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Lecture Comments (2)

1 answer

Last reply by: Professor Hovasapian
Mon Dec 29, 2014 7:35 PM

Post by e b on December 27, 2014

clip 30:46 ==> Should one of the rate be -ve? For example, -(delta S/delta P) or -(delta V/delta T), thus, = -V alpha

The Fundamental Equations of Thermodynamics

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 Fundamental Equations of Thermodynamics 0:44
    • Mechanical Properties of a System
    • Fundamental Properties of a System
    • Composite Properties of a System
    • General Condition of Equilibrium
    • Composite Functions & Their Differentiations
    • dH = TdS + VdP
    • dA = -SdT - PdV
    • dG = -SdT + VdP
  • Summary of Equations 12:10
    • Equation #1
    • Equation #2
    • Equation #3
    • Equation #4
  • Maxwell's Relations 20:20
    • Maxwell's Relations
    • Isothermal Volume-Dependence of Entropy & Isothermal Pressure-Dependence of Entropy

Transcription: The Fundamental Equations of Thermodynamics

Hello welcome to and welcome back to Physical Chemistry.0000

Today, we are going to talk about the fundamental equations of thermodynamics.0004

It is in this lesson that absolutely everything that we have talked about comes together.0008

In this lesson, it is going too close the circle on thermodynamics.0012

For me personally, no matter how often I present this particular material in this particular lesson,0017

I absolutely cannot get over just how unbelievably beautiful this stuff is.0023

The circle this closes in such an amazing way.0029

Everything that we have worked on is now going to come together and get this absolutely beautiful way.0033

For me personally, I hope that you feel like that also even just a little bit.0038

Let us get this started.0043

A system has two mechanical properties.0045

Let us go ahead and stick with black.0050

A system has two mechanical properties, they are the pressure and the volume so P and V.0054

A system has three fundamental properties.0076

These fundamental properties correlate to the laws of thermodynamics, the 0 law, the first law, and the second law.0087

They are of course the temperature, the energy, and entropy.0095

0 law, first law, second law are the fundamental properties.0102

A system has a three composite properties or compound properties if you will.0108

They are the enthalpy, Helmholtz energy and the Gibbs energy.0120

We are going to derive the fundamental equations of thermodynamics.0128

We are going to express the fundamental relations among all of these properties.0133

This is what I mean why how it all comes together.0138

As we mentioned before, we will restrict our discussion to systems that do only pressure, volume, work.0143

We are not going to be concerned with other work electrical, chemical, or otherwise.0179

In other words, DW other is always going to be 0 for us.0186

Let us recall a general condition of equilibrium.0193

The general condition of equilibrium we have - DU – P δ V + TDS - DW other is going to = 0.0200

Again this is going to be 0 for us so we are going to have – DU - PDV + TDS = 0.0233

Let us rearrange this a little bit and express it in terms of energy.0249

We will put the DU on the left is going to be DU = TDS – PDV.0253

Let us see here, this is nothing more than the fundamental equation of thermodynamics that we introduced when we talked about entropy.0269

We arranged it a little bit.0279

Let me do here in blue actually do in red.0285

Remember, we said that DS = 1/ T DU + P/ T DV.0288

If I multiply everything by T, I end up with TDS = DU + PDV and if I move this over there I end up 0296

with DU = TDS - PDV which is exactly what I have here.0307

There is nothing new about this equation, we have seen it before, we seen it in the context of entropy 0314

but now we have rearranged it so we can actually make it look more systematic with the other things that we are going to do.0320

This is just a fundamental equation of thermodynamics.0326

The fundamental equation, the one that relates energy with entropy and volume, pressure, and temperature.0328

Let us go ahead and call this equation number 1.0340

Let us go ahead and write our composite functions.0346

Let me go ahead and go back to this page, I will write it up here.0350

I have DU = TDS – PDV.0353

These are composite functions.0362

Let me go back to black, this is nice to do in black.0367

Our composite are H = U + PV.0376

Our A Helmholtz energy = U - TS and our Gibbs free energy = U + PV – TS.0387

Let us differentiate these three.0399

When we differentiate the three, we get the following.0413

I will go back to black.0417

We get DH = DU + PDV + VDP.0420

DA = DU - TDS – SDT, DG = DU + PDV + VDP - TDS – SDT.0435

This is the product rule, this times the derivative of that, that times the derivative of this.0461

This is where this is coming from.0465

Let us go back to blue.0470

In each of these equations right here, the differential equations for the H, A, and G, I'm going to put in the value of DU.0473

I'm going to put in T δ S and DU here and here and I will see what kind of relations that I find.0482

That is what I'm going to do.0493

Let us go ahead and do the first one.0497

Let us go ahead and do this in blue.0500

I’m going to take the DH first, I have DH DU + PDV + VDP.0510

For DU, I’m going to put in TDS - PDV so I have DH = TDS - PDV and I have + PDV + VDP.0522

This and this cancel and I'm left with DH = TDS + VDP.0540

We are going to call this equation number 2.0555

We are taking care of that one, now let us move on to the next one.0559

We have DA = DU -TDS – SDT.0564

We have DA = TDS -PDV-TDS – SDT, TDS and TDS go way I'm left with DA = - SDT and I'm going to rearrange this.0581

You are going to see why in a minute, - PDV.0611

This is equation number 3.0620

We will do DG = DU + PDV + VDP - TS – SDT, DG I will put in the value of DU.0622

DU was TDS – PDV, I have + PDV + VDP - TDS – SDT.0644

TDS cancels TDS, - PDV + PDV I'm left with DG = - SDT + VDP.0662

I’m going to call this equation number 4.0687

Equations 1, 2, 3, and 4, these are called the fundamental equations of thermodynamics.0690

They are not really for separate equations.0701

In fact of the matter is the first equation the DU, the TDS, the DU = TDS - PDV equation number 1, 0703

that is the fundamental equation of thermodynamics.0717

These other three are different ways of looking at the fundamental equation of thermodynamics 0719

through the eyes of the composite functions and the other things.0725

Let us go ahead and write them all in one.0731

Let me go ahead and write this way so we can take a look at we have DU = TDS – PDV.0736

We have DH = TDS + VDP.0748

I sure hope that I'm not making any mistakes as far as the + and - are concerned.0755

I hope that you will check these.0758

You have it in your book because there is lots of + and - and S and V, and all kinds of things flowing around here.0761

We have DA = - SDT - PDV and we have DG = - SDT + VDP these are the fundamental equations of thermodynamics.0769

Notice the symmetry, notice how we have arranged them.0795

We have energy and enthalpy, the energy of the system, the enthalpy of the system.0799

We have the Helmholtz energy and we have the Gibbs energy.0806

This is expressed in terms of TP is variables are S and V.0809

TDS here, TDS here is - PDV + VDP.0816

The DA and DT we have - SDT – SDT.0821

Here it is - PDV + VDP just like - VDP + VDP.0825

I’m not sure about the extent that you actually have to memorize these, you do not really.0832

Basically, all you need again is just the fundamental equation of thermodynamics which you get from the definition of entropy and 0836

the first law of thermodynamics and everything else to sort of falls out.0841

The rest all you are really doing is just manipulating the equations.0846

Let us see how unbelievably beautiful these are.0849

These equations express the relationships be among all these things, energy, temperature, entropy, volume, pressure, 0852

Helmholtz energy, Gibbs energy, volume pressure, pressure volume, entropy temperature, these ties everything together.0859

Let us go ahead and say a little bit more.0872

Equation number 1, a change in energy DU is related to a change in entropy DS and the change in volume DV.0875

Equation number 2, the change in enthalpy is related to a change in entropy and a change in pressure.0920

Nothing new here, we know this is the relationship energy, volume, enthalpy, pressure, and everything is coming together beautifully.0953

Equation number 3, a change in Helmholtz energy is related to a change in temperature and a change in volume.0964

Helmholtz energy relationship is temperature and volume are the variables for Helmholtz energy.0996

Equation number 4, the Gibbs energy, the change in Gibbs energy is related to change in temperature and change in pressure.1004

Let us go back to black here for a second because these equations are reasonably straightforward and simple.1048

Reasonably that is why we call them reasonably simple.1063

An expression, the variables of the differentials on the right hand sides of these equations, 1067

the variables on the right are called the natural variables for the particular property on the left.1083

Let us go ahead and go here, in other words.1127

Let me write out the equations again here just so I have them.1131

I’m going to write it here, I got DU = TDS – PDV, I got DH = TDS + VDP.1137

I got DA = - SDT - PDV and I have DG = - SDT + VDP.1152

In this particular, the case of energy, the natural variables are S and V.1166

In the case of enthalpy, the natural variables are S and P.1183

In the case of the Helmholtz energy, the natural variables are T and V.1188

In the case of Gibbs energy, the natural variables are T and P.1192

By natural variables, it is just these are the variables that they are expressed in.1198

Gibbs energies is a function of temperature and pressure.1202

Helmholtz energy is a function of temperature and volume.1205

DH enthalpy is a function of entropy and pressure.1209

Energy is a function of entropy and volume, that is all we are saying.1213

That is what we mean when we say the natural variables.1218

Here is work, it is really interesting and beautiful.1223

Since each of the expressions on the right is inexact differential for state properties, the mix partial derivatives of the coefficients are equal.1228

Let us do it one more time, there is no harm in doing so.1278

Let us go ahead and do this in blue actually.1282

We are going to do left and right.1285

Let us do DU = TDS - PDV an exact differential.1288

Therefore, the partial of this differential with respect to this variable = the partial of this 1298

with respect to this variable with the other variable being held constant.1305

What we mean is the following, we mean DT DV holding S constant = DP DS holding V constant.1309

This is -, we make this a better V.1327

The differential of this with respect to the other variable, holding this variable constant = the differential of this 1333

with respect to this variable holding this variable constant.1338

Mixed partial derivatives that is what this says.1343

From these relations I’m deriving these relations DT DV under constant entropy = - DP DS our constant volume.1346

With enthalpy DH = TDS + VDP from this we derive DT DP under constant S = DV DS under constant V.1358

The differential this with respect to this variable = the differential of this with respect to this variable.1386

We will do DA = - SDT - PDV so what you end up with is - DS DV holding T constant = - DP DT holding V constant.1395

Of course, the - disappear so I get that DS DV sub T = DP DT sub V.1431

We will do the Gibbs’ energy DG = - SDT + VDP, what you end up with is -DS DP holding T constant = DV DT holding P constant.1442

There you go.1473

These relations right here is partial derivatives relations are called Maxwell's relations.1475

Do not get these confused, those of you with Physics backgrounds you are not going to be confused with Maxwell’s equations for electro magnetism.1486

Those are the difference set of equations.1494

They form the foundation of electromagnetic theory, the classical electromagnetic theory.1497

These are Maxwell's relations for thermodynamics derived from the four fundamental the equations of thermodynamics.1502

The first and the second equation, these two relations right here, this relation and this relation, they are not going to concern us all that much.1514

It is the last two that are going to be very important for our particular purposes and here is why.1522

Let me go ahead and go to the next page.1529

Let me rewrite the last two again just so we have them.1541

I got DS DV under constant temperature = DP DT under constant volume and I have – DS DP under constant temperature = DV DT and it should be under constant pressure.1547

The last two, those are numbers 3 and 4.1578

These relations, they express are important because they relate the isothermal constant temperature, 1585

the isothermal volume dependence of the entropy, and the isothermal pressure dependence of the entropy.1609

In other words, when I change the volume how does the entropy change?1644

When I change the pressure how does the entropy change?1648

The rate of change.1650

This is the volume dependence of the entropy, this is the pressure dependence of the entropy.1652

These are expressed in terms of things that are easily measured.1658

If I have a system I will change the temperature and see how the pressure changes.1662

I change the temperature and see how the volume changes.1668

If I want to know how entropy changes when I change the volume, 1670

all I have to do is hold the volume constant, change the temperature and see how the pressure changes.1677

The number I get here that is this.1683

This is profoundly important.1684

These things on the right are very easily measurable.1687

These things on the left not easily measurable.1690

I get them this way that is what makes these important.1693

The relations are important because they relate the isothermal volume dependence of the entropy and1697

the isothermal pressure dependence of the entropy to easily measure quantities.1702

We have seen these equations before back when we discussed entropy.1721

DS DV = DP DT under constant V this is constant T we said that this = A/ K.1727

This is equal to α/ Kappa.1746

We already have seen this before.1748

This DS DP under constant T which is going to be DV DT under constant P = - V α.1750

We have seen these before.1776

If I know the entropy changes with respect to volume, all I really need to do is measure how the pressure changes when I change the temperature.1778

I already know that.1789

If I have the coefficient of thermal expansion and divide by the coefficient of compressibility that is what this is.1791

Here it is the volume of the system × the coefficient of thermal expansion.1797

All of these things are easily measurable.1800

We actually happen to calculate it, you do not have to measure them anymore 1802

for any particular gas and any solid, any liquid, that we have to be dealing with.1805

These all have been tabulated under standard conditions so that is what we have got here.1810

This and this.1816

These are the fundamental equations of thermodynamics and they are Maxwell's relations coming from the fundamental equations of thermodynamics.1818

This ties the mechanical properties, the fundamental properties, and the composite properties together.1827

This closes the circle on thermodynamics.1833

The rest of what we are going to be doing until we actually close out a full discussion of thermodynamics is just tying up loose ends.1836

There you have it, absolutely beautiful.1844

Thank you so much for joining us here at

We will see you next time, bye.1848