Chapter 7 – Periodic Properties of the Elements

5. Properties of Metals and Non-metals

Homework Problems: 5, 7, 9, 13, 17, 19, 21, 27, 33, 35, 41, 43, 45, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 90

Introduction

Chapter 6 deals with the properties of electromagnetic radiation and the electronic structure (that is the properties of the electrons) of atoms. This chapter describes at a very introductory level the fundamentals of quantum mechanics. The development of quantum mechanics and its implications is probably the most important and far reaching scientific discovery of the last 300 years! To a certain extent this is reflected by the fact that seven Nobel prizes (all in physics though) were awarded for the work discussed in Chapter 6:

• 1900 (1918) - Max Planck (Energy is quantized)
• 1905 (1921) - Albert Einstein (Explanation of the Photoelectric Effect)
• 1914 (1922) - Niels Bohr (Model of the Hydrogen Atom)
• 1925 (1929) - Louis DeBroglie (Wave properties of Matter)
• 1927 (1932) - Werner Heisenberg (Quantum Mechanics & The Uncertainty Principle)
• 1926 (1933) - Erwin Schrodinger & Paul Dirac (Wave Equation for Electrons)
• 1925 (1945) - Wolfgang Pauli (Exclusion Principle)

What is electromagnetic radiation? Electromagnetic radiation is a form of energy, sometimes called radiant energy or optical energy. The most familiar form of electromagnetic radiation is visible light. However, there are many other forms of electromagnetic radiation including:

• Visible Light
• X-rays
• Ultraviolet Light
• Microwaves
• Infrared Light

Our eyes are only sensitive to a small portion of the electromagnetic spectrum, but in a physical sense there is nothing that makes visible light unique from say the X-rays used at the hospital or the microwaves used in your oven.

What are some of the features common to all forms of electromagnetic radiation:

• They are all propagating (moving) forms of energy
• They all travel at the velocity (the speed of light)
• They all have wavelike characteristics

Lets expand upon this last point, what does it mean to behave as a wave? Waves have a periodic up/down motion to them. For example if a boat remains stationary while waves of water go past the height of the boat will alternately rise and fall, reaching a maximum at the peak of the wave and a minimum at the trough.

If we were now to take a picture of the wave, freeze the wave at one moment of time it would look something like the picture below:

This picture actually shows three cycles of the wave, that is it repeats itself three times in the above picture. The wave shown above is not moving, but both electromagnetic waves and water waves do not stand still, they move (propagate). Whereas, a stationary boat in the water is a good image for describing the periodicity of a wave, a surfer is a good image for describing the motion of a wave. A surfer travels with the wave, ideally staying at the same point on the wave as it moves (for example on just below the peak). Using these images and the picture above we can define the properties of a wave.

Wavelength (l) ® The distance between any point on the wave and the corresponding point where the wave begins its next cycle (for example from peak to peak).

Velocity (v) ® The speed with which any point on the wave moves through space (the speed of the surfer).

Frequency (n) ® The number of cycles that pass by a stationary point per second (If you were sitting on a tower sticking out of the surf, the frequency would be the number of surfers that would go past you every second, assuming 1 surfer per wave). Units of frequency are Hertz, Hz (1 Hz = 1 cycle/sec = 1 s-1).

Amplitude ® The height of the wave (as measured from the middle of the wave to the peak).

Going back to the surfer riding the crest of wave, lets consider how changes in the frequency and wavelength affect his speed. If we increase the frequency, more waves have to go by every second and his velocity increases. On the other hand if we keep the frequency constant and increase the wavelength the wave has to move faster in order to keep the same number of waves going by per unit time, so the surfers speed will increase. We can mathematically state this in the following expression:

v (m/s) = n (cylces/s) ´ l (m/cycle)

Recall that above we stated that all electromagnetic radiation travels at the speed of light, c = 3.00 ´ 108 m/s. Therefore, for electromagnetic radiation this expression reduces to:

c = n l

So for electromagnetic radiation if the frequency increases then the wavelength must decrease.

Example

What is the wavelength of microwave radiation with a frequency of 1 ´ 1010 Hz?

l = c/n = [3.00 ´ 108 m/s]/[ 1 ´ 1010 s-1] = 0.03 m = 3 cm

At this point you might ask what makes us think that light is a wave. A few of the ways light behaves as a wave are given below.

Refraction – The phenomenon where light bends upon passing from one substance to another. This is the reason why lenses and magnifying glasses work, it also explains why the surface looks closer than it really is when you are underwater.

Dispersion – The process of separating white light into its component colors. This is the principle upon which both prisms and rainbows work.

Diffraction – The interference patterns that are created when light passes through a series of two or more slits. It also explains why light bends when it goes around a corner, or spreads out when it goes through a hole.

For more detailed information on these phenomena see the following web sites:

Refraction and Dispersion

Science Hypermedia’s Page on Prisms & Electromagnetic Radiation

Diffraction

Dr. Grandinetti’s Chem 121 Notes on Chapter 6

Despite the successes of wave theory to explain the behavior of light, there were some phenomena that could not be explained by thinking of light as a wave. The two most prominent exceptions to the wave behavior of light were blackbody radiation and the photoelectric effect.

Most people have seen that when you heat a solid object in a fire it glows. As the temperature is increased the color given off by a heated solid shifts from red to orange to finally white (i.e. red hot is cooler than white hot).

As we increase the temperature we are putting more energy into the solid so it is natural to assume that more energy will be given off as light. In other words we would expect that as the temperature increases the intensity of the light will also increase (the light will get brighter).

Treating light as a wave we would expect this increased energy output to manifest itself as an increase in the amplitude of the wave. Instead, the shift in color (from red to white) implies that we are increasing the frequency rather than the amplitude.

If you are having trouble picturing this concept consider an analogy to waves at the beach. The tallest waves have the greatest amount of energy. Go stand in front of some waves if you don’t believe me, on second thought since we live in Ohio I guess you’ll have to take my word for it.

Photoelectric Effect

When light strikes the surface of a metal in certain cases it causes an electron to be ejected from the metal. This phenomenon is known as the photoelectric effect. In and of itself this behavior is not too surprising. The outermost electrons in a metal are bound rather loosely to the metallic nuclei, and the photoelectric effect is simply the transfer of energy from the light to the electron in order to break free of its attraction to the nucleus (the binding energy). Furthermore, if the energy of the light is in excess of the binding energy of the electron this extra energy will be used to increase the kinetic energy of the ejected electron.

Based on the wave description of light we would expect that the energy of low intensity light would be too small to eject an electron, however, as we increase the intensity of the light at a certain point we will begin to eject electrons. If we increase the intensity further the kinetic energy of the ejected electrons will show a corresponding increase. However, this is not what was observed experimentally.

Instead scientists found that if for light of a particular frequency and wavelength (i.e. infrared light) no matter how much the intensity was increased (10 Watt, 100 Watt, 1000 Watt bulb) no electrons were ejected. In order to eject electrons it was necessary to increase the frequency of the light (i.e. go from infrared light to visible light). Once electrons started to be ejected their kinetic energy increased linearly with further increases in the frequency. Increasing the intensity at this point increased the number of electrons ejected, but it did not change their kinetic energy.

Photons

In trying to explain these effects Max Planck and Albert Einstein developed theories that when put together led to the following principles.

• Light is made up of particles called photons.
• The energy of a photon is dependent only upon its frequency.

E = hn = hc/l

Where h is Planck’s constant (h = 6.626 ´ 10-34 J-s).

What this means is that if an object is giving off (or absorbing) light it is actually emitting (absorbing) photons. The energy of each photon is dependent only upon its frequency (or wavelength or color).

Example

What is the energy of a photon of light, with a wavelength of 515 nm?

E = hc/l = [6.626 ´ 10-34 J-s][3.00 ´ 108 m/s]/[515 ´ 10-9 m]

E = 3.86 ´ 10-19 J

The idea that light exists as a particles called photons explains both blackbody radiation and the photoelectric effect.

The light given off by a hot solid shifts to higher frequencies as the temperature increases, because as the frequency of the photons increases so does their energy.

The photoelectric can now be understood in the following manner.

If the energy of the photon (E = hn) is smaller than the binding energy of the electrons in the metal (called the work function, E0) then no electrons will be ejected.

If we increase the intensity of the light but leave the frequency constant the number of photons increases, but no photon has sufficient energy to eject an electron.

Increasing the frequency (n) of the light increases the energy of each photon, at the point where hn > E0, electrons will begin to be ejected.

Further increases in the frequency of the light will increase the energy of each photon, which will in turn lead to an increase in the kinetic energy of the ejected photons:

Ekin = hn - E0

Increasing the intensity of the light, when hn > E0, increases the number of photons but does not alter their individual energies. This will cause more electrons to be ejected, but leave the kinetic energy of the ejected electrons unchanged.

For a different wording of the photoelectric effect, complete with pictures click here to see Dr. Grandinetti’s 121 Notes.

Example

It requires 222 kJ/mol to eject electrons from potassium metal. (a) What is the minimum frequency of light necessary to induce emission of electrons from potassium via the photoelectric effect?

First I need to convert from the quantity of energy necessary to eject an 1 mole of electrons to the amount of energy required to eject a single electron.

E0 = [222 kJ/mol][1000 J/1 kJ][1 mol/6.022 ´ 1023 electrons]

E0 = 3.69 ´ 10-19 J

Next I calculate the frequency of a photon with this energy.

n = E/h = [3.69 ´ 10-19 J]/[6.626 ´ 10-34 J-s] = 5.56 ´ 1014 s-1

n = 5.56 ´ 1014 Hz

(b) What is the wavelength of this light?

l = c/n = [3.00 ´ 108 m/s]/[ 5.56 ´ 1014 s-1] = 5.39 ´ 10-7 m

l = 539 nm

(c) If potassium is irradiated with light of 350 nm, what is the maximum possible kinetic energy of the emitted electrons?

Ekin = Ephoton – E0

Ephoton = hc/l

Ephoton = [6.626 ´ 10-34 J-s][3.00 ´ 108 m/s]/[ 350 ´ 10-9 m]

Ephoton = 5.68 ´ 10-19 J

Ekin = 5.68 ´ 10-19 J - 3.69 ´ 10-19 J

Ekin = 1.98 ´ 10-19 J

Electromagnetic Radiation as a Particle and a Wave

The discussion above first says that electromagnetic radiation is a wave. Then I went on to say that electromagnetic radiation is a particle, which is it?

The answer is both. Electromagnetic radiation has properties of both a particle and a wave. This is called the wave particle duality of light. We shall see below that light is not unique in this aspect, but that matter can also behave as both a particle and a wave.

It is not easy to imagine something that is simultaneously both a wave and a particle. Personally I like to think of light as a particle that has some wavelike properties, such as diffraction and dispersion.

Electromagnetic Spectrum

We are now in a position to describe the electromagnetic spectrum from low (on the top) to high energy (on the bottom).

 Radiation Energy (J) Wavelength (m) Frequency (Hz) radio waves < 2.0´ 10-22 > 1 mm < 3´ 1011 microwaves 2.0´ 10-22 – 8.0´ 10-21 1 mm – 25 mm 3´ 1011 – 1.2´ 1013 infrared 8.0´ 10-21 – 2.7´ 10-19 25 mm –750 nm 1.2´ 1013 - 4´ 1014 visible 2.7´ 10-19 – 5.0´ 10-19 750 nm – 400 nm 4´ 1013 – 7.5´ 1014 ultraviolet 5.0´ 10-19 – 2.0´ 10-16 400 nm - 1 nm 7.5´ 1014 - 3´ 1017 x-rays 2.0´ 10-16 - 2.0´ 10-13 1 nm - 1 pm 3´ 1017 - 3´ 1020 gamma-rays > 2.0´ 10-13 < 10 pm > 3´ 1020

The visible portion of the spectrum can also divided up into regions.

Red (647 – 700 nm)

Orange (585 – 647 nm)

Yellow (575 – 585 nm)

Green (491 – 575 nm)

Blue (424 – 491 nm)

Violet (400 – 424 nm)

You may want to use the pneumonic ROY G. BiV to remember the order of the visible region of the electromagnetic spectrum.

For a pictoral representation of the spectrum see Science Science Hypermedia’s page on Electromagnetic Radiation.

Wave-Particle Duality of Matter

In 1925, Louis DeBroglie hypothesized that if light, which everyone thought for so long was a wave, is a particle, then perhaps particles like the electron, proton, and neutron might have wave-like behaviors.

In the same way that waves are described by their wavelength, particles can be described by their momentum, p

p = mv

where m is the mass of the particle and v is its velocity.

We can relate the velocity of a wave-particle with its wavelength by equating Planck’s relationship for the energy of a photon with Einstein’s Law of Relativity:

E = hn = hc/l

E = mc2

If we equate these two equations we get a relationship between momentum (a particle property) and wavelength (a wave property)

hc/l = mc2 = pc

p = h/l

Or by substituting mv for momentum we can wavelength of any object to its velocity and mass.

l = h/mv

The first real experimental proof of this relationship came from Davisson and Germer in 1925, who found that electrons will diffract and interfere like waves, just like X-ray photons (light).

Example

One application of the wave-particle duality of matter is electron diffraction, which is used to determine the distances between atoms in crystalline solids. Experimentally, this is done by accelerating a beam of electrons to the point where the wavelength of the electron beam is of the same order of magnitude as the distance between atoms (roughly 0.1 – 0.5 nm). How fast would an electron need to be travelling in order to have a wavelength of 0.1 nm?

l = h/mv

v = h/ml = [6.626 ´ 10-34 J-s]/{[9.109 ´ 10-31 kg][1 ´ 10-10 m]}

v = 7 ´ 106 m/s (over 16 million miles per hour)

So, both matter and light are composed of particles that have wave-like properties. The wave-like behavior is only observed on the subatomic length scales where the masses are small enough for the wavelengths to be detectable.

Quantum Mechanics and the Modern Picture of the Atom

As we now know today Bohr’s model of the atom was really an oversimplification of the way electrons actually behave in atoms. However, there were certain aspects of Bohr’s model that were correct.

What was correct about Bohr’s Model:

• Electrons reside in quantized energy levels.
• The Bohr model accurately and quantitatively predicts the energy levels of one electron atoms.

What was incorrect about Bohr’s Model:

• Electrons don’t orbit the nucleus in well defined circular orbits.
• Fails to accurately predict the energy levels in multielectron atoms.

Heisenberg Uncertainty Principle

Somewhat ironically a young scientist named Werner Heisenberg, who was working in Bohr’s lab as a postdoctoral researcher, developed the theory which showed that electrons do not orbit the nucleus in well defined circular orbits. His theory is called the Uncertainty Principle.

Uncertainty Principle ® It is not possible to precisely determine the momentum (hence the energy) and the position of a particle simultaneously. This is quantified in the mathematical expression:

Dx Dp ³ h/4p

Dx m Dv ³ h/4p

where Dx represents the uncertainty in the position of the particle and Dp represents the uncertainty in the momentum of the particle (p = mv).

The Heisenberg uncertainty principle can be understood from several points of view.

Pure waves are not localized in space. Consider a guitar string, we can tell the energy of the wave by the frequency of the sound given off by the string, however, it is not possible to define a precise location of the wave. In fact the wave exists along the entire length of the guitar string. We can extend this line of thinking to particles such as electrons because they behave in part as waves (as shown by DeBroglie).

To conceptualize simultaneous measurement of the position and momentum of an electron consider the analogy to photographs taken at night of a busy intersection. If we use high speed film a photograph will show us the positions of every car at a given moment, but it will not give us any information regarding their speed. On the other hand if we use a long exposure time then the moving headlights will show up a streaks. The length of the streak gives us a good estimate of each car’s speed (long streaks for fast moving cars, short streaks for slow moving cars), but now we can’t identify the exact positions of the cars.

You may think that our inability to measure both the position and momentum of an electron (or substitute your favorite subatomic particle) is caused by inadequate instrumentation. However, this is not true, the Uncertainty Principle is a fundamental law of physics. You can understand this by thinking about how we would go about making such a measurement.

Perhaps the ultimate way to measure the position and energy of an electron would be to take a picture. In a certain sense this is exactly what scientists do. They shine light on the electron and study how interacts (bounces back, is bent, etc.) with the electron. Now in order to very accurately measure the position and size of the electron we need light with a wavelength similar in size to that of the electron (this is a general principle of diffraction experiments). This would be an extremely small wavelength, thus the frequency and energy of such photons would be very high. However, if a very high energy photon collides with an electron it will alter the momentum of the electron (imagine firing a high powered pellet gun at a rolling billiard ball). Thus the measurement alters the momentum of the electron. Going in the opposite direction if we use a long wavelength photon, so that the energy of the photon is small, our measurement will not significantly alter the momentum of the electron. Unfortunately, the long wavelength will lead to a large uncertainty in the position of the electron. This is the physical reality of the uncertainty principle.

The uncertainty principle has important ramifications for our picture of the atom. In the Bohr model of the atom the energy (thus its momentum) of the electron and the radius of its orbit (thus its position) are precisely defined quantities. This is a direct violation of the Uncertainty Principle. Since the energy levels of the hydrogen atom had been shown to be experimentally well defined by Bohr’s model (Dp ~ 0), we are left with the conclusion that we know very little about the exact location of the electrons in space (Dx ~ infinity).

Example

An electron moving near an atomic nucleus is measured to have a speed of 6 ´ 106 m/s plus or minus 1%. What is the uncertainty in its position?

First I calculate the uncertainty in the velocity.

Dv = [6 ´ 106 m/s][0.01] = 6 ´ 104 m/s

Dx m Dv ³ h/4p

Dx ³ h/(4p Dv m)

Dx ³ [6.626 ´ 10-34 J-s]/{4p[6 ´ 104 m/s][9.11 ´ 1031 kg]}

Dx ³ 1 ´ 10-9 m = 1 nm

This result indicates that the uncertainty in the position of the electron is approximately 10 times larger than the diameter of the atom. Therefore, we really have no idea where in the atom the electron is located.

Schrodinger’s Wave Equation

Schrodinger looked at Bohr’s model of the hydrogen atom and retained the good points of the model, namely the presence of quantized energy levels for the electron. In light of the uncertainty principle he discarded the notion that electrons move about the nucleus in well defined circular orbits. Instead he described the properties of the electron in terms of a wave function, y . The wavefunction is a mathematical function that tells one the energy of the electron (if you work through the advanced math associated with quantum mechanics).

In terms of picturing where the electron is or how it moves the wavefunction doesn’t have any physical significance. However, the square of the wavefunction, y 2, expresses the probability that the electron will be found at a given point. If we plot the value of y 2 as a function of position we get an electron density distribution (see figure 6.18 in your book). The electron density distribution is somewhat like a photograph of the atom with a long exposure time. In the case of a hydrogen atom most of the time the electron is found rather near to the nucleus, but every once in a while it is found very far from the nucleus. This does not mean that the electron itself is smeared out into a spherical cloud of charge.

To better visualize the electron density distribution (y 2), picture the students and seats in our lecture hall. Now lets say that for every lecture 150 people show up, but there are 250 seats. If I take a picture at any given time I see that some of the seats are filled while others are empty. If I were now to take a time exposure photograph over say 1000 lectures, I would find that the occupied seats are not always the same. They change from one class to the next. Furthermore, some seats (such as rows 3 – 7) are occupied a high percentage of the time, while others (such as the balcony seats) are rarely occupied. We could then say that the distribution of students reaches its highest density at say the 5th row and drops off as we move the upper reaches of the classroom. In a similar way the electron is moving about the atom, spending most of its time at a comfortable distance from the nucleus, but every once in a while we find it either quite far or unusually close to the nucleus.

Orbitals and Quantum Numbers

Using the mathematics that Schrodinger developed one can obtain a set of wave functions, each with a corresponding energy, which satisfy the mathematical equations derived for the hydrogen atom. Each wavefunction corresponds to an atomic orbital. Each orbital has a specific energy and shape (as determined from its electron density distribution). It’s important to keep in mind that the "orbital" described by Schrodinger bear no resemblance to the the "orbits" developed by Bohr. An "orbit" was meant to describe a path followed by the electron as it moves around the nucleus, whereas an "orbital" is purely a mathematical solution to Schrodinger’s equation.

This is obviously not an upper level class in math or physics, so we cannot go into the details of the wavefunctions describing each orbital. Fortunately, much of the qualitative information contained in the quantum-mechanical description of the electron is contained in four numbers associated with each electron in the atom. These numbers are known as the quantum numbers.

In Bohr’s model there was one quantum number, n, which described both the energy and the size of the orbit. In the quantum-mechanical description there are three quantum numbers which describe the energy of the electron, and the size, shape and orientation of the electron density distribution. As implied by the name the quantum numbers can only take discrete values, and the values they can adopt are limited by the values of the other quantum numbers.

Principle Quantum Number (n)

Possible Values = 1, 2, 3, …

The principle quantum number defines the size of the orbital. As n increases we see the following trends:

• The orbital becomes larger
• The energy of the electron increases
• The electron is less tightly bound to the nucleus

Azimuthal Quantum Number (l)

Possible Values = 0 to n-1

The azimuthal quantum number defines the shape of the orbital. We will commonly use a letter designation to designate the value of l.

 Value of l 0 1 2 3 Letter Designation s p d f

The pictures in your book show the shapes of the electron density distributions associated with each value of l.

• s = sphere
• p = dumbbell
• d = double dumbbell

Magnetic Quantum Number (ml)

Possible Values = l, …, -l

The magnetic quantum number describes the orientation of the orbital.

 Subshell l Values of ml Orbitals s 0 0 s p 1 1, 0, -1 Px, py, pz d 2 2, 1, 0, -1, -2 dxy, dyz, dxz, dx2-y2, dz2 f 3 3, 2, 1, 0, -1, -2, -3

These three quantum numbers completely define the size, shape, and orientation of an orbital. However, there is a final quantum number which is an intrinsic property of the electron.

Spin Quantum Number

Possible Values = ½, -½

You can imagine the electrons as balls of charge spinning about their own axes, this creates a magnetic field that can be experimentally observed. For example all magnetic materials depend upon the magnetic field which originates on the electron for their magnetic properties.