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

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

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## Table of Contents

## Transcription

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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
- Work, Heat, and Energy
- 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
- 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
- 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

- 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

### Physical Chemistry Online Course

### 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

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