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

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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 5

This module is the most extensive in the course, so you may want to set aside a little extra time this week to address all of the material. We will encounter the First Law of Thermodynamics and discuss the nature of internal energy, heat, and work. Especially, we will focus on internal energy as a state function and heat and work as path functions. We will examine how gases can do (or have done on them) pressure-volume (PV) work and how the nature of gas expansion (or compression) affects that work as well as possible heat transfer between the gas and its surroundings. We will examine the molecular level details of pressure that permit its derivation from the partition function. Finally, we will consider another state function, enthalpy, its associated constant pressure heat capacity, and their utilities in the context of making predictions of standard thermochemistries of reaction or phase change. 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. We've been doing a lot of work with ha

ha. Work.

Thermodynamics. work with pressure and volume changes in

gasses and their influence on the path dependence of work, of heat.

And the path independence of, of internal energy.

So I want to look a little more carefully at pressure, this concept of pressure,

from a statistical mechanical viewpoint. And so, I'll remind you of the definition

of the internal energy, and that is a sum over all possible energy states within an

ensemble, probability weighted by their contribution to the ensemble.

And remember that in stat mech the probability is dictated by e to the minus

beta times the energy. Which in this case depends on number of

particles and volume, divided by the full partition function, the sum of all

probabilities. So, I'm interested in the first law in

the differential of the internal energy, dU.

So, if I apply the chain rule to this expression, I will get probability times

the differential of the energy. Plus energy times the differential of the

probability. Well, the energy, itself, depends on

number of particles and volume. And so, really, I need to chain rule

through a bit further. But, for the moment, I'm going to work

with a system with a constant number of particles.

No particles coming in, no particles going out.

In that case, I'll treat de as just the partial derivative of the energy, with

respect to the volume, with number of particles held constant.

And the reason that this differential expression is interesting is what if I

compare it to first law expression, and in particular I'm going to consider

reversible processes. Well I've got delta w reversible and

delta q reversible. These two have a relationship because

delta w reversible, what does this mean? It's an infinitesimal change in the

energy levels, with probability remaining the same times a dv.

That's why I want to equate this with work.

It's because it's multiplying a dv term And the differential work is associated

with a pressure times a differential volume.

Meanwhile over here, the other piece of the term, just by what's left in the

first law, must be associated with differential heat.

And what is that? Well, it's infinitesimal changes in the

probabilities of the levels with the energy of the levels remaining the same.

So it might not be obvious what, how do these really correspond, one to another?

It's actually a little helpful to take a look at it graphically, to help make this

more clear. And in particular, let me, let me think

about this first term. What's it really saying?

Work. So differential work is, small changes in

energy levels with constant probabilities.

So, that would be like, so here might be a distribution within energy space, so

every dot, the more dots the more probable.

And, so as I go up in energy, I've got less probability.

So, increasing the energy would be like pushing these levels further and further

apart. And, in a sense, what pressure does in,

say, reducing a volume in a box, for example, that would split the energy

levels apart, they would become wider. And if I don't change the probabilities,

the net energy is raised, and that's really sort of what this term is doing,

that's the work term. What about the heat term?

The heat term might even be a, a little bit more sensible in, in a way.

If I keep the energy levels the same, so these lines are spaced just the same as

these lines but I changed the probability of being in those levels.

So, to increase the energy, I'd move probabilities out of low levels, up into

high levels. That's like increasing the temperature,

it's adding heat. Decreasing the energy, obviously the

opposite direction. So, pressure then, I mean, given this

relationship of work, differential work is this probability times differential

energy. And the more common differential work is

just minus pdv. I'll get that the pressure is this

expression here, proceeding dv. It's the probability weighted, the

ensemble weighted values of partial e, partial v.

Or, if I want to compute pressure, it is minus.

So I've just been carrying this negative sign through.

Partial derivative of energy with respect to volume.

Keeping the number of particles constant, averaged over the ensemble.

So in video 3.4, I just sort of presented this as a trust me, this is the way to

compute pressure, and showed that if that was the way to compute pressure, then the

ideal gas partition function recovered the ideal gas equation of state.

Here's an actual way to appreciate why that's how we calculate pressure.

Because of this du being used on the partition function for energy.

Alright, well, that's as I said kind of a digression because I want to provide an

atomistic understanding of pressure which certainly to do thermodynamics.

You don't have to know why pressure is what it is, it's just some quantity you

can measure. But for a chemist it's useful to see how

that comes from molecular concepts. Next we're going to move on to a new

state function. And in particular, having worked a lot

with internal energy up to this point, we're going to look at something called

enthalpy.

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