This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.
Video 1.3 - Quantization of Energy
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Del curso dictado por University of Minnesota
Statistical Molecular Thermodynamics
163 calificaciones
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De la lección
Module 1
This module includes philosophical observations on why it's valuable to have a broadly disseminated appreciation of thermodynamics, as well as some drive-by examples of thermodynamics in action, with the intent being to illustrate up front the practical utility of the science, and to provide students with an idea of precisely what they will indeed be able to do themselves upon completion of the course materials (e.g., predictions of pressure changes, temperature changes, and directions of spontaneous reactions). The other primary goal for this week is to summarize the quantized levels available to atoms and molecules in which energy can be stored. For those who have previously taken a course in elementary quantum mechanics, this will be a review. For others, there will be no requirement to follow precisely how the energy levels are derived--simply learning the final results that derive from quantum mechanics will inform our progress moving forward. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts.
Conoce a los instructores
Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics
Chemistry
The universe behaves in a very counterintuitive manner.
When you get to a small enough size scale.
And in this video I want to talk about that.
I want to talk about the quantization of energy.
That is, I'll talk a little bit about quantum mechanics.
And you may be thinking to yourself, wait a minute.
I thought I was taking a course on thermodynamics, not a course on quantum
mechanics. Well, quantum mechanics provides the
foundational principles for all molecular processes.
Molecular statistical mechanics builds upon that foundation, and serves itself
as the basis for thermodynamics. So, in order to do thermodynamics, we
need to know something about quantum mechanics, so I won't be developing
quantum mechanics from the ground up, building quantum mechanical hammers, the
way we will thermodynamical hammers, but you do need to know a few of the results
from quantum mechanics, and that's what I'm going to use this video to talk
about. We're going to develop thermodynamics in
a molecular context. And, molecules, being very small, on a
size scale, behave in a quantum mechanical way, so we need to know those
results. I want to emphasize, it's not a
prerequisite, if you will, to follow this course, to have already have taken a
course in quantum mechanics. If you have, wonderful, you'll have seen
many of these results before. If you haven't, that's fine.
I'm going to present the key features that you'll need in order to accomplish
thermodynamics using some quantum mechanical details.
Quantum mechanics is a fascinating subject, I hope some day I get to teach
about quantum mechanics in this format. But, in any case we'll move on and just
look at the results we'll need. So, first off there is this curious
phenomenon that certainly seems counterintuitive.
Namely, that energy is quantized. So, imagine that your coffee could have
only certain temperatures. So, here is a lovely cup of coffee.
It's actually an espresso shot in a nice little espresso cup.
And if you think about your coffee getting cold, if you were actually going
to do an experiment, go into the laboratory and put a thermometer into
your coffee. You would discover that if the coffee was
initially hot, its temperature would drop, drop, drop as it gets closer and
closer to room temperature over time. And eventually it would be whatever the
ambient temperature is. And there would be a nice, smooth curve.
But what if I told you that, actually, the curve was not smooth?
It should be more staircase shaped. So, we have a certain temperature for a
while, and then it drops to another temperature, and then it drops to another
temperature. And only these discrete levels of
temperature are allowed. So if I were to make a connection between
temperature and energy. That would imply that there are certain
levels of energy that are allowed, and levels in between are not.
So, the real key issue here is the relative spacing of energy versus the
total amount of energy. So, in the case of macroscopic things
like a cup of coffee, a big thing. Well, they have a huge amount of energy
because there's so much stuff, and the spacing of the energy, the quantization
of the energy is really, really small by comparison.
And so, if you have little bitty steps, this is about as little as I can raw them
on this slide, but they should certainly be even littler than that.
Ultimately, it, it looks as though there is a continuum of energy levels available
and we see a nice smooth curve, just the way we observe from macroscopic
phenomena. But we need to do a small enough length
scale, the microscopic scale. Where you're at the level of molecules,
perhaps. Well now, the energy spacing is large,
compared to the total energy. And as a result, you can observe these
gaps in the energy. That are significant.
And we have quantized energy levels. And the impact of quantization is
absolutely decisive on the way matter behaves, on the way the universe behaves.
Energy itself is quantized by this physical constant that's labeled italic
h, and that's Planck's Constant. And so here's Max Planck, a picture from
1900, Max Planck is for whom the Planck Research Institute in Germany are now
named. And he was a physicist.
And he was the first to suggest that radiated energy would come in quantized
packets. And the size of those packets would be
Planck's constant, h, times nu, the frequency of the radiation.
That is he proposed a relationship between frequency of radiation and
energy. And h is the constant that can be thought
of as changing the units. It's the proportionality constant.
And so, if frequency is given in per second, and if energy is given in joules,
the SI unit, then Planck's constant is shown here.
It's 6.626 times 10 to the minus 34th, so that's a pretty small number,
joule-seconds. So you see that when I multiply
joules-seconds times per seconds, I get joules.
So E equals h nu, a very fundamental equation.
And as a result, when you talk about energy, one can now specify energy in
various ways, so for instance, if I give a frequency for radiation, then the
energy is related by multiplying times Planck's constant.
I might, instead of giving you the frequency, give you the wavelength.
And we know the relationship between frequency and wavelength, the product of
the two is the speed of light. So given that, if I replace nu, the
frequency, with c over lamda, which comes from this equation here.
I'll get that h times c divided by lambda gives the energy.
Speed of light in a vacuum, you can look that up.
It's 3.0 times 10 to the 8th meters per second.
And then finally, because people found it a little inconvenient to divide by
wavelength. Division always seems more difficult that
multiplication, I guess. There is another quantity, it's called
the wavenumber. So, the wavenumber is just 1 over the
wavelength. So, it's in units of reciprocal length,
could be reciprocal centimeters, or reciprocal meters, in which case, we go
from h c divided by lambda, to h c times, I've got a little nu tilde here, to
represent something that's in wavenumbers.
So those are all equivalent ways of expressing energy, because there's a
relationship between wavelength, frequency, wavenumber, and energy.
And Planck's constant is the proportionality factor.
So with that in mind, I'm going to give you an opportunity to assess yourself
how, how well you've followed that last concept.
You'll get a little quiz to come up in a moment, and you can look at the
explanation for why the correct answer is the correct answer.
And then we'll move on again. Alright, I want to wrap up now by
explaining how do we get these quantized energy levels.
How can we use quantum mechanics to predict allowed energies for systems.
And the answer to that is the Schrodinger Equation.
So here's Erwin Schrodinger, and he was a German physicist who first explored how
to take wave mechanics and use it to describe the properties of energy and
matter. And so, here is the one dimensional
Schrodinger equation. And it's not terribly important that you
memorize this equation. I'll just call out some key features of
it It says that there is this thing, it's called a wave function, psi, operating on
the wave function by taking its second derivative.
And it's one dimensional, so it depends on x, and I'm taking the second
derivative with respect to x. Multiplying by h bar squared.
So here's Planck's constant. But h bar is Planck's constant divided by
2 pi, it's just a short-hand way to keep track of a factor of 2 pi.
A mass associated with the system, whatever it be.
A potential energy term, so there's a kinetic energy term and a potential
energy term. Those are the two kinds of energy that
you might find in a physical system. And this wave function, when operated on
in this fashion, gives rise to a loud energy levels, that satisfy this
equation. So, what we do is solve the Schrodinger
equation for a given potential, and so depending on the nature of the system,
there are different potentials: particles confined in boxes, free particles,
particles rotating in a circle. That defines what the potential might be.
It also defines a set of boundary conditions.
So that's typically quite important in quantum mechanics.
Where do we know the system is not allowed to be.
So particle in a box for instance, can't be outside the box.
Given those potentials and boundary conditions we solve for the wave
functions, and each of those wave functions they're indexed by n, just some
integer, we count them, one, two, three, four, five.
And every one of the wave functions has an associated energy.
And those are the allowed energy levels, we say they are allowed.
If you're wondering what, what this wave function really contains within it, it
turns out that the wave function when you take its square modulus over a certain
volume, or this is one dimensional, so over a certain length.
You determine the probability that you'll find the system within whatever small
volume or length you're surveying. And ,it's an interesting concept and one
that troubles people in the sense that it says there's no objective reality until
you actually do an experiment. There's only a probability that something
may be somewhere, and when you do the experiment you either find it or you
don't. But it's not that it really was or wasn't
there ahead of time. It's just probabilistically distributed.
It exists in what's called a superposition of states.
So, some of you may have heard of Schrodinger's cat.
And this is a thought experiment that Schrodinger proposed.
That if you put a cat in a box with a radioactive nucleus that could decay
randomly at any moment, and when it decays it breaks a vial of poison and
that kills the cat. But until you open the box, you don't
know if the decay has happened or not. And so is the cat alive or dead?
It's neither. It's in a superposition of states.
And until you open it, you won't know. Now, I don't like cruelty to animals so
I'm going to just imagine that cat is alive.
But it's a fundamentally different way of thinking about the universe obviously.
And that's one of the fascinating things about quantum mechanics.
So, actually, defining the relevant Schrodinger equations and doing the
solutions to get the energy levels, that's what a course in elementary
quantum mechanics is about. We are not going to do that.
All we do need to do is be familiar with the allowed energy levels and for the
systems that we'll encounter. And so, we will be going through some of
those in the very near future. But before that, we're going to do a
demonstration that will help to illustrate the concept of quantized
energy levels. And in particular, we're going to look at
the hydrogen chloride cannon. So this is a reasonably exciting
demonstration I think. And I hope you'll enjoy it as much as I
enjoy doing it. In this demonstration, I want to show you
a chemical reaction that we will look at in more detail, to see how thermodynamics
helps us explain, in quantitative detail, what we observe qualitatively.
In this apparatus is a tube that has been filled with a mixture of hydrogen gas,
H2, and chlorine gas, Cl2. The tube is further encased in a clear
plastic sleeve, for reasons that will become clear shortly.
These two gases can react with one another to make a new gas hydrogen
chloride, dissolving hydrogen chloride in water is how hydrochloric acid is made.
Just as was true for the thermite reaction, demonstrated in this course's
introductory video, reaction of H2 and Cl2 to make HCl is favorable, but it
requires and initial kick to get the reaction started, and that kick is the
energy required to break the bond between two chlorine in Cl2, which amount of
energy is 242 kilojoules per mole. I'm holding here three different laser
pointers. One lazes red light.
One green light and one blue light. As we'll see in some exercises in an
early lecture, the energy of a photon, which is the fundamental unit of light,
is determined by its wavelength. The red, green, and blue laser pointers
emit wavelengths of 650, 532, and 405 nanometers respectively.
Let's see which, if any, of these wave lengths seem sufficient to break the
chlorine-chlorine bond. Here's red shining into the gas mixture.
Not much happening. Now, green.
Nothing. Now let's try blue.
[SOUND] Wow, something must've happened because that cork shot out of the inner
test tube at high speed. [SOUND] Actually.
As we'll work out soon the heat released from the reaction causes a pressure
increase of about 29 atmospheres inside the tube.
And that either causes the cork to be ejected or the glass tube to break.
And that's why we have it shrouded. We'll look at that calculation in a
cursory way very soon. Simply to illustrate that there is a
quantitative way to model it, but we'll cover all of its parts in detail before
we reach the end of the course. So you'll have a chance to fully
appreciate how our understanding of the molecular properties of the gases, lets
us predict the outcome of the reaction from first principles.
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