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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### The Relationship Between Cp & Cv

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 Relationship Between Cp & Cv
- For a Constant Volume Process No Work is Done
- For a Constant Pressure Process ∆V ≠ 0, so Work is Done
- The Relationship Between Cp & Cv: For an Ideal Gas
- The Relationship Between Cp & Cv: In Terms of Molar heat Capacities
- Heat Capacity Can Have an Infinite # of Values
- The Relationship Between Cp & Cv
- When Cp is Greater than Cv
- Constant P Process: 3 Parts
- Define : γ = (Cp/Cv)

- Intro 0:00
- The Relationship Between Cp & Cv 0:21
- For a Constant Volume Process No Work is Done
- For a Constant Pressure Process ∆V ≠ 0, so Work is Done
- The Relationship Between Cp & Cv: For an Ideal Gas
- The Relationship Between Cp & Cv: In Terms of Molar heat Capacities
- Heat Capacity Can Have an Infinite # of Values
- The Relationship Between Cp & Cv
- When Cp is Greater than Cv 17:13
- 2nd Term
- 1st Term
- Constant P Process: 3 Parts 22:36
- Part 1
- Part 2
- Part 3
- Define : γ = (Cp/Cv) 28:06
- For Gases
- For Liquids
- For an Ideal Gas

### Physical Chemistry Online Course

### Transcription: The Relationship Between Cp & Cv

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

*In the last couple of lessons we have introduced this thing called the constant volume heat capacity C sub V.*0005

*And we have introduced this thing called the constant pressure heat capacity C sub P.*0011

*Let us see if we can elucidate the relationship between the two.*0016

*First of all, know that numerically the constant pressure heat capacity is greater than the constant volume heat capacity.*0024

*The qualitative reason for that is, the intuitive reason is this, for a constant volume process no work is done because the change in volume is 0.*0032

*Remember, work is pressure times the change in volume.*0043

*The change in volume is 0, the work is 0.*0045

*Therefore, all of the heat transfer during the process goes just straight into the chaotic motion of the molecules of the system.*0049

*All of the heat transfer during the process accounts for the energy change that is why we have that*0059

*the heat transfer during constant volume process = a change in energy.*0066

*All of the heat goes to change in energy.*0073

*For constant pressure process, the change in volume is not 0 so work is done.*0077

*Therefore, the heat transfer to the process, some of this energy that is transferred as heat is used to perform work.*0083

*Heat goes in and a part of that heat is used to do work in the expansion leaving δ U to be less than expected.*0098

*Thus, the change in energy is equal to the heat transfer which is similar to this - the energy lost as pressure volume work.*0107

*The change in energy is going to be less than heat transferred or is for constant volume process, all of the heat accounts for the energy change.*0117

*If the change in energy is less than expected, the change in temperature is going to be*0127

*less than expected because energy and temperature are the same essentially.*0131

*That is what energy is a function of temperature so if energy is less that means the temperature of the system is less.*0137

*In order to achieve the same unit change in temperature, the same 1° increase in temperature,*0145

*more heat has to be transferred during the constant pressure process.*0151

*The more heat for the same change in temperature means a higher heat capacity.*0156

*The constant volume heat capacity, a certain amount of heat is transferred, it is all energy.*0162

*In a constant pressure process, a certain amount of heat is transferred of that heat, some of it is actually used to expand the gas.*0167

*My energy is less, my temperature is less.*0176

*In order to achieve the same change in temperature, I have to put more heat in.*0179

*Therefore, my constant pressure heat capacity is greater than my constant volume heat capacity.*0184

*Under conditions of constant pressure, I have to put more heat to achieve the same 1° change in temperature.*0190

*It is the qualitative reason so you should be able to qualitatively, intuitively, that is why it is true.*0199

*Let us go ahead and do a little bit of work here.*0207

*For an ideal gas, in other words let us find out what this relationship between CP and CV actually is.*0212

*Let us find an equation.*0219

*For an ideal gas it is really simple, there are several ways to get this.*0221

*I just decided to take what I consider to be the easiest, quickest approach.*0230

*Let us begin with the definition enthalpy.*0235

*Begin with the enthalpy is equal to the energy + PV.*0240

*For ideal gas, we know that PV = nRT so we can just go ahead and substitute that into that and we get that H =U + nRT.*0252

*Let us go ahead and take the derivative of both sides with respect to T.*0265

*We get DH DT = DU DT = nR.*0269

*We know what DH DT is DHDT is the constant pressure heat capacity, the change in enthalpy per unit change in temperature.*0279

*We know what DUDT is, the change in the energy vs. the unit change in temperature, this is the constant volume heat capacity.*0288

*Let us keep our = and + separate here, that is our relationship.*0304

*For an ideal gas, its constant pressure heat capacity is equal to its constant volume heat capacity + the number of mol × R or you can write it this way CP - CV = nr.*0311

*In terms of molar heat capacities, molar quantities is per mol, which means just divide everything by n, dividing everything by the number of moles.*0331

*When I divide everything by n, what I get is the following.*0341

*In terms of molar heat capacities, CP this is J/ K.*0346

*When we divide by mol we get J/ K/ mol, the one that we are used to seeing.*0360

*In terms of molar heat capacity, divide everything by n and what you get is CP/ N + CD/ n = R.*0366

*All molar variables just put a line over them.*0386

*I’m missing a bunch of + and - signs here, - there you go - CV = R, these are our relations right here.*0395

*You can think of it this way, it does not matter.*0407

*Any time you see a line over something it is a molar, that means that you just divide everything by n, the number of mol.*0409

*There you go.*0415

*To for an ideal gas, this is the relationship the constant pressure heat capacity is equal to the constant volume heat capacity + nr or CP - CV = nr.*0416

*Either one is fine depending on how you want to use it.*0427

*In general, if the system has a differential change in temperature that is associated with a change of state,*0433

*if it is going from state 1 to state 2, there is some change in temperature for the system.*0463

*Then Q, the amount of heat transfer, the amount withdrawn from the surroundings*0474

*then Q can have an infinite number of values and because heat Q is a path function.*0483

*Temperature is a state function.*0510

*If I want to take something from 10° C to 25° C, the 25° change I can do a whole bunch of ways.*0512

*I can just go straight to it, I can go off to 100 and other 25, I can go down to 100 then back to 15 then to 25.*0520

*The heat transfer as the heat changes the temperatures itself is a state function.*0529

*Q was a path function.*0538

*Therefore, the heat transfer can have an infinite number of values going from state 1 of one temperature to state 2 of different temperature.*0540

*There are a bunch of ways that heat can transfer.*0548

*The heat transfer can have an infinite number of values.*0551

*Heat capacity is defined as the change in heat per a change in temperature.*0559

*If the quantities for heat have an infinite number of values, the truth is for a given change in temperature*0570

*there are an infinite number of heat capacities.*0577

*Heat capacity can also have an infinite number of values for a particular temperature change because*0584

*they are a bunch of paths in order to get from one temperature to another.*0600

*Only two heat capacities are generally important, heat capacity under constant pressure and heat capacity under constant volume.*0606

*When we put these two constraints on there, those are the two that matter not the infinite variety of them.*0621

*In another words, we want to go from temperature 1 to temperature 2 under constant volume.*0629

*We want to go from temperature 1 to temperature 2 under constant pressure.*0633

*In both of those cases, there is heat capacity associated with both of the paths, with both of those processes.*0637

*Only 2 heat capacities are generally important.*0643

*But it is important for you to know again, that heat capacity there is an infinite number of them depending on the path that you take.*0650

*It depends on the heat transfer.*0656

*The heat capacity is defined as the heat transfer per unit change in temperature.*0658

*If there is an infinite number of heat, heat has to be transferred as an infinite number of heat capacities CP and CV.*0662

*There you go, constant pressure and constant volume heat capacity.*0677

*Let us go back.*0681

*Let us return to our mathematics, DU = DU DT VDT.*0684

*Let us see if I can keep it all straight, DU/ DV constant TDV.*0700

*We also have DU is equal to DQ - DW which is equal to DQ - P external DV.*0708

*The definition of work is pressure × volume.*0728

*We have our equation DQ - P external DV = this side is going to be DU/ DV constant V × DT + DU/ DV constant T × DV.*0734

*Let us see what we can do.*0760

*At constant pressure under conditions of constant P, the P of the system is equal to P external.*0763

*I put P over here and move it over the other side and I get the following.*0783

*I get DQ sub P because now we are at under constant pressure, DQ without the subscript it just means that heat transfer.*0788

*DQ sub P is under conditions of constant pressure is equal to DUDT sub V × DT was DUDV under TDV + P DV.*0797

*I just put this into here and moved it over to that side.*0824

*I’m going to go ahead and I have a DV here and here.*0828

*Let me combine some terms.*0831

*I have got DQP is equal to CVDT because the DUDT is equal to the constant volume heat capacity + DU DV T + P × DV.*0834

*Now I'm going to divide DT.*0857

*If I divide everything by DT, I get the following.*0864

*I get DQP DT this is just mathematical manipulation.*0868

*It is just people playing with symbols.*0877

*They did not know where they are going when they are doing it, they just started doing it.*0882

*It looks like we know what we are doing because we have the result of their fruitful work.*0886

*We have CV, DT DT is just 1 + DU DV sub T + P × DV DT under constant pressure conditions.*0894

*What it is that we have got, DQ DP DQ PDT this is a constant pressure heat capacity.*0929

*We have constant pressure heat capacity = constant volume heat capacity + DU DV T + P × DV.*0938

*I’m trying to keep this all straight is not easy.*0960

*There you go, this is the relationship that we are looking for.*0963

*This expresses a relationship between the constant pressure heat capacity and the constant volume heat capacity.*0966

*The constant volume heat capacity + something accounts for the constant pressure heat capacity.*0974

*Let us talk about what this something is.*0981

*You can move the CV over there to write in a different way.*0984

*I'm going to go ahead and actually multiply and distribute this over this and over this and write it this way.*0989

*CP =, let me go ahead and first to bring the CV over for our discussion.*0998

*I'm going to get DU DV under constant temperature conditions × DVDT, our constant pressure + P × DVDT under constant pressure conditions.*1010

*The amount by which CP is greater than CV has two components.*1036

*This is one component that is the other component.*1054

*I’m going to the second term first.*1064

*Let me actually rewrite it here so I have on top.*1067

*I have got CP - CV = DU/ DV under conditions of constant pressure × DVDT under conditions of constant pressure + PDVDT conditions of constant pressure.*1071

*That is the second term first.*1091

*This term right here, the second term.*1094

*This is the work that the system does on the surroundings per unit change in temperature at constant pressure, constant P.*1096

*Just take a look at the units.*1127

*We have pressure × volume divided by temperature.*1129

*Pressure × volume is work per unit temperature.*1136

*First term, the first term is a little stranger.*1143

*It is a little harder to wrap your mind around this one.*1148

*In this particular case, I’m going to ask to take it off.*1150

*This is the change in energy per unit change in volume under conditions of constant pressure × a change in volume*1154

*or the change in temperature under conditions of constant pressure.*1161

*This is the energy required to pull molecules apart against intermolecular attractive forces.*1166

*This term right here, this accounts for the amount of energy it takes to actually pull molecules apart against attractive forces*1207

*and separate them by a certain distance.*1217

*In the active separating at a certain distance, the gas is going to expand.*1219

*If you are taking a molecule, collection of molecules, are pulling further way from other molecules the gas is going to expand.*1223

*That expansion is going to do work on the surroundings.*1230

*Some of the energy is lost in actually pushing the atmosphere away as the gas expands.*1234

*That is the difference between the two heat capacities, two of the terms of the energy that transfers heat.*1241

*Some of the energy goes to pull the molecules apart against intermolecular forces.*1251

*Some of the energy goes to doing pressure and the process of pulling them apart against the atmosphere that you are pushing.*1258

*The rest of it is leftover, that is your CV component, that is the part that goes straight into the random motion of the molecules.*1266

*The temperature changes accounted by this term not by this term and not by this term.*1275

*We will say more about it in just a second.*1281

*Let me rewrite this way so CP = CV + DU DV constant pressure DVDT, what makes sense is the amount of energy to have to put the system*1286

*in order to change its volume or separate, that is what change in volume is.*1310

*You are separating gas molecules and then the amount of separation per unit change in temperature.*1315

*It is the amount of energy that takes per change in temperature was accompanied by a change in volume.*1323

*In other words, the amount of energy I have to take in order to take this molecule and pull them apart because it requires energy,*1329

*because of the inter molecular attraction.*1335

*Under constant pressure + the pressure × the change in volume over the change in temperature at constant pressure.*1339

*For a constant pressure process, for constant P process, the heat was transferred per unit T change in temp per unit T change.*1350

*The heat transferred per unit change in T, in other words the CP heat capacity is divided into three parts.*1393

*Part 1, did not do this in order for the equation is concerned.*1419

*So one part pulls the molecules apart.*1426

*Let us slow down, pulls the molecules apart and separates them.*1440

*Another part, does work on the surroundings in the process of separation, that is what we call expansion is.*1452

*The third part, is the actual increase in chaotic motion.*1488

*In other words the δ U, only this part reflects the temperature change.*1508

*Let me write the equation again, the constant pressure heat capacity is equal to the constant volume heat capacity + DU DV*1531

*under T and this is DVD T under P + P × DVDT under P.*1549

*The constant pressure heat capacity is divided into three parts, the amount of heat transfer per unit change in temperature.*1562

*One part of it does work on the surroundings.*1571

*One part pulls the molecules apart and separates them, that is this part.*1573

*That is the amount of energy it takes to actually pull molecules apart against their intermolecular forces.*1579

*Another part does work on the surroundings in the process of separation.*1588

*In the process it pulled the energy it takes to pull it apart and separate them, that separation, that pushing or pulling molecules away from each other,*1591

*in order to pull it this way, I have to push it against the atmosphere.*1600

*That ends up doing work on the surroundings.*1604

*What is left over the third part is the actual increase in the chaotic motion.*1607

*This is what accounts for the δ U, this is what accounts for the temperature increase.*1612

*This is not change temperature.*1618

*This is where the temperature change happens.*1622

*Therefore, in order to achieve the same amount of temperature increase as a constant volume process, the constant pressure process has to put all this.*1625

*You have to put this much + this much + this much, more heat is required in order to achieve*1634

*the same 1° temperature rise which is the definition of heat capacity.*1642

*This is the relationship.*1646

*It is very important.*1648

*Again, there is nothing here that is counterintuitive.*1650

*You get it now, you understand where this is coming from.*1652

*Under a constant volume process, I have to put a certain amount of heat under constant pressure process,*1656

*a certain heat is transferred but all the heat that is transferred some of the heat has to go to pulling this molecules apart.*1662

*In the process of pulling them apart, some of that heat has to be converted to work to actually push the atmosphere away so that we can pull the molecules apart.*1670

*The rest of it just goes to the energy of the system.*1678

*That is what is happening here.*1682

*Let us go ahead and finish up with a couple of definitions.*1688

*Excuse me, we are going to define this thing called gamma.*1691

*Let us make it a little bit better, you will see it every so often.*1700

*Gamma is equal to, it is a ratio of the constant pressure heat capacity to the constant volume heat capacity.*1705

*This is going to be greater than 1.*1712

*You knew this already because CP is greater than CV.*1714

*For gases, the difference between CP and CV is significant, of course it is, because gases expand.*1719

*For liquids and solids, because the change in volume is so small, it is not nearly 0.*1748

*It is small but we do not say it is close to 0 because it is reasonably significant.*1771

*Because the change in volume is so small CP is approximately equal to CV.*1776

*Tabulated values, the values that you see for heat capacity for liquid and solids in your books and in all the table that you read, that constant pressure heat capacities.*1783

*Tabulated values for liquids and solids are constant pressure heat capacities and the reason is it is very easy to measure.*1798

*We just do it under atmosphere conditions because these are easy to measure experimentally.*1818

*Not quite so easy to do constant volume for a liquid or solid.*1836

*This is kind of messy actually.*1844

*For an ideal gas, as we said the CP - CV = nr or for molar CP - CV = R, that is just heat capacity per mol.*1847

*This is actually good approximation for real gas as well.*1873

*When you are dealing with the real gas and if you are given the constant pressure heat capacity and*1891

*you need the constant volume heat capacity, just go ahead and solve this equation.*1897

*That is fine, for all practical purposes.*1902

*Real gases under conditions of low pressure and high temperature they behave ideally which is why we use the Pv=nRT unless, we are doing really precise work.*1906

*For a real gas, that is a good approximation for a real gas also.*1916

*There you have it, there is the relationship between the constant pressure heat capacity and the constant volume heat capacity.*1923

*Who know that there was a relationship?*1929

*You know you probably never thought that there is actually an infinite number of heat capacities but there are.*1931

*There are only two we are concerned with constant pressure and constant volume.*1935

*Thank you so much for joining us here at www.educator.com.*1940

*We will see you next time, bye.*1942

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