WEBVTT chemistry/physical-chemistry/hovasapian
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Hello welcome to www.educator.com and welcome back to Physical Chemistry.
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Today, we are going to talk about the fundamental equations of thermodynamics.
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It is in this lesson that absolutely everything that we have talked about comes together.
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In this lesson, it is going too close the circle on thermodynamics.
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For me personally, no matter how often I present this particular material in this particular lesson,
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I absolutely cannot get over just how unbelievably beautiful this stuff is.
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The circle this closes in such an amazing way.
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Everything that we have worked on is now going to come together and get this absolutely beautiful way.
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For me personally, I hope that you feel like that also even just a little bit.
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Let us get this started.
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A system has two mechanical properties.
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Let us go ahead and stick with black.
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A system has two mechanical properties, they are the pressure and the volume so P and V.
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A system has three fundamental properties.
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These fundamental properties correlate to the laws of thermodynamics, the 0 law, the first law, and the second law.
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They are of course the temperature, the energy, and entropy.
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0 law, first law, second law are the fundamental properties.
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A system has a three composite properties or compound properties if you will.
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They are the enthalpy, Helmholtz energy and the Gibbs energy.
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We are going to derive the fundamental equations of thermodynamics.
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We are going to express the fundamental relations among all of these properties.
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This is what I mean why how it all comes together.
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As we mentioned before, we will restrict our discussion to systems that do only pressure, volume, work.
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We are not going to be concerned with other work electrical, chemical, or otherwise.
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In other words, DW other is always going to be 0 for us.
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Let us recall a general condition of equilibrium.
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The general condition of equilibrium we have - DU – P δ V + TDS - DW other is going to = 0.
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Again this is going to be 0 for us so we are going to have – DU - PDV + TDS = 0.
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Let us rearrange this a little bit and express it in terms of energy.
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We will put the DU on the left is going to be DU = TDS – PDV.
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Let us see here, this is nothing more than the fundamental equation of thermodynamics that we introduced when we talked about entropy.
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We arranged it a little bit.
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Let me do here in blue actually do in red.
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Remember, we said that DS = 1/ T DU + P/ T DV.
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If I multiply everything by T, I end up with TDS = DU + PDV and if I move this over there I end up
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with DU = TDS - PDV which is exactly what I have here.
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There is nothing new about this equation, we have seen it before, we seen it in the context of entropy
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but now we have rearranged it so we can actually make it look more systematic with the other things that we are going to do.
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This is just a fundamental equation of thermodynamics.
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The fundamental equation, the one that relates energy with entropy and volume, pressure, and temperature.
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Let us go ahead and call this equation number 1.
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Let us go ahead and write our composite functions.
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Let me go ahead and go back to this page, I will write it up here.
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I have DU = TDS – PDV.
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These are composite functions.
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Let me go back to black, this is nice to do in black.
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Our composite are H = U + PV.
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Our A Helmholtz energy = U - TS and our Gibbs free energy = U + PV – TS.
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Let us differentiate these three.
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When we differentiate the three, we get the following.
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I will go back to black.
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We get DH = DU + PDV + VDP.
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DA = DU - TDS – SDT, DG = DU + PDV + VDP - TDS – SDT.
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This is the product rule, this times the derivative of that, that times the derivative of this.
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This is where this is coming from.
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Let us go back to blue.
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In each of these equations right here, the differential equations for the H, A, and G, I'm going to put in the value of DU.
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I'm going to put in T δ S and DU here and here and I will see what kind of relations that I find.
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That is what I'm going to do.
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Let us go ahead and do the first one.
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Let us go ahead and do this in blue.
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I’m going to take the DH first, I have DH DU + PDV + VDP.
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For DU, I’m going to put in TDS - PDV so I have DH = TDS - PDV and I have + PDV + VDP.
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This and this cancel and I'm left with DH = TDS + VDP.
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We are going to call this equation number 2.
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We are taking care of that one, now let us move on to the next one.
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We have DA = DU -TDS – SDT.
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We have DA = TDS -PDV-TDS – SDT, TDS and TDS go way I'm left with DA = - SDT and I'm going to rearrange this.
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You are going to see why in a minute, - PDV.
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This is equation number 3.
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We will do DG = DU + PDV + VDP - TS – SDT, DG I will put in the value of DU.
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DU was TDS – PDV, I have + PDV + VDP - TDS – SDT.
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TDS cancels TDS, - PDV + PDV I'm left with DG = - SDT + VDP.
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I’m going to call this equation number 4.
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Equations 1, 2, 3, and 4, these are called the fundamental equations of thermodynamics.
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They are not really for separate equations.
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In fact of the matter is the first equation the DU, the TDS, the DU = TDS - PDV equation number 1,
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that is the fundamental equation of thermodynamics.
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These other three are different ways of looking at the fundamental equation of thermodynamics
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through the eyes of the composite functions and the other things.
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Let us go ahead and write them all in one.
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Let me go ahead and write this way so we can take a look at we have DU = TDS – PDV.
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We have DH = TDS + VDP.
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I sure hope that I'm not making any mistakes as far as the + and - are concerned.
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I hope that you will check these.
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You have it in your book because there is lots of + and - and S and V, and all kinds of things flowing around here.
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We have DA = - SDT - PDV and we have DG = - SDT + VDP these are the fundamental equations of thermodynamics.
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Notice the symmetry, notice how we have arranged them.
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We have energy and enthalpy, the energy of the system, the enthalpy of the system.
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We have the Helmholtz energy and we have the Gibbs energy.
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This is expressed in terms of TP is variables are S and V.
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TDS here, TDS here is - PDV + VDP.
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The DA and DT we have - SDT – SDT.
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Here it is - PDV + VDP just like - VDP + VDP.
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I’m not sure about the extent that you actually have to memorize these, you do not really.
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Basically, all you need again is just the fundamental equation of thermodynamics which you get from the definition of entropy and
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the first law of thermodynamics and everything else to sort of falls out.
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The rest all you are really doing is just manipulating the equations.
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Let us see how unbelievably beautiful these are.
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These equations express the relationships be among all these things, energy, temperature, entropy, volume, pressure,
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Helmholtz energy, Gibbs energy, volume pressure, pressure volume, entropy temperature, these ties everything together.
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Let us go ahead and say a little bit more.
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Equation number 1, a change in energy DU is related to a change in entropy DS and the change in volume DV.
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Equation number 2, the change in enthalpy is related to a change in entropy and a change in pressure.
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Nothing new here, we know this is the relationship energy, volume, enthalpy, pressure, and everything is coming together beautifully.
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Equation number 3, a change in Helmholtz energy is related to a change in temperature and a change in volume.
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Helmholtz energy relationship is temperature and volume are the variables for Helmholtz energy.
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Equation number 4, the Gibbs energy, the change in Gibbs energy is related to change in temperature and change in pressure.
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Let us go back to black here for a second because these equations are reasonably straightforward and simple.
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Reasonably that is why we call them reasonably simple.
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An expression, the variables of the differentials on the right hand sides of these equations,
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the variables on the right are called the natural variables for the particular property on the left.
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Let us go ahead and go here, in other words.
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Let me write out the equations again here just so I have them.
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I’m going to write it here, I got DU = TDS – PDV, I got DH = TDS + VDP.
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I got DA = - SDT - PDV and I have DG = - SDT + VDP.
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In this particular, the case of energy, the natural variables are S and V.
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In the case of enthalpy, the natural variables are S and P.
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In the case of the Helmholtz energy, the natural variables are T and V.
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In the case of Gibbs energy, the natural variables are T and P.
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By natural variables, it is just these are the variables that they are expressed in.
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Gibbs energies is a function of temperature and pressure.
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Helmholtz energy is a function of temperature and volume.
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DH enthalpy is a function of entropy and pressure.
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Energy is a function of entropy and volume, that is all we are saying.
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That is what we mean when we say the natural variables.
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Here is work, it is really interesting and beautiful.
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Since each of the expressions on the right is inexact differential for state properties, the mix partial derivatives of the coefficients are equal.
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Let us do it one more time, there is no harm in doing so.
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Let us go ahead and do this in blue actually.
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We are going to do left and right.
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Let us do DU = TDS - PDV an exact differential.
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Therefore, the partial of this differential with respect to this variable = the partial of this
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with respect to this variable with the other variable being held constant.
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What we mean is the following, we mean DT DV holding S constant = DP DS holding V constant.
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This is -, we make this a better V.
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The differential of this with respect to the other variable, holding this variable constant = the differential of this
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with respect to this variable holding this variable constant.
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Mixed partial derivatives that is what this says.
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From these relations I’m deriving these relations DT DV under constant entropy = - DP DS our constant volume.
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With enthalpy DH = TDS + VDP from this we derive DT DP under constant S = DV DS under constant V.
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The differential this with respect to this variable = the differential of this with respect to this variable.
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We will do DA = - SDT - PDV so what you end up with is - DS DV holding T constant = - DP DT holding V constant.
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Of course, the - disappear so I get that DS DV sub T = DP DT sub V.
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We will do the Gibbs’ energy DG = - SDT + VDP, what you end up with is -DS DP holding T constant = DV DT holding P constant.
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There you go.
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These relations right here is partial derivatives relations are called Maxwell's relations.
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Do not get these confused, those of you with Physics backgrounds you are not going to be confused with Maxwell’s equations for electro magnetism.
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Those are the difference set of equations.
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They form the foundation of electromagnetic theory, the classical electromagnetic theory.
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These are Maxwell's relations for thermodynamics derived from the four fundamental the equations of thermodynamics.
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The first and the second equation, these two relations right here, this relation and this relation, they are not going to concern us all that much.
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It is the last two that are going to be very important for our particular purposes and here is why.
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Let me go ahead and go to the next page.
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Let me rewrite the last two again just so we have them.
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I got DS DV under constant temperature = DP DT under constant volume and I have – DS DP under constant temperature = DV DT and it should be under constant pressure.
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The last two, those are numbers 3 and 4.
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These relations, they express are important because they relate the isothermal constant temperature,
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the isothermal volume dependence of the entropy, and the isothermal pressure dependence of the entropy.
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In other words, when I change the volume how does the entropy change?
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When I change the pressure how does the entropy change?
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The rate of change.
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This is the volume dependence of the entropy, this is the pressure dependence of the entropy.
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These are expressed in terms of things that are easily measured.
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If I have a system I will change the temperature and see how the pressure changes.
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I change the temperature and see how the volume changes.
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If I want to know how entropy changes when I change the volume,
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all I have to do is hold the volume constant, change the temperature and see how the pressure changes.
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The number I get here that is this.
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This is profoundly important.
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These things on the right are very easily measurable.
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These things on the left not easily measurable.
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I get them this way that is what makes these important.
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The relations are important because they relate the isothermal volume dependence of the entropy and
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the isothermal pressure dependence of the entropy to easily measure quantities.
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We have seen these equations before back when we discussed entropy.
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DS DV = DP DT under constant V this is constant T we said that this = A/ K.
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This is equal to α/ Kappa.
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We already have seen this before.
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This DS DP under constant T which is going to be DV DT under constant P = - V α.
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We have seen these before.
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If I know the entropy changes with respect to volume, all I really need to do is measure how the pressure changes when I change the temperature.
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I already know that.
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If I have the coefficient of thermal expansion and divide by the coefficient of compressibility that is what this is.
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Here it is the volume of the system × the coefficient of thermal expansion.
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All of these things are easily measurable.
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We actually happen to calculate it, you do not have to measure them anymore
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for any particular gas and any solid, any liquid, that we have to be dealing with.
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These all have been tabulated under standard conditions so that is what we have got here.
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This and this.
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These are the fundamental equations of thermodynamics and they are Maxwell's relations coming from the fundamental equations of thermodynamics.
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This ties the mechanical properties, the fundamental properties, and the composite properties together.
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This closes the circle on thermodynamics.
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The rest of what we are going to be doing until we actually close out a full discussion of thermodynamics is just tying up loose ends.
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There you have it, absolutely beautiful.
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Thank you so much for joining us here at www.educator.com.
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We will see you next time, bye.