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

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From the course by 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.

From the lesson

Module 8

This last module rounds out the course with the introduction of new state functions, namely, the Helmholtz and Gibbs free energies. The relevance of these state functions for predicting the direction of chemical processes in isothermal-isochoric and isothermal-isobaric ensembles, respectively, is derived. With the various state functions in hand, and with their respective definitions and knowledge of their so-called natural independent variables, Maxwell relations between different thermochemical properties are determined and employed to determine thermochemical quantities not readily subject to direct measurement (such as internal energy). Armed with a full thermochemical toolbox, we will explain the behavior of an elastomer (a rubber band, in this instance) as a function of temperature. Homework problems will provide you the opportunity to demonstrate mastery in the application of the above concepts. The final exam will offer you a chance to demonstrate your mastery of the entirety of the course material.

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

Chemistry

Okay, let's take a look at the key concepts from this last week.

So, first the Helmholtz fee energy is defined as A equals U the internal

energy, minus T the temperature, x S the entropy.

At constant temperature and volume spontaneous processes occur to minimize

the Helmholtz free energy. goes until the system is at equilibrium,

at which point only reversible process, that is those for which dA equals 0 will

occur. An analgous quantity is the Gibbs free

energy, G and that's equal to internal energy U plus pressure P times volume V.

Minus temperature T times entropy S. And a constant temperature and pressure

by contrast to the Helmholtz free energy which is volume.

Spontaneous processes occur to minimize G until the system's at equilibrium at

which point only reversible processes for which dG equal 0 occur.

The maximum non PV work that can be extracted from a spontaneous process

occurring at constant temperature and volume or constant temperature and

pressure is delta A or delta G respectively.

Depending on which of those two conditions holds.

Either isothermal, isochoric is actually how you say constant volume.

So, constant T constant V is isothermal isochoric or isothermal, isobaric, that

is constant temperature, constant pressure.

Those correspond to Helmholtz and Gibbs free energies respectively.

Spontaneous free energy changes can depend on a balance of energetic and

entropic changes. They can go either way depending on the

temperature. When the temperature causes the entropy

to outweigh an energy or enthalpy preference.

We looked at Maxwell relations and Maxwell relations are determined through

the equality of the mixed partial derivative of thermodynamic state

functions. With respect to other thermodynamic

functions. And so, one example that we derive from

Helmholtz's free energy was that the partial derivative of pressure with

respect to temperature when holding volume constant is equal to the partial

derivative of entropy with respect to volume when holding temperature constant.

And the reason Maxwell relations are useful is that they can establish a

connection between thermodynamic state functions, which may be tricky to

measure, such as the entropy and PVT equations of state.

So, those are easy quantities to measure pressure, volume and temperature.

And that's what appears everywhere else in this particular Maxwell relation.

An example of non PV work, which can be extracted from a system is stretching a

rubber band a length delta l against a restoring force f, for example.

So, more work is required at higher temperatures because the entropy of the

rubber band decreases when it's stretched.

Natural independent variables for a given state function are those, which permit

derivatives of that state function with respect to those variables to be

expressed as simple thermodynamic functions.

For example, the natural independent variables of H are S and P, which leads

to partial derivative H with respect to S at constant pressure equals temperature.

And, the partial derivative of H with respect to pressure at constant entropy

is equal to volume. So, there you see these simple,

thermodynamic variables. The Gibbs free energy of an ideal gas as

a function of temperature and pressure can be related to the pressure and the

standard molar Gibbs free energy at one bar.

Using G over bar at a given temperature and pressure is equal to G superscript 0

at temperature. This is the standard molar Gibbs free

energy at a given temperature and it is defined as the standard molar Gibbs free

energy at 1 bar plus RT log P. So, I only know what pressure I'm going

to in order to know what the new free pressure will be, given a tabulated

standard molar free energy. The temperature dependence of the Gibbs

free energy can be expressed through the Gibbs Helmholtz equation.

And that is the partial derivative with respect to T of G over T, holding

pressure constant is minus H over T squared, the Gibbs-Helmhotz equation.

And then finally, the Gibbs free energy as a function of temperature can also be

assembled from the enthalpy at a given temperature.

Which is evaluated relative to enthalpy at 0 kelvin, by doing measurements of

heat capacities. Minus T times the third law entropy at a

given temperature. And again, that is evaluated from

measurements of heat capacities at various temperatures.

Except what's being integrated is CP over T, instead of just CP, as is true for

enthalpy. So, given those enthalpy and entropy

quantities, we can always determine a free energy.

All right, those are the key points and hopefully those will be helpful to you as

you address the homework. And we've actually come to the end of

statistical molecular thermodynamics, the eight weeks of this course on Coursera.

So, I appreciate you holding out to the end and I hope you found it enjoyable and

you've learned something along the way. And I also hope you'll indulge me.

One doesn't make one of these things all by oneself.

I'm going to create one last video where I roll credits and thank some people.

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