WEBVTT chemistry/physical-chemistry/hovasapian
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Hello and welcome back to www.educator.com and welcome back to Physical Chemistry.
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Today, I thought we would stop and do a little summary of what we have done with entropy so far.
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In the last couple of lessons, we have always done a lot of mathematical derivations.
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There has been a lot of partial derivatives, a lot of substitution, a lot of math and symbolism on the page.
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Oftentimes, when that happens it is easy to lose your way, to lose the fours from the threes.
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If I pullback and give us a nice global perspective of what is that we did and more importantly what was that we are actually after.
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To distill and reduce it to what is important but you want to take away from all these derivations and all these equations that we have been dealing with.
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Let us jump right on in.
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Basically, just like energy now we have this new state property that we call entropy.
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Again, we have not defined what entropy is aside from saying that the best way to handle it
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at this point is to just treat it like the general disorder of a system.
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It is the best way to handle it.
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I will go ahead and write that down.
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We have this new state property of the system that was very important, it is a state property.
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State property entropy which we been symbolizing with S and our empirical evidence with this state property we just to define it like this.
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Our empirical experience leads us to define the differential change in entropy of a system this way as DS = DQ reversible / T
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which just means that any process when you going from state 1 to state 2, if I actually follow a reversal path
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when I'm making this particular change then the amount of heat has gained or lost,
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Let us go ahead and take it from the perspective of the surroundings.
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The heat withdrawn from the surroundings divided by the temperature at which the change takes place is some numerical value
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that equals the change in this property in going from state 1 to state 2.
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The definition is reasonably straightforward, it is not altogether different than when we defined DU = DQ – DW.
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We defined energy in terms of the heat and the work.
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Here the entropy is defined only in terms of the heat along a reversible path.
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We wanted to investigate entropy’s behavior especially with respect to the state variables temperature, pressure, and volume.
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We begin by combining the first and second laws, this is the second i will go ahead and write this down.
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This is the second law and this was the first law, first law of energy and second law is entropy.
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We combine those and we came up with something called the fundamental equation for thermodynamics.
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The fundamental equation of thermodynamics was our beginning point.
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We just put these two together and we moved this over here DQ, we put it here and we are rearranged it to express it like this DS = 1 / T DU + P / T DV.
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Basically, what this does it expresses when I combine these we found that this thing expresses
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the change in entropy of a system with respect to a change in energy or change in volume.
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If I want to change the entropy of the system, I can do it in two ways independently.
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I can either change the energy, change the volume, or change both.
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This is a relationship that expresses how it changes.
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Notice that all the variables are represented here, energy, entropy, temperature, pressure, volume,
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that is why it is called the fundamental equation of thermodynamics.
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This equation is one you absolutely have to memorize.
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You have to know this.
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As far as doing the problems are concerned, we are going to use the equations that we derived from this but this is very important.
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You do not have to memorize it.
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I mean you have this definition and this definition, you just substitute this in and you get this.
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It is not a problem, it is very easy.
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If you know these two, you already know this.
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We start with that.
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Basically, all we have to do is investigate for a particular system we are running our experiments,
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we have to change the volume and the energy and we are going to see how the entropy changes.
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Experimentally, we do not usually control the energy of the system.
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We can control the volume but we do not normally control the energy.
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We decided to take this equation and fiddle with it mathematically to see if we can somehow express
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the change in entropy with respect to the other variables temperature, pressure, and volume.
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That is what we wanted, that is the overarching goal.
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That is why we went through all those mathematical derivations that we did.
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When we did that under conditions of temperature and volume, here is what we came up with.
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We first considered entropy as a function of temperature and volume to get the total exact differential expression
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which is this one DS = a partial with respect to the first variable holding the second variable constant × its differential + the partial of S with respect to the second variable.
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The total differential expression is always the same, holding the other variable constant × its differential.
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This is the basic mathematical expression.
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This is the one of the equations if you want to memorize this, that is fine.
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We actually identified these differential coefficients with something that easily measurable
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which is ultimately what we want but this is the mathematical expression for this.
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This is how we begin.
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We first consider this to get this fundamental equation and now with some mathematical manipulation we were able to identify
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these differential coefficients that says how does entropy change when I change temperature?
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How does entropy change when I change volume?
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Holding the other variable constant we were able to come up with the following.
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In the case of DS DT, we had DS DT V = the constant volume heat capacity divided by the temperature.
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For the change in entropy with respect to any change in volume holding the temperature constant,
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we ended up with A / K which is the coefficient of thermal expansion divided by the coefficient of compressibility.
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When we rewrite this equation using these things, putting these in there we get DS = CV/ T DT + A/ K DV.
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This is one of the equations that is absolutely necessary and one of the ones that you want to begin all of your problems with
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because it expresses the change in entropy with respect to a change in temperature and a change in volume base on things that we actually can measure or look up.
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This is one of the things that we did.
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Again, this is one of the equations that you must memorize.
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This is the one that you want to bring to the table not the derivation, not all those partial derivatives, this one.
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S as a function of temperature and volume.
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If you hold volume constant this term is 0.
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If you hold temperature constant this term is 0 it just becomes that.
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This is true for all systems, liquid, solid, and gases.
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This is the general, the most general equation.
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We also do this for temperature and pressure.
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For temperature and pressure we came up so let us go ahead.
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We considered entropy as a function this time of temperature and pressure.
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What happens when I change the temperature and what happens when I change the pressure?
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The total differential expression mathematically is DS DT holding P constant × its differential + DS DP
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'k holding the temperature constant × its differential that the basic mathematical expression.
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Again, with some mathematical manipulation which constituted the previous lessons with all the derivations, we were able to identify these differential coefficients with the following.
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In this case, DS DT under constant pressure is the same, one of the constant volume that is equal to this times
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the constant pressure heat capacity divided by the temperature.
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Notice the pattern, I’m holding volume constant, its constant volume divided by pressure.
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If I’m holding pressure constant, it is going to be constant pressure heat capacity divided by the temperature.
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This is generally true.
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This one the DS DP, holding temperature constant that will end up being equal to the volume of the system × the coefficient of thermal expansion.
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We got DS = CP/ T DT – V A DP this is the other equation you want to memorize.
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Under conditions of temperature and pressure, this is the equation we want to bring to the table.
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Under conditions of temperature and volume the other equation is what we want to bring to the table.
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These are the two fundamental equations that you need to be memorized.
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Let us see what came next.
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You remember, these equations these are analogs to the equations that we did for energy.
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Back to energy we essentially said that there are two different expressions, the total differential expressions
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that you need to memorize in order to solve the problems for energy.
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Those are the equations that we always started off with.
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These are the analogs for entropy.
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We are doing the same thing, we have already done it.
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We resolve all those problems, all those example problems.
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We end up doing the same thing here, we are going to be doing a ton of example problems in a couple of lessons.
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We are just reducing it so we can make sense of what can see the four from the trees from a global perspective.
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This is what we wanted.
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At the previous equation that was what we wanted.
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These equations are analogous to what we did for energy.
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You remember those equations were as follows DU = CV DT + DU DV that was the equation for the change in energy with respect to temperature and volume.
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And then we had DH under conditions of constant pressure, we did not consider the energy of the system but we consider the enthalpy of the system.
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Because the enthalpy of the system accounts for any pressure, volume, work that was done so we end up with the following.
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The change in enthalpy = CP DT + DH DP constant T DP.
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This equation is analogous to this equation and this equation is analogous to the previous one for temperature and volume.
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That is it, we are doing the exact same thing.
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We want to find out how the state variable changes with respect to changes in temperature, pressure, volume.
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We have seen two important equations for how entropy behaves.
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We have seen DS = CV/ T DT + A / K DV this is temperature and volume dependents.
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We have seen DS = CP / T DT – VA DP this is a temperature and pressure dependents.
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In each case, the temperature dependence was simple.
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In fact, it is very simple, it is just the appropriate heat capacity for the particular variable divided by the temperature.
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It is just the appropriate heat capacity divided by T.
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In other words, this and this.
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When we hold V constant that goes to 0.
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The temperature dependence of the entropy under the conditions of constant pressure, that is it, there is nothing going on here.
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If for any reason the state of the system is described by T and some other variable, let us call it M.
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In this case, it was T and V, in this case it was T and P.
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If let us say, you are doing some kind of work and you are actually describing the system not with temperature volume, temperature and pressure,
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but now by temperature and some other variable M, whatever that variable happens to be
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then the heat capacity of the system at constant M is just the change the DM / DT.
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It is just a change in that particular, let us do it this way.
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DQ is the change in heat under the condition of constant M divided by the temperature change, this is just the definition of heat capacity.
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I will write that down.
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This is just the definition of heat capacity which is just heat per unit temperature.
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That is the fundamental definition of heat capacity.
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Now if we have, let us go ahead and move to next page here.
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If I have the constant M heat capacity for the system = DQ M/ T and if we have the definition of entropy
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which is DQ reversible / T which implies if I move this T over here I get T DS = DQ reversible.
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I will just go ahead and put this into here and I get CM = T DS/ DT that I divide by T and I get the following.
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I get CM/ T = basically, what this says is that the change in entropy per unit change in temperature,
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if I'm holding variable M constant is just the constant M heat capacity divided by the temperature.
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I already know that with respect to M being volume and N pressure.
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When N is volume well the DS DT is CV/ T.
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When M is pressure, it is the DS DT = CP/ T.
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N is any other variable, whatever it happens to be, the change in entropy per unit change in temperature
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holding that particular variable M constant is just the constant M heat capacity of the system which I can measure from this divided by temperature.
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This is just a general expression that is all that is going on here.
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Let us go ahead and write that down.
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Under constraint of any state variable dependents of entropy is just we will call it the appropriate heat capacity divided by T.
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In practice, we generally just hold either the volume constant or the pressure constant because those are the easiest thing
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for us to do experimentally which is why the constant volume heat capacity and the constant pressure heat capacity are the most important heat capacities.
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For any reason there are some other variable that we are dealing with, that we are holding constant it is going to be the constant M heat capacity.
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That is all that is going on here.
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In terms of pressure and volume, let us go ahead and do this.
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DS DT under constant volume = CV / T and of course we have DS DT under constant pressure = CP / T.
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I'm going to go ahead and move these T's over.
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I’m going to rewrite this as CV = T × DS DT V and CP = T × DS/ DT constant P.
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These last two equations, if I want to now that I have entropy I can go ahead and take this temperature × DS DT
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under constant V or temperature × DS DT under constant P, I can take these as the definitions of heat capacity as opposed to defining heat capacity.
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Let us say the constant volume heat capacity as the change in heat as I hold volume.
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This was the definition of heat capacity.
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We defined in terms of heat.
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The heat that has lost or gained during the particular process.
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If I want, because of these particular dependents of entropy on the heat capacity, I can actually define my heat capacities differently.
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These two, this and this, they are alternate definitions for what a constant volume heat capacity is or what a constant pressure heat capacity is.
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Not altogether that important but if I want to, I can use these relations.
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There are just some other relations and definition that I can use if I need to.
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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.