WEBVTT chemistry/physical-chemistry/hovasapian
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Hello and welcome to www.educator.com and welcome back to Physical Chemistry.
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Today, we are going to talk about the general thermodynamic equations of state.
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Let us jump right on in.
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I will go ahead and stick with black today.
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The equations of state for gas the PV = nRT, the Van Der Waals equation and number of other equations that you may have seen.
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These are relations between pressure, volume, and temperature that have been derived empirically.
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We have run the experiments, we derived these equations empirically.
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This is how gases behave.
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Any other equations, maybe we took some equations and we modified them and derived different equations
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based on certain assumptions about atomic and molecular size, or maybe attractive forces.
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These are all empirically derived, the equations for the gas.
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The equations of state for liquid and solids was pretty much the same thing.
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It is the empirical observations and empirical derivations.
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The equation of state for liquids and solids was expressed or is expressed, I should say, via experimentally determined coefficients,
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the coefficients of thermal expansion and compressibility.
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Let me go ahead and write the equation.
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It is V = V 00 × 1 + Α × T - T0 × 1 - Κ × P -1, where V00 is the volume of the system at 0°C and 1 atm pressure.
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This is the equation state just like PV = nRT is the equation state for an ideal gas and
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the Van Der Waals equation is an equation of state for a Van Der Waals Gas.
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A little bit more of a real gas although not quite, it is a little bit better than the ideal gas.
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This is the equation of state for liquid and solids.
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It just says that the volume at any given moment happens to equal this initial volume
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which is the volume of the system 0°C and 1 atm pressure the × this.
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This is the initial temperature, this is a new temperature that you have to be measuring and the speed is the pressure.
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At a particular pressure and at a particular temperature this is the volume.
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It is a relationship between pressure, temperature, and volume.
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This is empirically derived that is based on these coefficients, the coefficient of thermal expansion and the coefficient of compressibility.
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This is just another equation of state.
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We have this equation state for gases, we have this equation of state for liquids and solids.
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Is there a way to find one equation of state that applies to all the way across the board?
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The most general equation of state that relates pressure, volume, temperature, the answer is yes.
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Here is how we do it.
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The equation of states, just write few more things here.
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These equations apply to systems at equilibrium.
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We now have a more general condition for equilibrium, we have done it to the last few lessons.
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We have talked about this general condition of spontaneity, a general condition of equilibrium.
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For equilibrium we have the following.
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Let me write this out actually but we now have a more general condition for equilibrium.
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The condition is DU = TDS - PDV a fundamental equation of thermodynamics.
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This is the relation for a system at equilibrium.
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The change in its energy = TDS - PDV the relationship between energy, entropy, volume, pressure, temperature.
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This is the general condition of equilibrium.
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It is what we have been doing in the last couple of lessons.
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From this, let us see if we can actually derive a more general equation of state
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that applies to solids, liquids, gases, any phase, any system, anytime under any conditions.
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Let us see what we can do.
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Let us go ahead and write this equation again.
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I actually go back but this equation that I just wrote is DU = TDS – PDV.
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Let us go ahead and recall where this actually comes from.
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We said that general condition of equilibrium is the following.
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The general condition, I’m just going to a quick derivation again.
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Just take a couple of lines here, it is - DU - PDV - D work other + TDS = 0 this is the general condition
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but we are not concerned with other work.
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We are only concerned with this and this and this.
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When you rearrange this, you get that.
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That is where this comes from, this is the general expression.
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This is just a rearranged version of it.
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Let us go ahead and start, we have DU = TDS – PDV.
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Let us let any change happen at a constant temperature.
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Let us go ahead and hold the temperature constant.
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When we do that, we can write this as the following.
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We can write them as DU partial T = T DS T this is subscript just means we are holding the temperature constant - P DV T.
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We are going to go ahead and divide everything by this term right here by the DV and we get the following.
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We get DU DV at constant T = T DS DV at constant T - P.
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Let me rearrange this, I end up with pressure = T DS DV at constant T - DU DV.
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Let us make this a little bit more clear, my apologies, DV at constant T.
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I have this equation, S and U, the entropy and the energy they are functions of temperature and volume.
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We know this already.
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They are functions of temperature and volume.
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We have an equation that relates P the pressure to functions of temperature and volume, pressure, temperature, volume.
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This is an equation of state that relates P to functions of temperature and volume.
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You have an expression that relates to temperature to pressure and volume, you have an equation of state.
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We have P = T DS DV - DU DV at constant T, this is our equation of state.
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The first of our general equations of state that relates pressure to volume and temperature.
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We call it Maxwell's relations, one of Maxwell's relations said that the DS DV T is actually equal to DP DT at constant V.
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Therefore, I’m going to go ahead and put this into here and I'm going to get P = T × DP DT at constant V - DU DV at constant T.
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This is our equation of state, this is the first of our general equations of state.
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That is going to be equation number 1.
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Let us look at the second fundamental equation.
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The second fundamental equation, let us do this in red.
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Let us look at the second fundamental question of thermodynamics.
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For the first equation was DU = TDS - PDV this one we ended up with
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that equation of state pressure expressed in terms of functions of temperature and volume.
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Pressure, temperature, and volume, this is an equation of state.
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It is a general equation of state, this equation right here applies to any system, anytime, any temperature, any volume, any phase.
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Liquid, solid, gas this holds, this is true in general.
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The second equation is DH = TDS + VDP.
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Let us go ahead and take care of this isothermally.
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Isothermally, I can go ahead and write it this way, I can write DH = T DS constant T + V DP constant T.
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I'm going to go ahead and divide by this term DP and I end up with the following.
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I end up with DH DP constant T = T DS DP at constant T + the volume.
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I'm going to go ahead and rearrange so I'm going to go ahead and write volume = - T DS DP constant T + DH DP at constant T.
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I have Maxwell's relation, it tells me that this thing the DS DP at constant T is actually equal to – DV DT at constant volume, temperature at constant P.
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I’m going to go ahead and put this in for that and I end up with the following.
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I end up with volume = T × DV DT at constant P + DH DP at constant T, this is my second general equation of state.
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Volume is a function of pressure and temperature.
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Enthalpy is a function of pressure and temperature.
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These are functions of pressure and temperature.
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Volume is expressed in terms of functions of pressure and temperature.
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Volume, pressure, and temperature, this is an equation of state.
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It relates the volume of the system to the pressure of the system to the temperature of the system.
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This is the second general equation of state.
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One is P in terms of V and T, this is V in terms of P and T.
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They are essentially the same but they are just different ways of looking at it.
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One is pressure and one is volume.
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Let us see, this is our general and applied to any substance, in any phase.
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We do not have an equation of state just for gases, an equation of state for liquid and solids, we had just two general equations of state.
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Of course, in the most practical conditions you are going to be using one of the other two.
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Again, if we want to we can use these equations of state here, perfectly good and perfectly valid and perfect general that is what makes this beautiful.
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Let us go ahead and actually talk about some applications, how can we use this?
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What can we do here?
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Let us go ahead and rewrite our equations.
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We have P = T × DP DV under constant V - DU DV under constant T and we have volume = T DV DT under constant P + δ H/ δ P under constant T.
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This is equation 1 and this is equation 2.
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If we knew what this is or if we knew what this is, let me just go ahead and put the values in and
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we would immediately have an equation of state.
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It would be absolutely beautiful but most of the time we actually do not know what this and this are.
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These things are not easy to measure.
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Remember back when we are discussing energy, this change in energy with respect to a change in volume
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when we are talking about Joules law and this change in enthalpy with a change in pressure, when we are talking about the Joule-Thompson effect.
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These things are not easy to measure.
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If we knew then it would be really great, but we do not know then so let us see what else we can do.
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Let us go ahead and actually rearrange these equations.
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Since under most circumstances we do not know what this and this are, let us see if we can find ways to find out what they are.
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Let us play around with these equations.
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That is what we are going to do, we are just going to play around with the equations.
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Let me go ahead and take the first equation.
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I'm going to take P = this and when we rearrange it I'm going to write it as.
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Let me go ahead and do on the next page.
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I’m going to write it as, let me write the original P = T × DP DT sub Z - DU DV sub T.
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I’m going to move this over here and move P to the right.
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I’m going to write this as DU DV with constant temperature = T × DP DT V – P.
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For an ideal gas I know that PV = nRT.
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Therefore, P is nRT/ V.
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If I take the partial of P with respect to T DP DT holding V constant, I end up with nR/ V.
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DP DT holding V constant is nR/ V.
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I can go ahead and put this in here so I get DU DV/ T = T × nR / V – P.
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nRT/ V is P so it ends up being P - P and ends up being 0.
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This confirms that Joules law.
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Remember, we said that from ideal gas, Joules law, this DU DV at constant T is actually equal to 0.
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This confirms it, it just came around the other way and was a lot easier to do so.
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0 which confirms Joules law, that is nice.
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That was a nice little application which confirms Joules law which for an ideal gas the change in energy with respect to change in volume is actually 0.
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Let us see what else we can do here.
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We have DU DV sub T = T DP DT sub V – P.
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We already know what DP/ DT is, DP/ DT under constant V is actually equal to Α/ Κ.
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If we put that in here, we get the following.
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We get DU DV sub T = T × Α/ Κ – P.
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Finding DU/ DV is not a very easy thing to do.
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However, T, Α, K, and P these are all very easily measurable quantities.
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I do not have to worry about measuring this, I can go ahead and measure these things, come up with this,
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and put back in here to get my equation state.
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We were able to find a nice expression in terms of easily measurable quantities for something that is not quite so easily measurable.
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Something that is reasonably elusive and actually difficult to do.
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This is what we always want to do, this is what we have been doing the entire time.
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Trying to find ways of expressing things that are not so easily measurable in terms of things that are easily measurable,
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that is what we have done here.
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For equation number 2, C we have the volume = T DV DT at constant P + DH DP at constant T.
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When I rearrange this I end up with the following.
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I’m going to move this over to the other side so I get this DH DP which is again as a very difficult thing to measure.
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T is going to equal V - T DV DT under constant pressure.
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However, we know what DV D T or P is, DV DT sub P is actually equal to V × Α.
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Therefore, when I put this in for here I end up with the following.
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I end up with DH DP sub T = V - TV Α.
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I have found a way to take something that is not easily measurable.
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This was related to the Joule-Thompson coefficient, in terms of things that are very easily measurable.
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I have the coefficient of thermal expansion, the volume, the temperature, the volume.
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This is really nice.
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We have ways of finding DU DV T and DH DP sub T from easily measurable quantities is always a great thing to achieve.
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Let us go back a bit.
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Remember when we are talking about energy we had these two relations.
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We had DU = the constant volume heat capacity DT + DU DV sub T DV.
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Remember, these differential equations and we had one for enthalpy as well.
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We had DH = the constant pressure heat capacity DT + DH DP at constant T × DP.
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Whenever we want to find the change in energy of the system this was our general equation that we started off with.
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Whenever we want find the enthalpy of the system, this was the general equation that we started off with.
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For ideal gases, these equal 0 so you have just this part right here.
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Again, these are the general equations that we always start off with.
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These are the equations that we want to memorize.
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We will look at what we have done, we have just found an easy expression for that and we found an easy expression for that.
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Let us go ahead and put these expressions into here.
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I’m going to go ahead and do this in blue.
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We can finally express the change in energy of the system and the change in enthalpy of the system entirely in terms of things that are easily measurable.
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This is an incredible achievement.
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We are expressing the change in energy and change in enthalpy of the system completely in terms of things that are easily measurable.
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Let us go ahead and write that out.
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We have DU = CV DT.
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Let me make it feel a bit much clear, DT + DU DV that is nothing more than T Α/ Κ - pressure × DV.
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The change in energy, the energy of the system is a function of temperature and volume.
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The change in energy is expressed in terms of the constant volume heat capacity, temperature, Κ, Α, and pressure.
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All these things are easily measurable for a system, very easily measurable.
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This is fantastic, this is beautiful, absolutely beautiful.
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This is what we want, something that is very abstract, energy.
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We are expressing it in terms that are easily measurable than laboratory.
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Let us do one for enthalpy.
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We have DH equal to the constant pressure heat capacity × the differential change in temperature + DH DP.
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DH DP is V – TV A, V - TV Α × DP.
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This is the other equation.
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Before, we have to leave it as this and this, but now that we are able to from our fundamental equations of thermodynamics,
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from our conditions of equilibrium, from the fact that we are able to derive
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a couple of general equations of state we were able to find easy ways of finding these two values.
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We put them in and now we have closed the very important circle, absolutely beautiful.
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Let us go ahead and round this out with a couple of more applications.
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We call the general expression for the difference in the heat capacities the C sub P - C sub V.
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Remember, for an ideal gas we said that = RN.
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The general expression for the difference of heat capacities was the following.
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We had CP - CV = this reasonably complicated looking thing, DU DV sub T × DV DT sub P.
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This is the general expression for the difference in the heat capacities of any system, of any substance no matter what it is.
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Let us see what we have got.
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CP - CV we know what DU DV is.
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We just figured it out in terms of Α and Κ and P.
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We also know what DV DT is, we knew that already as V Α.
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Let us go ahead and put all of those values in.
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We get P + DU DV is T Α / Κ – P × V Α.
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The P cancels the P, therefore, in general the constant pressure heat capacity of a substance - its constant volume heat capacity,
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in other words the difference of heat capacities = T × V × A²/ Κ.
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It is that not the most beautiful thing you have ever seen your life?
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That is absolutely extraordinary.
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The difference of heat capacities, constant pressure, and constant volume is related to the temperature volume,
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the square of the coefficient of thermal expansion divided by the coefficient of compressibility.
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All of these things are easily measurable or you just look them up in a book, this is absolutely fantastic.
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This applies to any system, any substance, any time, any phase, that is what makes this amazing.
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Let us keep going.
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How about the Joule-Thompson coefficient?
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Let us do this one and go back to black.
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For the Joule-Thompson coefficient we had the following relation.
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We had - C sub P × Joule-Thompson coefficient = DH DP T.
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We know what that is, - CP × μ which is the Joule-Thompson coefficient.
00:31:29.600 --> 00:31:36.600
This DH DP T = V - TV Α.
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Let us go ahead and rearrange this.
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We end up with the Joule-Thompson coefficient = TV Α - V/ CP.
00:31:51.700 --> 00:31:54.700
Absolutely extraordinary, all of these things are easily measurable.
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Constant pressure heat capacity, volume, Α, we have a way of actually finding
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the Joule-Thompson coefficient for a substance from things that we already know.
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This is absolutely extraordinary.
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Clearly, these are much more easily measurable than this is.
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Finding the Joule-Thompson coefficient is not an easy thing to do.
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Let us keep going, the Joule-Thompson inversion temperature.
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The Joule-Thompson coefficient changes sign, it goes from positive to negative.
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In other words, the Joule-Thompson coefficient has to equal 0.
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At a temperature it can go from positive to negative, it equal 0.
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When we set this equal to 0, 0 =TV Α - V/ CP.
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We end up with the following.
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We end up with TV Α - V = 0.
00:33:14.200 --> 00:33:20.100
Let me make it a little bit clear here so it does not look like U = 0.
00:33:20.100 --> 00:33:22.300
We go ahead and we end up with.
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Let us go ahead and factor out V, we get T Α -1 = 0.
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We get T A -1 = 0 and we end up with the inversion temperature, the Joule-Thompson inversion temperature = 1/ Α.
00:33:45.300 --> 00:33:52.000
Absolutely fantastic, from easy things to measure, easy things to look up,
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we are able to find the Joule-Thompson coefficient we are able to find the inversion temperature, all kinds of things.
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That is what makes this amazing.
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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.