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 1 answerLast reply by: Professor HovasapianFri Mar 25, 2016 8:32 PMPost by John Charpentier on March 23, 2016Hello. 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.

### 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 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

### 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 equations0035

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 and0166

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 volume0189

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 state0338

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 condition0388

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 with0685

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 and0995

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 volume1014

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 derive1667

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 finding1915

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