November 1, 2011

CH-111

Chapter 6

Electronic Structures of Atoms

 

Electronic Structure: the arrangements of electrons in atoms.

Refers to the number of electrons in the atom as well as their distribution around the nucleus and their energies.

 

Electromagnetic Radiation: aka: radiant energy, carries energy through space, a form of energy that exhibits wave-like behavior as it travels through space.

 

Electromagnetic radiation moves through the vacuum of space at the speed of light: 3.00X10^8m/s (299,792,458m/s! Thank you, Professor Brian Cox and "Why Does e=mc^2".) and all types have wave-like characteristics. Wavelengths are periodic, meaning they peak and trough in a pattern that repeats itself at regular intervals. The distance between two adjacent peaks is what we call a wavelength.

 

Wavelength: the distance between two adjacent peaks (or two adjacent troughs) on a cross sectional view of a wave.

Frequency: the number of complete wavelengths, or cycles, that pass a given point each second.

 

These characteristics, as well as all other wave characteristics of electromagnetic radiation (EMR), are due to the periodic oscillations in the intensities of the electric and magnetic fields associated with the radiation.

 

All EMR moves at the same speed, 3.00 x 10^8m/s, light speed. Longer wavelengths = fewer cycles of the wave pass a given point per second = low frequency. High frequency waves are inversely proportionate where shorter wavelengths = more cycles passing a given point each second = high frequency. This inverse relationship between frequency and wavelength of electromagnetic radiation is expressed by the equation:

c=ʎv

c is the speed of light, ʎ is Greek lambda for wavelength, and v is frequency.

PS: windows character map, discovered!

Different types of electromagnetic radiation have different properties because of their differing wavelengths.

Angstrom = 10^-10 = X-Ray

Nanometer = 10^-9 = Ultraviolet, visible

Micrometer = 10^-6 = Infrared

Millimeter = 10^-3 = Microwave

Centimeter = 10^-2 = Microwave

Meter = 1 = Television, radio

Kilometer = 1000 = Radio

 

It is important to remember here that in the spectrum of waves, the only ones we as humans can actually perceive account for a small portion compared to the amount of waves that exist. Visible light, for us, is only 400-750nm. The range of waves spans from 10^-11 (gamma rays) to 10^3 (radio frequencies). Look at this awesome chart that I stole:

 

 

That chart makes a pretty powerful point in showing us how very little we can actually perceive in the world of electromagnetic radiation. So, so small.... siiiigh. Also this chart kind of makes me regret all that tanning I used to do in tanning machines when I was younger (or, more accurately, last summer). Siiiiigh. So gross.

 

Back to science.

 

Frequency is expressed in cycles per second, or more commonly known as herz (Hz), now you can understand all those silly values you're looking at when picking out your flatscreen. Yaaay. The frequency unit is usually given per second, or /s or s^-1 as in 820 kHz = 820,000 Hz = 820,000 s^-1 = 820,000/s <--- all the same.

 

Quantized Energy & Photons:

 

Blackbody radiation: the emission of light from hot objects (called blackbody radiation because the objects studied appear black before heating).

Photoelectric effect: the emission of electrons from metal surfaces on which light shines.

Emission Spectra: the emission of light from electronically excited gas atoms.

 

The three phenomena listed above cannot be explained by the wave model.

 

Hot objects and the Quantization of Energy

Wavelength distribution is dependent on temperature, for instance a red-hot object is cooler than a white-hot object. There is a relationship between the temperature and the intensity and wavelength of the emitted radiation. Planck gave the name quantum to the smallest quantity of energy that can be emitted or absorbed as electromagnetic radiation. He proposed that energy, E, of a single quantum equals a constant times the frequency of the radiation, or: E=hv

The h represents Planck's constant.

 

Planck's Constant: 6.626 x 10^-34 J/s (Joules per second)

 

According to Planck's theory, matter can emit and absorb energy only in whole number multiples of hv, ie: 2hv, 3hv, etc. Ex: if the quantity of energy emitted by an atom is 3hv, we say that three quanta of energy have been emitted. Quanta is the plural for quantum. Because energy can only be released in specific amounts, we say that the allowed energies are quantized- their values are restricted to certain quantities.

 

The Photoelectric Effect and Photons

Photoelectric Effect: light shining on a clean metal surface causes the surface to emit electrons. A minimum frequency of light, different depending on type of metal, is required for the emission of electrons to occur.

 

Einstein explained the photoelectric effect in 1905... because he was a baller.He explained it by assuming that the radiant energy striking the metal surface behaves like a stream of tiny energy packets, each like a particle of energy, called photons. Remember photons? Yay! Einstein extended Planck's quantum theory. He deduced that each photon must have an energy equal to Planck's constant times the frequency of the light:
Energy of photon = E=hv

Thus, radiant energy itself is quantized.

 

Given the right conditions, photons striking a metal surface will transfer their energy to electrons in the metal and with a certain amount of energy, referred to as work function, the electrons overcome the attractive forces holding them to the metal. This results in an intensity of light that is  related to the the number of photons striking the surface per unit time but not to the energy of each photon.

 

Einstein won the Nobel, because he is a baller.

 

All of this is well and good and obviously hurray for Einstein for being such a smart dude, but the photoelectric effect also poses a dilemma. Is light a wave or does it behave like a particle? To resolve this, we have to consider light as both wave-like and particle-like in nature and will behave more like one or the other depending on the situation.

 

Line Spectra and the Bohr Model

Monochromatic: radiation composed of a single wavelength.

Polychromatic: radiation containing many different wavelengths.

Spectrum: produced when radiation from a source is separated into its component wavelengths.

 

When high voltage is applied to tubes containing different gases under reduced pressure, the gases emit different colors of light, as in neon signs.

 

Line Spectrum: a spectrum containing radiation of only specific wavelengths.

 

Rydberg equation: 1/ʎ = (RH)(1/n1^2 - 1/n2^2)

Where ʎ is the wavelength

RH is Rydberg Constant = 1.096776 x 10^7m^-1

n1 and n2 are positive integers with n2 > n1

 

The remarkable simplicity of the equation took 30 more years to explain.

 

Bohr's Model

According to classical physics, a charged particle (such as an electron) moving in a continuous orbit (circular path) should continuously lose energy. As the electron orbiting the positively charged nucleus (proton) loses energy, it would be sucked into the nucleus, as it's negative charge lessened and ceased to maintain the energy necessary to repel it away. However, this never happens. Bohr solved this  adopting Planck's idea that energies are quantized. Bohr based his model on three postulates:

1. Only orbits of certain radii, corresponding to certain specific energies, are permitted for the electron in a hydrogen atom.

2. An electron in a permitted orbit is in an "allowed" energy state. An electron in an allowed energy state does not radiate energy and, therefore, does not spiral into the nucleus.

3. Energy is emitted or absorbed by the electron only as the electron changes from one allowed energy state to another. This energy is emitted or absorbed as a photon that has energy E=hv.

 

So basically:

1. Electrons hold orbits of certain energy levels, this is the only way they can exist in an atom.

2. The electrons that are in these orbits are in a fixed state of energy, meaning they do not lose any energy and thus can continue to orbit around the nucleus without getting sucked in.

3. The electron only changes its orbit (increasing or decreasing in radius) when energy is added or removed. This is the only time an electron will absorb or give off energy, which will occur as a photon with energy being equal to the equation E=hv.

 

The Energy States of the Hydrogen Atom

I forgot to mention the important little fact that all of what was just stated above was determined by Bohr's dealings with the H atom. He was attempting to explain the line spectrum of hydrogen, which he does by showing that it has four allowed energy levels and thus 4 wavelengths and thus, emits 4 lines in a line spectrum.

 

Holy mackarole, it's about to get harder.

 

E = (-hcRH)(1/n^2) = (-2.18 x 10^-18J)(1/n^2)

What the **** is that?

WELL! h represents Planck's Constant, remember? (hint: 6.626 x 10^-34J/s)

And c represents the speed of light, 3.00 x 10^8m/s

And RH represents Rydberg's Constant, 1.096776 x 10^7m^-1

 

 

The integer n, which can have whole-number only values from 1 to infinity and is called the principal quantum number. Each orbit corresponds to a different n value. An increase in radius of orbit results in a larger n value. So the two are related. The orbits begin closest to the nucleus and continue outward, increasing in energy as they are located further and further from the nucleus.  The lowest, n = 1, and so forth. The lowest energy state (n=1) is called the ground state of the atom. Note that in the equation

E = (-hcRH)(1/n^2) = (-2.18 x 10^-18J)(1/n^2)

the values for n are negative. The lower (ground state, more negative) the energy is, the more stable an atom is. When an electron is in a higher energy state (n=2 or higher) the atom is referred to as being in an excited state.

It's really hard to summarize this chapter because literally every single sentence is important. I feel like typing word-for-word. But I won't. That's cheating. Doing this is only useful to me if I'm forced to describe it in my own words, that being impossible to accomplish without some basic grasp and understanding of the concepts. Okay I'm done complaining.

When the orbit radius and the energy of n become infinitely large the electron completely separates from the nucleus. When n= ∞ the electron is completely separated from the nucleus and the energy of the electron is zero:

 

E=(-2.18 x 10^-18J)(1/∞^2) = 0     It equals ZERO!!

 

This state, in which the electron is removed from the nucleus, is called the reference or zero-energy state of a hydrogen atom.

 

Bohr also assumed (in his third postulate) that if an electron absorbs or emits the exact amount of energy that corresponds to the difference between orbits, it will move to a higher or lower energy state:

ΔE = Ef - Ei = Ephoton = hv

Where change in energy is equal to the initial state of energy subtracted from the final state of energy which is equal to the photon energy which is equal to Planck's Constant (h) multiplied by frequency (v).

 

Basically we're reiterating what we considered before but showing it as an equation. Only the specific frequencies of light satisfy the equation above can be absorbed or emitted by the atom.

So now we're doing that thing where we stick all the various equations together to get one big bad equation. Remember that v=c/ʎ or frequency=speed-of-light/wavelength, and recalling our first equation for calculating energies (way up top, highlighted green), we mash them all together in a chemical physics squish of doom!

 ΔE = hv = hc/ʎ = (-2.18 x 10^-18J) (1/nf^2 - 1/ni^2)

Head hurting yet?

Well, the (-2.18 x 10^-18) corresponds to the green equation where -hcRH = -2.18x10^-18 (really, I promise, if you multiply those constants by the speed of light you'll get 2.18x10^-18, I know because I was lost and didn't understand where tf -2.18x10^-18J was coming from so I backtracked and put it all together. Those constants multiply out to 2.18x10^-18 and remember the negative sign in front of hcRH and BAM, there it is!) Then we look back at our whole number integer (principal quantum number) and incorporate their initial and final energies, which is 1/nf^2 - 1/ni^2 and that's how we make sense of the equation above.

Not done yet.

If nf is smaller than ni, the electron moves closer to the nucleus and ΔE is a negative number. Think, an electron moving close to the nucleus is losing energy, therefore it's final energy state would be less than its initial. Conversely, if the electron was gaining energy and moving away from the nucleus, nf would be greater than ni. If nf < ni the atom is releasing energy. If nf > ni the atom is absorbing energy. Let's plug in some numbers and play with this equation a little so that we can SEE what is happening rather than just saying it:

ΔE = (-2.18 x 10^-18J)(1/1^2 - 1/3^2)

ΔE = (-2.18 x 10^-18J)(8/9)

ΔE = 1.94 x 10^-18J

We see, firstly, that nf was less than ni (nf=1 ni =9), which means the electron is losing energy (emitting energy) and moving closer to the nucleus. By this calculation, it is emitting 1.94x10^-18J as a photon. Knowing the energy of the emitted photon, we can now determine its frequency or wavelength using the following calculations:

For wavelength:

ʎ = c/v = hc/ΔE

ʎ = (6.626 x 10^-34J-s)(3.00 x 10^8m/s)/1/94 x 10^-18J

Where wavelength (ʎ) is equal to speed of light multiplied by frequency or Planck's Constant multiplied by speed of light divided by change of energy (which we already determined in our previous equation. See all that highlighting? That took me forever. Longer even than it took me to understand that equation. But the highlighting separates and focuses each point. I like it. SO yeah......

1.02 x 10^-7m = (6.626 x 10^-34 J-s)(3.00 x 10^8m/s)/1.94 x 10^-18 J

We don't include the negative sign because wavelengths are always expressed as positive values, but we do indicate that a photon of wavelength 1.02 x 10^-7 has been EMITTED. This ensures that we know the direction of energy flow.

The Bohr model explains well the line spectrum of the hydrogen atom, however it cannot explain the spectra of other atoms, at least not in an exact way. Bohr also just assumed that the negatively charged atom wouldn't fall into the nucleus of the atom, thereby avoiding an actual explanation of why it does not happen. Finally, an acceptable electronic structure model must accommodate the fact that electrons also exhibit wavelike properties. Bohr's model was an important step along the way, but it fell short of a being completely comprehensive. His model, however, introduced these two very important ideas:

1. electrons exist only in certain discrete energy levels, which are described by quantum numbers.

2. energy  is involved in the transition of an electron from one level to another. 

The Wave Behavior of Matter

Depending on the circumstances, radiation appears to have either a wave-like or particle-like (photon) character. Louis de Broglie extended this idea by suggesting that an electron moving about the nucleus of an atom behaves like a wave and therefore has a wavelength.  He proposed that the wavelength of an electron, or any other particle, depends on its mass (m) and its velocity (v), which is a different (v) than the (v) we used above for frequency. I couldn't find the stupid symbol for the v my book used in the previous equation in Windows character map so I just used a normal v. Rest assured, THIS (v) is actually denoted by a normal (v) where the aforementioned (v) was actually the lowercase (v) for the Greek letter "Nu", which is ridiculously hard to find a symbol for so I just used a v. I'm being redundant. Moving on....

ʎ = h/mv

Where wavelength is equal to Planck's Constant divided by mass multiplied by velocity, or momentum (which is mass x velocity = momentum).

De Broglie used the term "matter waves" to describe the wave characteristics of material particles. His hypothesis is applicable to all matter, however we see that the larger the matter, the smaller the associated wavelength and therefore, the things we see have wavelengths that are imperceptible to us.

Sample Exercise!

Q: What is the wavelength of an electron movie with a speed of 5.97 x 10^6m/s? The mass of the electron is 9.11 x 10^-31kg.

A: We're basically given everything we need and therefore really only need to plug the values into the equation ʎ = h/mv

ʎ = 6.626 x 10^-34J/s / (9.11 x 10^-31kg)(5.97 x 10^6 m/s)

ʎ = 6.626 x 10^-34J/s /  5.43867 x 10^-24

The answer must have units cancelled out to be in the correct unit (m) so

6.626 x 10^-34J/s / (9.11 x 10^-31kg)(5.97 x 10^6m/s) X (1kg - m^2/s^2/1J)

So the J-s and the m/s will cancel out leaving us with just m, so the final answer is:

ʎ = 1.22 x 10^-10m

Now I'm not really sure why my book goes further and converts it into Angstroms because the question asked for wavelength and wavelength is given in m, I guess because Angstrom is given by 10^-10 so this wavelength could be expressed as 1.22 Angstroms and then referred back to a specific table in my book that divides the type of electromagnetic radiation by unit. Angstroms are X-rays. The question wasn't really asking for the type, just the wavelength so I don't think it's entirely necessary to take it that far, but it is interesting anyway.

The Uncertainty Principle

Ah, yes, Heisenberg's Uncertainty Principle. I've been inundated with it since I started watching The Science Channel something like 10+ years ago.

Here we start to see that for every answer we get in the science we are posed ten new questions.

A wave extends in space and its location is not precisely defined, therefore we might anticipate that it is impossible to determine exactly where an electron is located at any given instant. Heisenberg stated that the dual nature of matter places a fundamental limitation on how precisely we can know both location and momentum of an electron at a given time. This becomes extremely important at the subatomic level. Heisenberg related the uncertainty in position, Δx, and the uncertainty in position, Δ(mv), to a quantity involving Planck's Constant (it's everywhere!):

Δx · Δ(mv) ≥ h/4π

Q: The electron has a mass of 9.11 x 10^-31kg and moves at an average speed of about 5 x 10^6m/s in an H atom. Assuming we know the speed to an uncertainty of 1% (that is 0.01 of 5 x 10^6m/s, which is 5 x 10^4m/s), and that this is the only source of uncertainty in the momentum, calculate the uncertainty in position of the electron.

A: We use that ballin' equation above to do as such by plugging in values, twisting it around a bit first like this:

Δx ≥ h/4πmΔv        <----although this looks wicked scary, it's really not so bad after we input the values:

Δx ≥ 6.626 x 10^-34 J-s / 4π(9.11 x 10^-31 kg)(5 x 10^4 m/s)

Δx ≥ 1 x 10^-9 m

That means the uncertainty of the position of an electron in the H atom is 1 x 10^-9m. Consider that the diameter of an H atom is only about 1 x 10^-10m and we can see that we have essentially no idea whatsoever where the electron is located. If we plugged in a mass of an observable size, like a cup of coffee or something like that, we'd see the uncertainty decrease to an inconsequential number.

Quantum Mechanics and Atomic Orbitals

Our friend Schrodinger, famous for his cat, proposed an equation known as Schrodinger's wave equation, which incorporated both wave-like and particle-like behaviors of the electron. His new approach marked the beginnings of what we know as quantum mechanics or wave mechanics. According to my book, Schrodinger's equation requires advanced calculus so we're not going to explore it mathematically :( or rather :'( which is an emoticon of me crying about that. Well, math was never my strong suit anyway so continuing on.......

Schrodinger treated the electron in an H atom like a save on a plucked guitar string. Because these waves do not travel in space, they are called standing waves. Just as the plucked guitar string produces a standing wave that has a fundamental frequency and higher overtones (harmonics), the electron exhibits a lowest-energy standing wave and higher-energy ones. Furthermore, just as the overtones of the guitar string have nodes, points where the amplitude of the wave is zero, so do the waves characteristic of the electron. Solutions to Schrodinger's equation lead to a series of mathematical functions called wave functions, represented by the symbol ψ which is Greek psi.

Although wave function has no direct physical meaning, the square of the wave function, ψ^2, provides information about the electron's location when it is in an allowed energy state. For the H atom, these correspond to the allowed energies predicted by the Bohr model. The square of the wave function, ψ^2, represents the probability that the electron will be found at that location.

Probability Density: or electron density: ψ^2

Electron density distributions plot dots in the areas where electrons are most likely to be found. You can look at the Google image results for "electron density distributions" by clicking the link. 

Orbitals and Quantum Numbers

Schrodinger's equation yields a set of wave functions called orbitals.

Orbitals: a characteristic shape and energy around an atom in which an electron may occupy given its energy state. It is not the same as an orbit, as it describes electrons in terms of probabilities visualized as "electron clouds" rather than an "orbit" because the motion of an electron in a atom is undetermined (Heisenberg).

Quantum mechanical model uses three quantum numbers: n, l, and Ml to describe an orbital.

1. n is the principal quantum number, and can have positive integral values 1, 2, 3... with increase of n resulting in increase in orbital and the electron spending more time farther away from the nucleus and thus a higher energy and therefore less tightly bound to the nucleus.

 

2. l (lower case L) is the angular momentum quantum number, and can have integral values from zero to (n-1) for each value of n. This number defines the shape of the orbital and the value of l is generally designated by the letters s, p, d, f

Value of L            Letter used

0                                 s

1                                 p

2                                 d

3                                 f

3. Ml is the magnetic quantum number and can have integral values between -l and l, including zero. The quantum number describes the orientation of the orbital in space.

Subshell: set of orbitals that have the same n and l values. Each subshell is designated by a number (the value of n) and a letter (s,p,d,f corresponding to the value of l).

For example: an orbital having the principal quantum number of 3 and the angular momentum quantum number of 2 are called 3d orbitals because n=3 and 2 corresponds to the letter d for the l value. 

I'd love to put the chart from my book here but I can't find a comparable version on Google so I give up. If it's not on Google it doesn't exist.

The restrictions on possible values give rise to the following very important observations (word for word from my book....):

1. The shell with principal quantum number n consists of exactly n subshells. Each subshell corresponds to a different allowed value of l from 0 to (n-1). Thus, the first shell (n=1) consists of only one subshell, the 1s (l=0); the second shell (n=2) consists of two subshells, the 2s (l=0) and 2p (l=1) and so forth. 

Khan academy does an excellent tutorial on electron orbitals, although I don't think it's in depth enough to be an acceptable alternative to learning from the book, it is helpful. 

2. Each subshell consists of a specific number of orbitals. Each orbital corresponds to a different allowed value of Ml, ranging from -l to +l. Thus, each s (l=0) subshell consists of one orbital; each p (l=1) subshell consists of three orbitals; each d (l=2) subshell consists of five orbitals, and so on.

3. The total number of orbitals in a shell is n^2, where n in sthe principal quantum number of the shell. The resulting number of orbitals for the shells 1, 4, 9, 16 are related to a pattern seen in the periodic table: we see that the number of elements in the rows of the periodic table, 2, 8, 18, and 32, equals twice these numbers.

Dude. I made that table.  **** yeah, table skills.

So that's basically all I'm doing on this topic. The rest of the chapter details orbital characteristics and while they are super super cool, I'd rather read than write about them. Up next: Periodic Properties of the Elements!

Can't. Freaking. Wait.