which has the higher frequency, red light or violet light? Which has the longer wavelength? Which as the greater energy?Problem 5.32 (p. 196):This is fairly simple. Red light has a lower frequency than does violet light. Since frequency is inversely proportional to wavelength, red light has the longer wave length and violet the shorter. Energy, on the other hand is directly proportional to frequency (E = hn). Thus violet light has greater energy than does red light.
What is the wavelength (in meters) of ultraviolet light with n= 5.5 x 1015 s-1?Problem 5.34 (p. 196):We do this by first writing the relevant formula and then putting in the numbers and solving. Note that, since c is stored in my calculator, all numbers--even when not strictly necessary--are shown!
(I slipped in nm at the end to give you a feeling for the best units to use for UV and visible wavelengths!)
Calculate the energies of the following waves (in kJ/mol) and tell which member of each pair has the higher value.(a) An FM radio wave at 99.5 MHz and an AM radio wave at 115.0 kHz.
(b) An X ray with l = 3.44 x 10-9 m and a microwave with l= 0.0671 m.At this point it is wise to calculate a useful conversion factor which we can use in this an some other problems. First, from
we see that frequency can be considered an energy unit. In this context, we see that
Now, it is easy to convert this into J/mol. We do this now:
Or, we can write this directly as the conversion factor,
(This might be another number to store in your calculator!)
Now we can do the problem!
(a):
(b):and
Clearly, the FM radio wave has the higher energy.
In this next problem we are given wavelengths. The easiest way to do this is to replace n with c /l. Doing this, and keeping the same conversion factor, we come up with the following solutions for the X ray and microwave respectively:
and
Clearly, the X ray is far more energetic than is the microwave. (So, would things cook faster in an X-ray oven...?) Note that, in the above work, we use 1Hz = 1s-1 without much ado.
Problem 5.40 (p. 196):
What is the de Broglie wavelength (in meters) of a baseball weighing 145 g and traveling at 156 km/hr?Problem 5.44 (p. 196):This is about a 94 mph fastball and, hence, is one heck of a pitch! Be that as it may, the wavelength is quite small. (If this were a curve, it was a curve for far different reasons than quantum mechanics!) Anyway, here is the calculation:
Note that, when doing dimensional analysis with Joules that
Note that, to save time, I wrote 145 g as 145 x 10-3 kg and 156 m as 156 x 103 m.
(As to the spacer bar you see below, what can I say other than "What is baseball without bats?")
According to the equation for the Balmer line spectrum of hydrogen, a value of n = 3 gives a red spectral line at 656.3 nm, a value of n = 4 gives a green line at 486.1 nm and a value of n = 5 gives a blue line at 434.0 nm. Calculate the energy (in kJ/mol) of the radiation corresponding to each of these spectral lines.Problem 5.54 (p. 197):This is quite easy if you remember that n = c /l and multiply each of the wavelengths by 10-9 to get them immediately into meters. We do each of these without further comment.
(As if this weren't enough, notice the chorus line of red, green, and blue guys... .)
What is meant by the term effective nuclear charge, Zeff , and what is it due to?Problem 5.56 (p. 197):The answer I give here is a little different from that of the book but is in the same ballpark. Electrons further from the nucleus are partially shielded by electrons closer to the nucleus. That is, the total charge of the nucleus, Z, is partially screened by the other electrons. Also, the inner electrons repel the outer electrons (which is, essentially, another way of saying that the effective charge of the nucleus is reduced). In any event, outer electrons experience less of the nuclear charge than do the inner electrons. Any way you look at this, the effective nuclear charge is reduced to what we call Zeff .
For comparison, here is the text book's answer:
Part of the electron-nucleus attraction is canceled by the electron-electron repulsion, an effect we describe by saying that the electrons are shielded from the nucleus by the other electrons. The net nuclear charge felt by an electron is called the effective nuclear charge, Zeff . Zeff = Zactual - electron shielding.
Give the allowable combinations of quantum numbers for each of the following electrons:Problem 5.58 (p. 197):
(a) A 4s electron (b) A 3p electron (c) A 5f electron (d) A 5d electron Before getting started, we first note that l = 0, 1, 2, and 3 for s-, p-, d- and f-electrons, respectively. Next, we note that |ml| <l or, alternatively, that ml can go from - l to + l in integer steps. Finally, we note that n is fixed for these examples (and that the l-values are legal for the given n-values) and that ms is just +1/2 no matter what else is going on. Given these restrictions, we can now produce a table of allowed values for the types of electrons listed above.
Problem Electron Type n l ml ms (a) 4s 4 0 0 +1/2 -1/2 (b) 3p 3 1 -1 0 +1 +1/2 -1/2 (c) 5f 5 3 -3 -2 -1 0 +1 +2 +3 +1/2 -1/2 (d) 5d 5 2 -2 -1 0 +1 +2 +1/2 -1/2 This is all quite straightforward and easy (I think...)!
Tell which of the following combinations of quantum numbers are not allowed. Explain your answers.(a) n = 3, l = 0, ml = -1
(b) n = 3, l = 1, ml = 1
(c) n = 4, l = 4, ml = 0Here are the answers:
(a): Not allowed since |ml| cannot exceed l.
(b): These are allowed. l < n - 1 and |ml| = l.
(c): Note allowed since l cannot be greater than n - 1.
Problem 5.64 (p. 197):Why does the number of elements in successive perioids of the periodic table increase by the progression, 2, 8, 18, 32?Problem 5.68 (p. 197):Before you look at my answer, look at the one in the back of the book on p. A-24. Does that make sense compared to what I am about to say?
To put things in perspective, and to save you from having to ravish your fingers thumbing through pages, here is what the book says:
The principal quantum number n increases by 1 from one period to the next. As the principal quantum number increases, the number of orbitals in a shell increases. The progression of elements parallels the number of electrons in a particular shell.Now, exactly what does this say? Is what I say below maybe a little bit more informative?Dr. J's essay on the whichness of the why with the periodic table:
Before we say anything else, please note that each orbital can have two electrons, one of spin +1/2 and the other of spin -1/2. Now given that, note that for each principal quantum number value, n, we can have l range from 0 to n - 1 and mlrange from -l to +l. Now, what does all this mean?First, for n = 1, l = 0 is the only possibility and thus ml can only have one value, namely 0. But, ms can take two values, as it can in any single orbital, namely +1/2. This essentially explains why the first row of the periodic table contains just two elements.
Second, we look at n = 2. Here, l = 0 and l = 1 are allowed. l = 0 allows just 1 value for ml but l = 2 allows 3 (-1, 0, +1). Now we note that
and that 4 = 1 + 3
4 x 2 = 8. With n = 3, we come up with similar logic and just give the results:
and that 9 = 1 + 3 + 5
9 x 2 = 18. You can see a pattern here, I hope:
This is a longer but, I believe, a less vague answer than that in the book.
- The sum of the first n odd digits is just n2.
- Thus, the length of any row in the periodic table must be given by 2n2, where n = 1, 2, 3, 4, ... .
- Thus, possibles lengths of rows are just 2, 8, 18, 32, 50, ... .
According to the aufbau principle, which orbital is filled immediately after each of the following in a multielectron atom?
(a) 4s (b) 3d (c) 5f (d) 5p The simplest way to do this is to just look at the periodic table. If you do, the answers are fairly easy:
(a) 4s is followed by 3d.
(b) 3d is followed by 4p.
(c) 5f is followed by 6d.
(d) 5p is followed by 6s.
Problem 5.72 (p. 197):Draw orbital filling diagrams for the following atoms. Show each electron as an up or down arrow and use the abbreaviation of the preceding noble gas to represent inner-shell electrons.(a) Rb (b) W (c) Ge (d) Zr
These are fun to draw. Let's see if they are fun to type...
(a): Rubidium: (Element #37; Kr is #36)
(b): Tungsten: (Element #74; Xe is #54)
(c): Ge: (Element #32; Ar is #18)
(d): Zr: (Element #40; Kr is #36)
Problem 5.80 (p. 198):At what atomic number is the filling of a g orbital likely to begin?Looking at the periodic table, the best guess would be gained by first looking at the inert gas, element 118 (this probably would be a gas). Two s-electrons come after this and, then one would expect element 121 to be the first one with a g-orbital filling (but, we must emphasize that this is a guess).
Problem 5.82 (p. 198):Why do atomic radii increase gong down a group of the periodic table?As you go down a column, the number of electrons is increasing and each shell is further away. Here, you don't have a shrinkage like from left to right because you are adding electrons outside pre-existing shells. (This can be done more precisely mathematically, but we shall not attempt that here. Take CHM 4411 if you are interested in these things.)
Problem 5.83 (p. 198):Why do atomic radii decrease from left to right across a period of the periodic table?The following graph gives part of the explanation:
Here is further explanation in words. Zeff is increasing and, hence, electrons are pulled closer to the nucleus and, of course, the atoms get smaller!
