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.