This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Из курса от партнера University of Minnesota

Statistical Molecular Thermodynamics

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This introductory physical chemistry course examines the connections between molecular properties and the behavior of macroscopic chemical systems.

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Module 7

This module is relatively light, so if you've fallen a bit behind, you will possibly have the opportunity to catch up again. We examine the concept of the standard entropy made possible by the Third Law of Thermodynamics. The measurement of Third Law entropies from constant pressure heat capacities is explained and is compared for gases to values computed directly from molecular partition functions. The additivity of standard entropies is exploited to compute entropic changes for general chemical changes. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts.

- Dr. Christopher J. CramerDistinguished McKnight and University Teaching Professor of Chemistry and Chemical Physics

Chemistry

All right. This week of the course had a little less

material in it than some of the others. But maybe people were ready for a little

bit of a break as we come into the home stretch.

But, nevertheless, let's take a look at what the most important concepts were.

So, first, the entropy variation associated with a change in temperature

can be expressed as the, heat capacity. Either at constant volume or at constant

pressure, depending on what kind of a system you're looking at, divided by T.

And just as integrating heat capacity can be used to determine entropy, the

integration of heat capacity divided by temperature is used to determine entropy.

The Third Law of Thermodynamics says that at non-zero temperatures, all substances

have positive entropies, at 0 Kelvin the entropy of a perfect crystal is equal to

zero. An entropy of exactly zero depends on

there being a single non-degenerate ground state.

So if the ground state has some degeneracy it still might be very, very

small but it won't be exactly zero. Exactly zero requires a non-degenerate

ground state for which case W, the measure of disorder, is equal to 1.

And also the probability, of being in a given state, j, is the kronecker delta

ij, where i index is the ground state. So I actually used zero as that index

earlier this week, but in any case, there is one state with probability 1, and all

other states with probability 0. I told you about the research of

professor William Giauque, who generated temperatures very near to absolute 0

using adiabatic demagnetization. And that process allows you to go beyond

the lowest temperature that's achievable from adiabatic gas expansion.

And really come arbitrarily close to absolute zero.

Calculating the entropy through this integration process that I, I just

mentioned from the last slide, can be written mathematically as.

Entropy at a given temperature T, is the integral from zero to the melting point

of a heat capacity of a solid. Plus an entropy change associated with

melting, plus another integral from the melting point to the boiling point of the

heat capacity of the liquid. Plus another phase change, now

vaporization, entropy change. And then finally to an arbitrary

temperature t of a gas Integrating from the boiling point to that temperature the

heat capacity of the gas over the temperature.

Rather than having to start exactly at absolute zero, this process can be

facilitated by starting at a somewhat higher temperature, still cold, but

higher. And noting that the molar entropy at that

temperature ought to be equal to the constant pressure heat capacity at that

temperature over 3. Because of the Debye T3 law.

Entropies that are measured in this fashion, from experimental measurements,

are in near quantitative agreement with results.

That would be predicted from the partition function using S equals k log

partition function plus kT partial log Q, partial T, a constant number and constant

volume. Something I don't know that I said when I

presented all this by the way is, we tended to only to make this experiment

for gases. And that's because, we actually have a

pretty good way to construct the partition function for gasses.

Many of them behave nearly ideally. You might remember that you saw a

correction for Nitrogen that was very, very small

So at the temperature and pressure we were looking at it was pretty close to

ideal. Now, while we would still expect to see

this near quantitative agreement say, for a liquid.

The trouble is we might have a much more difficult time coming up with a good

partition function for the liquid. And that is not the subject of this

course, but suffice it to say that can be quite difficult.

But in any case, we get into the gaseous regime, we are golden with statistical

molecular thermodynamics. In terms of the degrees of freedom that

contribute to entropy. In ordered by quantitative importance the

translational energy levels and the ability to access many of them,

contributing to disorder. Contribute the most to the entropy of a

substance the rotational levels are next in importance, vibrational are lower and

electronic excitation even lower. these last 2, and many of the examples we

looked at, made virtually no contribution at all so this last greater than symbol

is. Somewhat notional in a way, but

translation dominant rotation often very important.

And again really gasses we're talking about here in other condensed phases we'd

need to talk about other phenomena associated with associated with motion.

As particle mass increases, the translational entropy also increases and

it does do in a logarithmic, logarithmically with the mass.

Very stiff, insulating solids, like diamond, have very low entropies, near

zero kelvin. And conductors get to low entropies much

less rapidly, because they're conductors. Because they have electronic motion that

needs to be considered as well. We also looked at, larger molecules than

just say monatomic and diatomic gases and in general, the more atoms a molecule

has. The greater its entropy at a given

temperature because generally that additional number of atoms will bring in

increased mass. Larger moments of inertia and perhaps

degrees of vibrational freedom where the vibrational temperature is low enough.

That they begin to contribute some disorder of their own.

Residual entropy is a phenomenon that can be associated with a system that fails

experimentally to access a perfect crystal at 0 kelvin.

And I used the example of carbon dioxide as one system that exhibits that kind of

phenomenon. Justice was true for entropy or is true

for any state function. The entropies of reaction are additive

and derive from the entropies of products minus entropies of reactants.

And last of all, entropies of gases are much, much greater than those of their

corresponding condensed phases. And we saw that when we looked at some

specific examples of entropies of reaction.

Alright, well, we have reached a stage now where we have actually covered all

three of the laws of thermodynamics, the first the second and the third.

In the last week of the course, we're going to take advantage of all of these

laws to look at two additional thermodynamic state functions.

And the relationships between all these different state functions, and what we

can do, taking advantage of these relationships.

So the first of those state functions that we look at will be the Helmholtz

Free Energy, and we'll get started on that next week.

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