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Looking Back Over Everything: All the Equations in One Place

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
  • Work, Heat, and Energy 0:18
    • Definition of Work, Energy, Enthalpy, and Heat Capacities
    • Heat Capacities for an Ideal Gas
    • Path Property & State Property
    • Energy Differential
    • Enthalpy Differential
    • Joule's Law & Joule-Thomson Coefficient
    • Coefficient of Thermal Expansion & Coefficient of Compressibility
    • Enthalpy of a Substance at Any Other Temperature
    • Enthalpy of a Reaction at Any Other Temperature
  • Entropy 8:53
    • Definition of Entropy
    • Clausius Inequality
    • Entropy Changes in Isothermal Systems
    • The Fundamental Equation of Thermodynamics
    • Expressing Entropy Changes in Terms of Properties of the System
    • Entropy Changes in the Ideal Gas
    • Third Law Entropies
    • Entropy Changes in Chemical Reactions
    • Statistical Definition of Entropy
    • Omega for the Spatial & Energy Distribution
  • Spontaneity and Equilibrium 15:43
    • Helmholtz Energy & Gibbs Energy
    • Condition for Spontaneity & Equilibrium
    • Condition for Spontaneity with Respect to Entropy
    • The Fundamental Equations
    • Maxwell's Relations
    • The Thermodynamic Equations of State
    • Energy & Enthalpy Differentials
    • Joule's Law & Joule-Thomson Coefficient
    • Relationship Between Constant Pressure & Constant Volume Heat Capacities
    • One Final Equation - Just for Fun

Transcription: Looking Back Over Everything: All the Equations in One Place

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

Today, we are going to look back over absolutely everything that we have done in this thermodynamics portion of Physical Chemistry.0004

It is going to be all the equations in one place.0013

Let us just jump right on in and get started.0016

You can consider this as a quick general review.0020

We started off with the definition of work and we set the work is equal to the external pressure.0025

The change in work DW is equal to the external pressure × the change in volume.0031

With the definition of energy so it is DU = DQ – DW.0039

This is absolutely very important.0044

I'm probably going to go ahead and put circles around the equations that I think are the ones that you need to memorize,0046

the ones that you need to bring to the table when you solve a particular problem.0052

Let me go ahead and do that in red actually.0058

This is a very important equation, the definition of work is absolutely fundamental and of course the definition of energy.0062

Recall, that we took heat from the systems point of view.0072

In other words, if heat goes into the system heat is positive.0075

If heat leaves the system, it is negative.0079

We took work from the surroundings point of view.0082

If work is done on the surroundings then work is positive.0085

If work is done on the system then work is negative.0091

It is the reverse of what some people do, often in chemistry we see this DQ + DW.0095

Again, this is the reason for the minus sign above equation that is different from what you are used to seeing or for what you do see.0101

Absolutely everything else is completely the same.0108

The only thing that this minus sign does is, when I get value of work and let us say it is positive, the only thing you have to do is change the sign.0111

That is the only thing because all you are doing is changing a particular perspective.0120

It is the magnitude that actually matters.0125

Should I say the direction does not matter.0128

The direction matters but in terms of the equation, it is just a question of point of view.0130

In chemistry we take the systems point of view for work and for heat.0134

What I have done here is take the systems point of view for heat, the surroundings point of view from work.0141

And if you go back to those lessons where I discussed it, I talked about why I actually did that. 0146

But as far as your problems are concerned, nothing changes.0152

You are going to use the same equations.0155

It is just your sign for work is just going to be the opposite of what you get here.0157

The definition of enthalpy.0165

The enthalpy of the system is a measure of the energy + the energy of the system + the pressure of the system × the volume of the system at a given point.0166

It is just a combined, it is a derived unit.0178

It is a compound unit H = U + PV.0180

We wanted to find the heat capacities.0185

The constant volume heat capacity is the change in heat at constant volume ÷ the change in temperature 0188

which happened to be DU/ DT at constant volume, the change in energy per unit change in temperature.0195

Constant pressure heat capacity, the same thing, is the amount of heat that is transferred 0205

under conditions of constant pressure ÷ the differential change in heat or it is the DH DT at constant P.0210

These are the definitions of the heat capacities.0217

A relationship between the two heat capacities for an ideal gas CP - CV = RN.0223

I think this is absolutely important one to remember.0229

I should have circled the ones for the heat capacities but that is okay.0232

Heat and work are path properties.0237

Remember, their values depend on the path taken to go from some initial state to some final state.0240

They actually change the values of the heat and work change depending on the path that you take.0245

Energy is a state property, along with all the other thermodynamic properties.0250

H is a state property, G is a state property, S is a state property.0256

Energy is a state property also called the state function.0263

Its value does not depend on the path taken to go from an initial state to a final state.0266

It only depends on the initial and final states.0271

It is actually pretty extraordinary given the definition of energy.0274

You have energy is equal to DQ – DW.0279

Heat and work are path functions.0282

How is it that the difference of two path functions ends up with a state function that is actually very extraordinary and very profound.0285

State properties are exact differentials.0293

It has profound consequences for the mathematics.0296

We want to express changes in the energy of a system in terms of properties of the system.0307

We decided on temperature volume and temperature pressure.0313

I think there is a little bit of typo here, that is not a problem I will fix it.0320

The energy differential is this, this is the total differential.0323

Just a straight mathematical form for the total differential.0326

DU DT being the definition of constant volume heat capacity, you end up with this equation.0330

One of the important equations to know.0339

As for enthalpy is concerned, enthalpy was going to be for temperature and pressure so I apologize here.0341

Let me go to black, let me erase this, this should be DP.0348

It looks like I have DP here so it was just a little typo there.0354

This is the differential expression for the δ H.0359

The DH DT being the constant pressure heat capacity, we end up with this equation.0365

If I change the temperature, if I change the pressure, how does the enthalpy of the system change?0374

We went on to discuss Joules law.0384

The Joules law for an ideal gas DU DV = 0, this is the second term in the equations that we just saw for energy.0386

The change in energy per unit change in volume for an ideal gas that is 0.0394

The Joule Thompson coefficient, change in enthalpy per unit change in pressure = -CP × the Joule Thompson coefficient.0399

For an ideal gas, the DH DPT that was equal to 0.0411

This is the second term in the previous equation that we saw for the enthalpy differential.0416

The coefficient of thermal expansion, these are very important to know as it keep coming up.0422

Coefficient of thermal expansion is Α 1/ DV DT.0430

Basically, it measures the change in volume per unit change in temperature at constant pressure.0434

The coefficient of compressibility measures the change in volume per unit change in pressure at constant temperature.0439

The enthalpy of a substance at any other temperature, besides the temperature that we know, we have tabulated enthalpy.0451

In the tables of thermodynamic data, they are done at 25°C or 298°K.0458

If you want to know the enthalpy of a particular substance at any other temperature, this is what you would use.0463

You would take the enthalpy of the standard temperature and 0470

then you would integrate the constant pressure heat capacity with respect to temperature from 298 to the new temperature.0473

The enthalpy of reaction at any other temperature, the enthalpy of a reaction is the δ H.0482

The enthalpy of the products - the enthalpy of the reactants.0488

That is again, the δ H at 298 + the integral 298 δ CP.0491

This is a little different, this is just the sum of the heat capacities of the products - the sum of the heat capacities of the reactors including these coefficients.0498

You do the same thing that you do with δ H, δ G, δ S.0511

Products – reactants, just make sure to include these coefficients.0513

Of δ H for elements and δ G for elements is 0.0520

You remember from general chemistry that is not the case here.0523

There is always going to be some number for the heat capacity even for an element, it is never 0.0526

We are going to discuss entropy and our definition of entropy was DS = DQ reversible/ T.0536

The differential change in entropy of the system is equal to the heat transferred by reversible process ÷ the temperature at which that takes place.0543

The Clausius inequality says DS is greater than DQ irreversible/ T.0552

For any spontaneous process this has to be satisfied.0557

This is for reversible process, this is at equilibrium.0561

Any particular process, an irreversible process, the heat transfer ÷ the temperature 0566

at which the transformation takes place is going to be less than the entropy change.0573

This is for a spontaneous process.0581

Entropy changes in an isothermal system, we said that an isothermal system temperature stays the same.0586

The temperature actually comes out of this one, we integrate these two equations and we get the following.0592

The change in entropy of a vaporization process is equal to the δ H of the vaporization ÷ the boiling temperature.0598

And the δ S of fusion is equal to the δ H of fusion ÷ the melting temperature.0605

The fundamental equation of thermodynamics expressed as DS.0614

This expresses the relationship between entropy, temperature, energy, pressure, and volume.0618

That is why it is called the fundamental equation of thermodynamics.0625

We will see it again in another form expressed in terms of DU rearranged.0628

It is DU over here alone a little bit later.0633

Let us see what we have got and how far are we.0640

We did the same thing with entropy that we did with energy.0644

We want to express it in terms of properties of the system.0647

Once again, we do temperature volume and we do temperature pressure.0650

This is the differential expression that ends up being this,0656

a very important equation, these right here, the Α/ Κ, let us go ahead and leave this alone.0662

This is the change in entropy when I change temperature volume, the change in entropy when I change temperature pressure.0672

These are the two equations that are really important to know.0679

Entropy changes in the ideal gas, you have this equation and you have this equation.0684

These equations can be derived from these just using PV = nrt.0690

That takes care of that.0699

Third law entropies.0701

Basically, the entropies that you see, the entropy values that you see in your table of thermodynamic data, these are third law entropies.0706

The only thing that you need to know is that if I want to measure the entropy of the substance at any particular temperature, 0712

depending on what it is, whether it is in the gas state, liquid state, the solid state. 0720

Let us say you know the entropy at 25°C from the table of thermodynamic data and you want to find the entropy at let us say 75°C.0728

It is 50° higher.0744

What you actually ended up doing is, because you are not changing state all you are going to do is 0746

integrate the change in the particular heat capacity for the solid, for the liquid, for the gas ÷ the temperature, going from one temperature to the next.0753

Essentially what this says is that going from Z to some melting temperature 0°K, this is going to be my entropy.0767

I have to include the entropy of the melting process.0777

I have to include the entropy of going from the melting temperature to the boiling temperature.0781

I have to include the entropy of the vaporization process.0786

I have to include the entropy in going from the boiling temperature to another temperature.0790

What you are going to be doing is, if you want to find the entropy at some temperature, 0798

you are going to take the entropy of some temperature that you know 0804

and you are just going to be integrating from the initial temperature to 0809

your new temperature of the constant pressure heat capacities ÷ T DT whatever state that is in.0812

If it is in a liquid state and you want to find the entropy of liquid water 25°, 75°.0819

In that range, the 25 to 75 is still liquid water so you would use the constant pressure heat capacities of liquid water.0827

You have to account for every single phase change and any temperature difference.0835

The entropy change in a chemical reaction, same sort of thing.0843

For chemical reaction, we have products – reactants.0847

It is going to be the entropy that you know at a particular temperature which for us is 25°C.0850

And you are going to integrate from that 25°C to the next temperature of this δ CP.0858

And again this is just the sum of the heat capacities of the products - the sum of the heat capacities of the reactants.0864

We will go on to discuss entropy from a statistical point of view.0876

The statistical definition of entropy was this, S = KB LN O.0881

KB is the Boltzmann constant.0885

The ω for the spatial distribution was this one.0888

Here N, this is the number of spaces available.0892

It is N sub A which is the number of particles.0904

A good approximation when the number of particles is a lot less than the number of spaces available.0912

A gas in a big volume is this, this is a good approximation to that.0919

The ω for the energy distribution is the one that we are actually going to use later on after we have discussed quantum mechanics.0924

When we come back and talk about statistical thermodynamics.0932

This is the one we are going to be concerned with not so much this one.0936

This is just the ω for the energy distribution, that is all.0940

Spontaneity and equilibrium, we went on to define this thing called the Helmholtz energy that was U – TS.0946

We did the definition of Gibbs energy.0953

The Gibbs energy was defined as the energy of the system + the pressure × the volume - the temperature × the entropy.0955

This is a compound thermodynamic properties made up of these three things.0963

U + PV happen to equal H as enthalpy.0969

G is also equal to H - TS or it is equal to A + PV.0972

All of these three are all definitions, this is the definition of G.0977

You can also use these two, if you need to.0981

Now under conditions of constant temperature and pressure, the condition of spontaneity was that δ G.0985

A little bit of problem here, it is not supposed to be greater than 0.0991

I’m thinking about entropy here, sorry about that.0997

For conditions of spontaneity, in order for a particular process, a particular reaction to be spontaneous, we need the δ G to be less than 0.1001

The condition for equilibrium is that DG or δ G is equal to 0.1012

These are very important to know.1017

One the most important equations and probably the most significant in chemists δ G = δ H - T δ S.1022

Your δ G in enthalpy term, in entropy term, and the temperature term.1029

The δ G for a reaction is the maximum amount of energy above and 1036

beyond expansion of work that can be extracted from a spontaneous process and harnessed to do useful work.1041

That is what δ G is, it gives you an upper limit on the amount of energy that you can actually use to do useful work.1049

Think of δ G as ordered energy, all the rest of the energy of the system is spent on the entropy, it is disordered energy.1057

Δ G is ordered energy that you can actually use to do useful work if δ G happens to be negative.1068

The conditions of spontaneity with respect to entropy.1079

The δ S of the universe = δ S of the surroundings + the δ S of the system.1081

The δ S of the surroundings is - the δ H of the system ÷ T.1087

The δ S of the universe was - δ G ÷ T this was the relationship.1092

Δ G' is for spontaneity, δ G has to be less than 0.1098

It is the same as δ S of the universe having to be greater than 0.1102

The fundamental equations.1111

These set of equations here and on the next page, they are the ones that basically tie everything together.1114

DU = T DS – P DV.1121

DH = T DS + V DP.1125

DA = - S DT – PDV.1128

DG = - S DT + V DP.1132

Energy, enthalpy, Helmholtz energy, Gibbs energy.1136

Maxwell's relations, these establish relationships between the rates of change from the fundamental equations that we just saw.1146

We have this one which is the rate of change of temperature per unit change in volume at a constant entropy 1155

is equal to the negative of the rate of change of pressure per unit change in entropy at constant volume.1162

These are relationships that exist and these relationships that we actually end up using as substitutions.1169

For example maybe we have this one here and perhaps this one here if it shows up in an equation, 1178

because it is equal to this we go ahead and we use this one because this is actually really easy to measure, volume, temperature, pressure.1187

Entropy, pressure and temperature entropy, things involving entropy are difficult to deal with so it is nice that we have these relationships.1194

Anytime something like the shows up, we can use this one.1200

If something like this shows up we can use this one.1203

That is exactly what we are going to do.1206

The thermodynamic equations of state are profoundly important.1209

You do not necessarily need to memorize them but basically instead of PV = nrt or the Van Der Waals gas law or the equation of state for a liquid, the equation of state for a solid.1213

Instead all of these equations, these equations they apply to every single state and every single situation.1226

These are the thermodynamic equations of state.1234

These are the most general expressions of the state of the system.1237

Rearranging these and using Α and Κ from above, remember the coefficient of compressibility and the coefficient of thermal expansion, 1243

we end up finding that the DU DV is the term that showed up in the energy differential expression.1253

The DH DP is the one that showed up in the enthalpy differential, it is actually equal to this.1259

You do not have to memorize these are good to now.1265

Now, you can substitute these values back in the equations for energy and enthalpy and you end up with this.1270

What makes these extraordinary is that these equations express changes in the energy and enthalpy of 1279

the system entirely in terms of values that we can either measure or obtain from a table.1283

Measurable measurable, table table.1290

That is fantastic, easily measurable quantities or easily something that I can look up in a table.1301

I can tell you that if I change the temperature and volume, or if I change the temperature and pressure of a system, 1307

I can tell you what the change in energy or the change in enthalpy is.1311

Profoundly beautiful.1315

Joules is DU DV =0 for an ideal gas.1321

The Joule Thompson coefficient, this one right here and the DH DPT is 0 for an ideal gas.1327

When we substitute from above, what we just got regarding the DU DV, we end up with this.1337

We end up with this equation CP nrt = Α TV – V.1344

Once again, in terms of something which is measurable, something that you can look up, I can find out what the Joule Thompson coefficient is.1349

That is absolutely extraordinary.1360

We have expressed a very important quantity, the Joule Thompson coefficient in terms of easily measurable and or easily retrievable quantities.1365

That is the running theme, this is why we have manipulated the equations the way that we have 1373

because we want to express these thermodynamic properties.1378

These really esoteric things in terms of things that we can measure, volume, temperature, heat capacity are the running theme.1381

That is what we have done all this mathematics.1392

The general expression for the relationship between the constant pressure and constant volume heat capacities was this equation.1396

You absolutely do not have to know that but again using values for the partial derivatives from above, you come up with this.1402

For an ideal gas we said that CP - CV = nr.1410

For any other thing, this is just TV Α²/ Κ, that is the relationship between the heat capacities.1417

This is profoundly important for relationship between the constant pressure and constant volume heat capacities.1423

Once again, we have expressed a very important relationship in terms of easily measurable and or easily retrievable quantities.1429

Much of science is dedicated to these, taking things that are abstract and esoteric and 1435

expressing them in terms of things that we can touch, that we can measure.1440

We actually come to the end here.1448

One final equation just for fun.1451

We expressed entropy in terms of temperature and volume.1454

We had a differential expression in terms of temperature and pressure.1459

Mixing and matching and using all these partial derivative relations we have between Maxwell's relation and Α and Κ,1465

We are actually able to express the entropy change in terms of pressure and volume.1472

If for some reason, I wanted to do that and there you go, this is the differential expression 1480

and this is the expression based on all the things that we can measure and or look up.1485

It ends up looking like this.1490

You absolutely do not have to memorize this, I just want to throw in there to let you know 1492

that now we have close the circle on all of this beautiful thermodynamics, energy, entropy, 1497

temperature, volume, pressure, free energy, Helmholtz energy, and enthalpy.1504

All of these come together really beautifully.1513

Thank you so much for joining us here at www.educator.com.1516

We will see you next time, bye.1519