Hot Topics in Physics:The Basics of Matter

A Conceptual Overview of How Matter Behaves Leading up to

Chemistry

Originally Presented at Mesa Community College

November 6, 2013

Suppose that a ball is rolling along a flat surface. In 10 seconds it goes 5 feet. How far would you expect it to travel over the next 10 seconds?

A falling object drops 16 feet in one second. In the next second would you expect it to fall more, less or the same amount?

Some physics questions

With no interference, another 5 feet

More than 16 feet

In science we seek to understand and make sense of the world around us.

We like to have things predicable, or even mechanical.

When it comes to understanding the matter that makes up everything around us, we need to focus on the building blocks. This means dealing with atoms and their components.

When we talk about atoms we must first begin with what makes up an atom.

Central part, called the nucleus. This is where almost all the mass is. The nucleus holds the atom together.

Tiny electrons buzz around the nucleus, filling in most of the space.

Overlapping electrons from neighboring atoms trap the atoms together.

When atoms are trapped together we say that there is a Bond between them.

So understanding how atoms bond together means understanding electrons.

But there are two big complications!

First: At the basic level matter shares the same level of predictability as people.

One person is difficult or to impossible to predict in terms of future behavior.

Groups of people are much easier to predict.

No problem, we just look for trends of behavior or chances that certain types of behavior will occur.

This leads to complication #2

The ordinary way we go about determining the chances of different outcomes is different for matter at the basic level than it is for our everyday world. Example: Suppose that one shots blasts from a single or double-barrel shotgun at a target. We would expect to see patterns similar to these.

If we imagine a strange shotgun where the shot from a single cartridge gets divided up between two barrels, the pattern should be similar.

Double BarrelSingle Barrel

Now apply that same logic to a beam containing electrons, atoms and so forth.

Split it up

Target

This looks like the shotgun scenario, so we should have a similar outcome.

Beam hits target via 2 routes

Source

However…. This is not always what we see in reality

We could see our familiar friend ..

Or if we simply adjust the speed of the matter being examined we can see something new.

or

A real example: Shooting electrons through a powdered crystal

We need to adjust our thought process to give us wiggle-room for the different outcomes.

We give ourselves the needed wiggle-room by working with what are called

Probability Amplitudes

Simply put, instead of talking about the chances for a certain outcome to occur, we work with numbers that, when squared, gives us the chance for the outcome.

Here is how the game is played: An example

What is the chance that, if you roll a die, you will roll a 5?

1 possible way to get 5

6 total possible outcomes

What is the chance that, if you roll a die, you will roll a 4?

What is the chance that, if you roll a die, you will roll a 4 or 5?

This tells us to expect a roll of a 4 or 5, on average, once every 3 rolls.

How do you play the game with Probability Amplitudes?

What is the probability amplitude that, if you roll a die, you will roll a 5?

What is the probability amplitude that, if you roll a die, you will roll a 4?

What is the chance that, if you roll a die, you will roll a 4 or 5?

We have two possible answers, depending on whether we add or subtract the numbers.

Using amplitudes gives us wiggle room to resolve our problem. In fact probability amplitudes can also be complex numbers, giving added flexibility.

Working through the mathematics of Probability Amplitudes is the business of Quantum Mechanics.

The mathematics can be formidable.

Schrodinger Equation:

Dirac Equation:

For example when the equations are applied to the electron in a Hydrogen atom one gets equations like these.

Don’t Panic, Remain calm

All that we are interested in as far as the math goes is that it produces a family of equations. Without worrying about the exact details of these equations we will consider the consequences.

As with any family, we need a way to distinguish one member from another.

For our equation family, this means introducing a set of numbers called Quantum Numbers.

n, l, ml

Also it is convenient to include a 4th Quantum Number, ms

So what values can n, l, ml and ms have?

• n can be any natural number: 1, 2, 3, …

• l can be any whole number less than n: 0, 1, 2, … n-1

Example: If n = 5 then l can equal 0, 1, 2, 3 or 4

• ml can be any integer between –l and +l

Example: If l = 3 then ml can equal -3, -2, -1, 0, 1, 2 or 3

• ms can only be - ½ or + ½

So let us summarize where we’ve been

• Matter is fundamentally unpredictable. We need to devise new rules to predict the chances of different outcomes being realized.

• We deal in “probability amplitudes.” The equations dealing with these can turn out to be nasty.

• When applied to a Hydrogen atom we get a family of solutions distinguished by a set of four “quantum numbers,” n, l, ml, ms.

• Historically the results for Hydrogen have been generalized to describe the electrons in atoms with multiple electrons.

So what has all this bought us?

We cannot predict where an electron for an atom is located, so let us suppose that we did countless experiments measuring where electrons are and plotted them. What might we see?

Consider the case where n = 1, l = 0, ml = 0 and ms = ± ½

We see that the electrons will tend to be found grouped around the nucleus.

While we cannot tell where the electron will be, and so cannot define its path as an orbit, we can describe the region it is most likely to located as an orbital.

For all solutions where l = 0 the convenient region is just a spherical region called an s-orbital.

How about the other solutions? Consider the cases where: l = 1, ml = 0 and ms = ± ½

l = 1, ml = ± 1 and ms = ± ½

For the ml = 0 case we see that the electrons tend to group in clusters on either side of the nucleus.

For the ml = ± 1 case the electrons form a doughnut type shape around the nucleus. This is no good for atoms with multiple electrons as it would tend to put them too close together. However if we add or subtract them we get the same clustering in perpendicular directions.

So for the three l = 1 solutions we also have orbitals. Only now it is 3 dumbbell shaped orbitals in perpendicular directions.

We can keep this up ad nauseum. However the pattern is already beginning to form. Each l value has a set of orbital(s) describing the most possible locations to find the electron.

l = 0 (one s orbital) ml = 0

l = 1 (3 p orbitals) ml = -1, 0, 1

l = 2 (5 d orbitals) ml = -2, -1, 0, 1, 2

But how does all this relate to the arrangement of atoms in a molecule?

Remember that atoms bond via the sharing of electrons. The atoms must be arranged to reflect where the electrons are. These are where the orbitals are.

Example: Hydrogen molecule (Two 1s orbitals)

Example: Chlorine molecule (Two 3p orbitals)

In our previous examples we saw molecules in nice straight lines where only one pair of electrons being shared.

Yet often times our previous arrangement does not work, the electrons will try to spread themselves out more than is permitted by the orbitals we have developed.

Example: Methane (The carbon’s 2p orbitals cannot account for this)

The problem is that we generalized our results for a hydrogen atom (one electron) to all atoms. The solution is to mix around the existing orbitals into a new group of orbitals called Hybrid Orbitals.

The end result is that we have a set of orbitals that allow for atoms to be arranged such that the electrons are spread out as far from each other as possible.

Example: Linear (CO2) Example: Trigonal planar (BCl3)

Example: Trigonal pyramidal (NH3 – Nitrogen has two unseen pairs of electrons)

One other aspect an electron in an atom is its energy.

Energy, like any aspect of matter in a Quantum World, is not always predictable.

However our family of solutions is built under the basis that energy is fixed, with values that would repeatedly be found in an actual measurement.

What this indicates that the electron is viewed as being restricted to the energy values allowed it by the family of solutions. These values are called Energy Levels.

When an electron gives off light it does so by going from one energy level to another.

This means that the wavelengths of light that are given off are restricted to a set of possibilities.

Full spectrum

Hydrogen

Helium

Empirical

Examples:

Let us return to our earlier Physics question.

Suppose that a ball is rolling along a flat surface. In 10 seconds it goes 5 feet. How far would you expect it to travel over the next 10 seconds? How can we incorporate what we have seen to this question?

•We cannot guarantee that we will know will it will be. But we can use probability amplitudes to determine the likelihood of it being found in different locations.

•The probability amplitudes must be able to shift how they are distributed.

In order to allow amplitudes to shift around they must be viewable as quantities can that move about.

These movable amplitudes are describable as waves.

Usually matter itself gets described as a wave itself. This marriage of convenience where matter is considered as a mixture of particle properties with wave properties is called wave-particle duality.

The core idea between the wave-particle philosophy:

• Prior to taking a measurement, one can view matter as being a wave.

• At the point of measurement the matter becomes a particle.

Thank-You!!