For more information, please see full course syllabus of Physical Chemistry

For more information, please see full course syllabus of Physical Chemistry

### The General Thermodynamic Equations of State

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- The General Thermodynamic Equations of State
- Equations of State for Liquids & Solids
- More General Condition for Equilibrium
- General Conditions: Equation that Relates P to Functions of T & V
- The Second Fundamental Equation of Thermodynamics
- Equation 1
- Equation 2
- Recall the General Expression for Cp - Cv
- For the Joule-Thomson Coefficient
- Joule-Thomson Inversion Temperature

- Intro 0:00
- The General Thermodynamic Equations of State 0:10
- Equations of State for Liquids & Solids
- More General Condition for Equilibrium
- General Conditions: Equation that Relates P to Functions of T & V
- The Second Fundamental Equation of Thermodynamics
- Equation 1
- Equation 2
- Recall the General Expression for Cp - Cv
- For the Joule-Thomson Coefficient
- Joule-Thomson Inversion Temperature

### Physical Chemistry Online Course

### Transcription: The General Thermodynamic Equations of State

*Hello and welcome to www.educator.com and welcome back to Physical Chemistry.*0000

*Today, we are going to talk about the general thermodynamic equations of state.*0004

*Let us jump right on in.*0008

*I will go ahead and stick with black today.*0012

*The equations of state for gas the PV = nRT, the Van Der Waals equation and number of other equations that you may have seen.*0015

*These are relations between pressure, volume, and temperature that have been derived empirically.*0023

*We have run the experiments, we derived these equations empirically.*0030

*This is how gases behave.*0033

*Any other equations, maybe we took some equations and we modified them and derived different equations*0035

*based on certain assumptions about atomic and molecular size, or maybe attractive forces.*0045

*These are all empirically derived, the equations for the gas.*0050

*The equations of state for liquid and solids was pretty much the same thing.*0056

*It is the empirical observations and empirical derivations.*0074

*The equation of state for liquids and solids was expressed or is expressed, I should say, via experimentally determined coefficients,*0077

*the coefficients of thermal expansion and compressibility.*0109

*Let me go ahead and write the equation.*0124

*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.*0127

*This is the equation state just like PV = nRT is the equation state for an ideal gas and*0166

*the Van Der Waals equation is an equation of state for a Van Der Waals Gas.*0174

*A little bit more of a real gas although not quite, it is a little bit better than the ideal gas.*0180

*This is the equation of state for liquid and solids.*0186

*It just says that the volume at any given moment happens to equal this initial volume*0189

*which is the volume of the system 0°C and 1 atm pressure the × this.*0195

*This is the initial temperature, this is a new temperature that you have to be measuring and the speed is the pressure.*0200

*At a particular pressure and at a particular temperature this is the volume.*0206

*It is a relationship between pressure, temperature, and volume.*0210

*This is empirically derived that is based on these coefficients, the coefficient of thermal expansion and the coefficient of compressibility.*0214

*This is just another equation of state.*0221

*We have this equation state for gases, we have this equation of state for liquids and solids.*0223

*Is there a way to find one equation of state that applies to all the way across the board?*0227

*The most general equation of state that relates pressure, volume, temperature, the answer is yes.*0232

*Here is how we do it.*0241

*The equation of states, just write few more things here.*0242

*These equations apply to systems at equilibrium.*0251

*We now have a more general condition for equilibrium, we have done it to the last few lessons.*0267

*We have talked about this general condition of spontaneity, a general condition of equilibrium.*0271

*For equilibrium we have the following.*0277

*Let me write this out actually but we now have a more general condition for equilibrium.*0281

*The condition is DU = TDS - PDV a fundamental equation of thermodynamics.*0306

*This is the relation for a system at equilibrium.*0319

*The change in its energy = TDS - PDV the relationship between energy, entropy, volume, pressure, temperature.*0321

*This is the general condition of equilibrium.*0330

*It is what we have been doing in the last couple of lessons.*0332

*From this, let us see if we can actually derive a more general equation of state*0338

*that applies to solids, liquids, gases, any phase, any system, anytime under any conditions.*0344

*Let us see what we can do.*0353

*Let us go ahead and write this equation again.*0356

*I actually go back but this equation that I just wrote is DU = TDS – PDV.*0364

*Let us go ahead and recall where this actually comes from.*0372

*We said that general condition of equilibrium is the following.*0376

*The general condition, I’m just going to a quick derivation again.*0381

*Just take a couple of lines here, it is - DU - PDV - D work other + TDS = 0 this is the general condition*0388

*but we are not concerned with other work.*0406

*We are only concerned with this and this and this.*0411

*When you rearrange this, you get that.*0414

*That is where this comes from, this is the general expression.*0417

*This is just a rearranged version of it.*0418

*Let us go ahead and start, we have DU = TDS – PDV.*0421

*Let us let any change happen at a constant temperature.*0432

*Let us go ahead and hold the temperature constant.*0446

*When we do that, we can write this as the following.*0448

*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.*0451

*We are going to go ahead and divide everything by this term right here by the DV and we get the following.*0473

*We get DU DV at constant T = T DS DV at constant T - P.*0480

*Let me rearrange this, I end up with pressure = T DS DV at constant T - DU DV.*0492

*Let us make this a little bit more clear, my apologies, DV at constant T.*0507

*I have this equation, S and U, the entropy and the energy they are functions of temperature and volume.*0513

*We know this already.*0525

*They are functions of temperature and volume.*0529

*We have an equation that relates P the pressure to functions of temperature and volume, pressure, temperature, volume.*0538

*This is an equation of state that relates P to functions of temperature and volume.*0558

*You have an expression that relates to temperature to pressure and volume, you have an equation of state.*0573

*We have P = T DS DV - DU DV at constant T, this is our equation of state.*0589

*The first of our general equations of state that relates pressure to volume and temperature.*0606

*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.*0612

*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.*0632

*This is our equation of state, this is the first of our general equations of state.*0653

*That is going to be equation number 1.*0658

*Let us look at the second fundamental equation.*0661

*The second fundamental equation, let us do this in red.*0665

*Let us look at the second fundamental question of thermodynamics.*0675

*For the first equation was DU = TDS - PDV this one we ended up with*0685

*that equation of state pressure expressed in terms of functions of temperature and volume.*0694

*Pressure, temperature, and volume, this is an equation of state.*0702

*It is a general equation of state, this equation right here applies to any system, anytime, any temperature, any volume, any phase.*0705

*Liquid, solid, gas this holds, this is true in general.*0713

*The second equation is DH = TDS + VDP.*0720

*Let us go ahead and take care of this isothermally.*0728

*Isothermally, I can go ahead and write it this way, I can write DH = T DS constant T + V DP constant T.*0731

*I'm going to go ahead and divide by this term DP and I end up with the following.*0746

*I end up with DH DP constant T = T DS DP at constant T + the volume.*0753

*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.*0766

*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.*0790

*I’m going to go ahead and put this in for that and I end up with the following.*0812

*I end up with volume = T × DV DT at constant P + DH DP at constant T, this is my second general equation of state.*0817

*Volume is a function of pressure and temperature.*0844

*Enthalpy is a function of pressure and temperature.*0850

*These are functions of pressure and temperature.*0855

*Volume is expressed in terms of functions of pressure and temperature.*0859

*Volume, pressure, and temperature, this is an equation of state.*0864

*It relates the volume of the system to the pressure of the system to the temperature of the system.*0868

*This is the second general equation of state.*0873

*One is P in terms of V and T, this is V in terms of P and T.*0876

*They are essentially the same but they are just different ways of looking at it.*0881

*One is pressure and one is volume.*0885

*Let us see, this is our general and applied to any substance, in any phase.*0890

*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.*0914

*Of course, in the most practical conditions you are going to be using one of the other two.*0923

*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.*0928

*Let us go ahead and actually talk about some applications, how can we use this?*0940

*What can we do here?*0945

*Let us go ahead and rewrite our equations.*0946

*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.*0949

*This is equation 1 and this is equation 2.*0989

*If we knew what this is or if we knew what this is, let me just go ahead and put the values in and*0995

*we would immediately have an equation of state.*1004

*It would be absolutely beautiful but most of the time we actually do not know what this and this are.*1007

*These things are not easy to measure.*1012

*Remember back when we are discussing energy, this change in energy with respect to a change in volume*1014

*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.*1020

*These things are not easy to measure.*1029

*If we knew then it would be really great, but we do not know then so let us see what else we can do.*1031

*Let us go ahead and actually rearrange these equations.*1039

*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.*1042

*Let us play around with these equations.*1051

*That is what we are going to do, we are just going to play around with the equations.*1053

*Let me go ahead and take the first equation.*1056

*I'm going to take P = this and when we rearrange it I'm going to write it as.*1060

*Let me go ahead and do on the next page.*1066

*I’m going to write it as, let me write the original P = T × DP DT sub Z - DU DV sub T.*1073

*I’m going to move this over here and move P to the right.*1092

*I’m going to write this as DU DV with constant temperature = T × DP DT V – P.*1095

*For an ideal gas I know that PV = nRT.*1109

*Therefore, P is nRT/ V.*1116

*If I take the partial of P with respect to T DP DT holding V constant, I end up with nR/ V.*1121

*DP DT holding V constant is nR/ V.*1137

*I can go ahead and put this in here so I get DU DV/ T = T × nR / V – P.*1141

*nRT/ V is P so it ends up being P - P and ends up being 0.*1156

*This confirms that Joules law.*1163

*Remember, we said that from ideal gas, Joules law, this DU DV at constant T is actually equal to 0.*1169

*This confirms it, it just came around the other way and was a lot easier to do so.*1176

*0 which confirms Joules law, that is nice.*1186

*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.*1189

*Let us see what else we can do here.*1207

*We have DU DV sub T = T DP DT sub V – P.*1211

*We already know what DP/ DT is, DP/ DT under constant V is actually equal to Α/ Κ.*1228

*If we put that in here, we get the following.*1239

*We get DU DV sub T = T × Α/ Κ – P.*1243

*Finding DU/ DV is not a very easy thing to do.*1265

*However, T, Α, K, and P these are all very easily measurable quantities.*1269

*I do not have to worry about measuring this, I can go ahead and measure these things, come up with this,*1278

*and put back in here to get my equation state.*1284

*We were able to find a nice expression in terms of easily measurable quantities for something that is not quite so easily measurable.*1288

*Something that is reasonably elusive and actually difficult to do.*1297

*This is what we always want to do, this is what we have been doing the entire time.*1301

*Trying to find ways of expressing things that are not so easily measurable in terms of things that are easily measurable,*1305

*that is what we have done here.*1312

*For equation number 2, C we have the volume = T DV DT at constant P + DH DP at constant T.*1318

*When I rearrange this I end up with the following.*1346

*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.*1349

*T is going to equal V - T DV DT under constant pressure.*1362

*However, we know what DV D T or P is, DV DT sub P is actually equal to V × Α.*1372

*Therefore, when I put this in for here I end up with the following.*1387

*I end up with DH DP sub T = V - TV Α.*1393

*I have found a way to take something that is not easily measurable.*1407

*This was related to the Joule-Thompson coefficient, in terms of things that are very easily measurable.*1412

*I have the coefficient of thermal expansion, the volume, the temperature, the volume.*1419

*This is really nice.*1423

*We have ways of finding DU DV T and DH DP sub T from easily measurable quantities is always a great thing to achieve.*1433

*Let us go back a bit.*1471

*Remember when we are talking about energy we had these two relations.*1473

*We had DU = the constant volume heat capacity DT + DU DV sub T DV.*1477

*Remember, these differential equations and we had one for enthalpy as well.*1493

*We had DH = the constant pressure heat capacity DT + DH DP at constant T × DP.*1497

*Whenever we want to find the change in energy of the system this was our general equation that we started off with.*1511

*Whenever we want find the enthalpy of the system, this was the general equation that we started off with.*1516

*For ideal gases, these equal 0 so you have just this part right here.*1521

*Again, these are the general equations that we always start off with.*1526

*These are the equations that we want to memorize.*1529

*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.*1532

*Let us go ahead and put these expressions into here.*1537

*I’m going to go ahead and do this in blue.*1541

*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.*1545

*This is an incredible achievement.*1555

*We are expressing the change in energy and change in enthalpy of the system completely in terms of things that are easily measurable.*1561

*Let us go ahead and write that out.*1568

*We have DU = CV DT.*1570

*Let me make it feel a bit much clear, DT + DU DV that is nothing more than T Α/ Κ - pressure × DV.*1577

*The change in energy, the energy of the system is a function of temperature and volume.*1593

*The change in energy is expressed in terms of the constant volume heat capacity, temperature, Κ, Α, and pressure.*1599

*All these things are easily measurable for a system, very easily measurable.*1608

*This is fantastic, this is beautiful, absolutely beautiful.*1612

*This is what we want, something that is very abstract, energy.*1617

*We are expressing it in terms that are easily measurable than laboratory.*1623

*Let us do one for enthalpy.*1628

*We have DH equal to the constant pressure heat capacity × the differential change in temperature + DH DP.*1630

*DH DP is V – TV A, V - TV Α × DP.*1639

*This is the other equation.*1654

*Before, we have to leave it as this and this, but now that we are able to from our fundamental equations of thermodynamics,*1660

*from our conditions of equilibrium, from the fact that we are able to derive*1667

*a couple of general equations of state we were able to find easy ways of finding these two values.*1673

*We put them in and now we have closed the very important circle, absolutely beautiful.*1679

*Let us go ahead and round this out with a couple of more applications.*1686

*We call the general expression for the difference in the heat capacities the C sub P - C sub V.*1692

*Remember, for an ideal gas we said that = RN.*1722

*The general expression for the difference of heat capacities was the following.*1725

*We had CP - CV = this reasonably complicated looking thing, DU DV sub T × DV DT sub P.*1729

*This is the general expression for the difference in the heat capacities of any system, of any substance no matter what it is.*1748

*Let us see what we have got.*1758

*CP - CV we know what DU DV is.*1760

*We just figured it out in terms of Α and Κ and P.*1765

*We also know what DV DT is, we knew that already as V Α.*1769

*Let us go ahead and put all of those values in.*1774

*We get P + DU DV is T Α / Κ – P × V Α.*1777

*The P cancels the P, therefore, in general the constant pressure heat capacity of a substance - its constant volume heat capacity,*1790

*in other words the difference of heat capacities = T × V × A²/ Κ.*1800

*It is that not the most beautiful thing you have ever seen your life?*1811

*That is absolutely extraordinary.*1813

*The difference of heat capacities, constant pressure, and constant volume is related to the temperature volume,*1815

*the square of the coefficient of thermal expansion divided by the coefficient of compressibility.*1822

*All of these things are easily measurable or you just look them up in a book, this is absolutely fantastic.*1827

*This applies to any system, any substance, any time, any phase, that is what makes this amazing.*1835

*Let us keep going.*1846

*How about the Joule-Thompson coefficient?*1847

*Let us do this one and go back to black.*1849

*For the Joule-Thompson coefficient we had the following relation.*1855

*We had - C sub P × Joule-Thompson coefficient = DH DP T.*1869

*We know what that is, - CP × μ which is the Joule-Thompson coefficient.*1882

*This DH DP T = V - TV Α.*1889

*Let us go ahead and rearrange this.*1896

*We end up with the Joule-Thompson coefficient = TV Α - V/ CP.*1900

*Absolutely extraordinary, all of these things are easily measurable.*1912

*Constant pressure heat capacity, volume, Α, we have a way of actually finding*1915

*the Joule-Thompson coefficient for a substance from things that we already know.*1919

*This is absolutely extraordinary.*1924

*Clearly, these are much more easily measurable than this is.*1927

*Finding the Joule-Thompson coefficient is not an easy thing to do.*1929

*Let us keep going, the Joule-Thompson inversion temperature.*1938

*The Joule-Thompson coefficient changes sign, it goes from positive to negative.*1954

*In other words, the Joule-Thompson coefficient has to equal 0.*1963

*At a temperature it can go from positive to negative, it equal 0.*1969

*When we set this equal to 0, 0 =TV Α - V/ CP.*1973

*We end up with the following.*1987

*We end up with TV Α - V = 0.*1988

*Let me make it a little bit clear here so it does not look like U = 0.*1994

*We go ahead and we end up with.*2000

*Let us go ahead and factor out V, we get T Α -1 = 0.*2002

*We get T A -1 = 0 and we end up with the inversion temperature, the Joule-Thompson inversion temperature = 1/ Α.*2009

*Absolutely fantastic, from easy things to measure, easy things to look up,*2025

*we are able to find the Joule-Thompson coefficient we are able to find the inversion temperature, all kinds of things.*2032

*That is what makes this amazing.*2041

*Thank you so much for joining us here at www.educator.com.*2043

*We will see you next time, bye.*2045

1 answer

Last reply by: Professor Hovasapian

Fri Mar 25, 2016 8:32 PM

Post by John Charpentier on March 23, 2016

Hello. A minor mistake occurs at 15:56. T(dP/dT)_V is written as T(dP/dV)_V. This is corrected when the equation is rewritten at 18:26. Not a big deal but thought I would point it out.