good afternoon to all of you last class we discussed the behavior of ideal gases which obeys the equation of state as T into V the pressure and the volume equals to mass into the characteristic gas constant and temperature and in another form if we express in terms of moles the amount of the gas then P times V becomes equal to the number of moles times the universal gas constant is constant for all gases and the temperature and following this equation of state as the first line of the definition of ideal gas we derived a number of corollaries one of those that enthalpy and internal energy are functions of temperatures then at the same time we assumed and that the specific heats at constant pressure and volume are constant for an ideal gas sometimes well for an ideal gas it will also tell the specific heat at constant pressure and volume may vary with the temperature but usually we assume for an ideal gas these are constant and in that cases in those cases the ideal gases are called calorically ideal gas kinetically perfect yes then we derived the expression relating to pressure volume pressure temperature and volume temperature from isentropic process and with the use of these equations that enthalpy and internal energies are functions of temperatures and the equation of state we derived certain equations relating to work transfer in the displacement mode work that is P DV and all these things and also the entropy change in a process how it relates the pressures and temperatures at the two equilibrium states executed by an ideal gas in a processing today I will discuss first what are real gases so gases in reality also we also realize this thing the gases in reality may not always obey the ideal gas States the the ideal gas equations are fairly accurate for all real gases when they are it a they are at a extremely superheated or rarefied state which can be theoretically stated that the limiting value of the pressure and volume as pressure tends to zero or volume tends to infinity becomes equal to R Bar T that means we start from this point today that for real gases we can write for real gases that means the gases in reality this is rather the equation that P let us consider the well the molar volume PV and as P tends to zero or molar volume tends to infinity we get R Bar T or in terms of this specific volume same thing limit PV as P tends to 0 or V tends to infinity is Archaea where R is the characteristic gas constant is constant for a particular thing so therefore we see under one limiting conditions this equation that PV is equal to R Bar T and PV is equal to RT are valid but not in all ranges of pressure and temperatures now it has been found from experiments that for real gases where the pressures are not very low and it is not at a very high superheated condition the product of pressure volume sorry the pressure volume temperature relationship is not followed by that of the ideal gases and at the same time it has been found experimentally that a unique relationship between pressure volume and temperature is very difficult to be obtained in a wider range of pressure and temperatures that is why some theories like empirical equations you know when you develop empirical equations for any process ok or the performance of a system that output parameter as a function of input parameters or operating parameters so these relationships are valid within the range of the experiments including certain range of operating parameters similarly different functional relationships of pressure and volume are obtained at different ranges of pressure and volume n temperatures for real gases so it is not like that an ideal gas at all ranges of pressure volume and temperature we can substantially take that equation of state PV is equal to RT or R Bar T depending upon whether it is specific volume or molar volume but there are equations which only which are only valid or which is only valid within a range of certain pressure and volume and usually in practice when we do with real gases and solve for problems relating to what and heat interactions in a process change of entropy in a process change of internal energy in a process or evaluating enthalpy so we are supplied with the equation of states within certain range this equation is valid within certain length this so you do not have to remember any equations in any book you will see there are different equations proposed by different scientists so which are valid for certain gases they specify these are the gases within these ranges these they obey this equation that means you have to know simply that equation of state is a functional relationship between pressure volume and temperature and it is easily obtained from experiments and these are being supplied if I know this relationship that then we can apply this equation just is the mathematical operations to evaluate all other thermodynamic parameters as I have told the heat work interactions change in entropy and all these things so one such equation of state was proposed by a scientist van der waals and this is known as van der Waals gas and the gas which obeys so I use first write van der so it is beyond the scope of this study van der Waals gas or van der Waals equation of state same thing that means the gas which obeys this equation of state it is beyond the scope of this studies that we will deal with in details with all types of real gases and what are the probable equations and all these things because I mean plenty of empirical equations the different ranges but I will tell you that how usually the real gases are being specified in terms of their equation of State one very important real gas is van der valk as it has been found that in a wider range of pressure and temperature many real gases follow a particular equation of state let me first write this equation of state then I will explain the equation of state is like this if P is the pressure in terms of the molar volume I write P plus a by v square into V this is a molar volume minus B is equal to R Bar T this R Bar is the universal gas constant whose value is same for that of ideal gas where a and B are parametric constant defining this vander waal equation so therefore you see this relationship between PV T even if you draw this curve in a pivot aplay Nazz a surface or in PV for different values of T the parameters a and B are there that means different vendor different gases obeying Vander Waals equation will have different values of a and B which are typical to the particular gas obeying the vendor hose that means Vander Waal gas represents a class of gases a class of gases over a certain range of pressure and volume and temperature which are being distinguished one from other by the different values of a and B this is this has been verified experimentally and it has been proposed from experimental results but van der Waals initially deduced this equation of state from kinetic theory of matter the way the ideal gases were developed by the kinetic theory of matter with the assumption that the molecules move in a rectilinear path there is no force of attraction and the volume of the molecules are negligible with respect to the volume of the container in which they exist then from that he relaxed actually they said he made the restrictions for these two assumptions one is the volume of the molecules were taken into account that means the volume of the molecules were not considered to be negligible and at the same time the molecular cohesive forces were taken into account because in real gases molecular cohesive forces will be there so he developed in this fashion I am NOT going to make a full proof of it that the pressure which is exerted now since the volume of the molecules are taken into account it is not negligible compared to the volume of the container so molecules get less space for their movement see this B so the number of collisions that the molecule makes for example in the wall for which we define the pressure as the change of momentum because of these collisions the number of collisions increases so therefore the pressure exerted by a real gas on the wall becomes more than that of the ideal gases and this factor van der Waal showed that this is taken to be a part of it is taken to be as a kinetic pressure which is so it should be proportional to like this that R Bar T by V is the pressure for an ideal gas R Bar T by sorry V Bar is the pressure given by the exerted by the ideal gas it is proportional to this value where V is a volume parameter what V physically indicates V indicate B indicates the volume excluding the molecular motions you understand that means if you explore the molecular motion what is the volume that is B which means that V minus B is actually the volume that is the restricted volume which is in access to the molecules for his motion and he proved that it is in linear proportion to that that means this is less restricted volume and so therefore the kinetic pressure that means the pressure which it should give is equal to this that means this is more than this value this is the ideal gas values apart from that he told that actual pressure will be less by this kinetic pressure by an amount which takes care of the molecular cohesiveness you have read in fluid mechanics that molecules are always being pulled by other molecules that is the equation forces force of attraction by its neighboring molecules that means if you concentrate your attention on a particular molecule and you will see a molecule on the bulk which is far from the free surface far from the wall is being pulled by all molecules from neighboring all neighboring molecules from outside that is outside that molecules their neighboring molecules so that a molecule almost in a balance but if we go very near to the wall or very near to the free surface you see there is a dis balance force you know that the molecules are being pulled by more number of molecules if you concentrate your attention on a single molecule by more number of molecules from the bulk of the fluid because there are very few ER molecules near the free surface or near the wall so therefore if you consider a molecule near the wall or near the free surface you will see there is a net attractive force of the molecule towards the bulk of the system bulk of the fluid okay bulk of the gas in this case which causes a diminished diminishing which causes a reduction rather which causes a reduction in the press you understand the way the surface is being stretched that means always there is a surface is stretched condition as we have read in fluid mechanics that to create a surface we require energy because the molecule has to be brought to the surface against the inward attraction of the molecule similar is the case if a molecule has to be brought to the wall he has to work has to be done that means what happens when a molecule comes very close to the wall a net pull on the molecule is there towards the bulk of the gas so for which a reduction in the pressure that is from the kinetic pressure is there and he proved that this is proportional to the it is directly proportional to the molar density will F be proportional to the molar density of and common sense you can guess the number of moles per unit volume is more so this will be the force will be more so that the pressure will be less so that it is inversely proportional to the molar volume so if this be the thing and if you substitute PK this value here you will get a relationship like that it is not a detail probe but I thought that this will be sufficient enough to know that the origin or the basic jeunesse is from which this comes that means here you see that a by v square terms accounts for the molecular cohesiveness or the molecular cohesive forces which were neglected in case of an ideal gas and B is the parameter or term which takes care of the volume of the molecules in case of an ideal gas this is zero and this is zero and PV is equal to R Bar T so this equation of state is known as van der Waals equation and in many real gases obey this van der Waals equation of state over a certain range of pressure and temperature now the next thing is that you see this van der Waal gas if you again write this that p plus a by v square into V minus B is R Bar T so in this case you see if you solve for V you see V will give three routes it's cubic in V this equation that means for a given pressure and temperature so V will be having three roots but it has been found if you solve this that this among these three roots only one is real at high temperature at high temperature one root is real at high temperature one root is real I am NOT writing everything one root one root is real and at low temperature there are three real roots but out of that one root is in a metastable state only this information will be sufficient another root which is real but it is not feasible that root will show will lie on a curve which gives a negative compressibility I told that always for all substances if you increase the pressure volume decreases so therefore one root one of the roots one is metastable state another root represents a physically impossible value so therefore only if you solve this we always get one real and physically possible value of V so this is mostly all about the van der Waals equation now after that I will tell this V real form V real form of real gas v real form of equation of state equation of state of a real gas virial form of equation of state of a real gas now real gas now like van der Waals equation there are a number of equations which you can express I as I have told for different gases obey at different pressures and temperatures but one scientist kamerlingh onnes a li ng h is a german scientist owns he proposed that all real gases should be expressed in this fashion some constant a into 1 plus B does in a power series of P in a power series in an infinite power series of either this two A's are same that I will explain now or 1 by V bar I can use BCD here or - it does not matter this is the usual convention that I am following then D by like this so we can express so that if we can express these in power series then we can closely approximate all real gases by properly evaluating these values of B - C - D - and BCD now here you see this a B - C - D - or ABCD are all constants and they are functions of temperatures for example here this AB determine that that means at a constant temperature these are giving numerical values so a B - C - D - which are the coefficients of this power series either in P or in V bar they are functions of temperatures now if you recall this limit PV bar as P tends to 0 or V tends to infinity always satisfies this relationship you get a becomes equal to R Bar T so therefore this equation can be expressed as P V Bar R Bar T is equal to 1 plus then what happens B - e + 3 - P Square + D - PQ + like that similarly in terms of molar volume b + c v square plus d vq + this infinite series so therefore you see for an ideal gas B - C - D - BCD becomes 0 and this P V - by RT if you represent this as J then we can write J is equal to 1 plus B - p + C - P Square not a B - C does is zero I am sorry I have told wrong I am sorry when foreign real gas B - C - D - B C D 0 that is hypothesis that is not wrong what I want to mean is that when P tends to 0 and V tends to infinity obviously this terms becomes 0 but for a real gas I am correct that this coefficients are 0 now this PV by RT v bar this is known as compressibility factor this Jade is known as compressibility factor this compressibility should not be confused with the compressibility of the fluid mechanics but this gives almost the same concept physically why because this is something to doing with the compressibility is the compression here you see this is known as compressibility factor first of all you see mathematically that Jade is compressibility factor whose value for ideal gas is 1 but for real gases are different from 1 and why it is compressibility factor because of the compression a gas at a rarefied State it satisfies the ideal gas laws but when you compress it at high pressure then it deviates from the ideal gas laws so J DS that is why they defined as the compressibility factor ok when j is equal to 1 that compressibility factor is equal to 1 that represents an ideal gas okay now this jade ideal gas the compressibility is usually a function of pressure and temperature may be a function of volume and temperature what is done in practice in solving problems that the compressibility either table or figure in figure Jade versus P at different values of T's as parameter like this I am showing a particular curve like that the curve is like this so at different temperature t1 t2 these are plotted and this is known as compressible these are in many compressible lit is charged that means if one can find out at a particular state point what is the value of jet from the compressibility chart or compressibility table just like your steam table as you use you know the steam properties at a particular pressure or temperature either saturated state or super tested and corresponding tables you know all the properties similar if you know the compressibility factor then you write the equation of state for any gas at jade r but the Jade is a function of P and T for example so that means you just mod is a multiplying factor that means PV bar is equal to j dot r bar t where j is equal to 1 for neidell gas and for a real gas we have to find out the value of jet corresponding to that particular pressure and temperature J is a function of the state point obviously because state point you will decide how much deviation is there from the ideal gas is clear so this is the definition of the compressibility and this is the compressibility chart now similar to the concept of fluid mechanics as we have seen that it has been found that if we now see for a Vander Waal gas that has example a particular gas for Vander Waal gas Vander Waal gas you see P plus a by v square into now you see for different values of a and B it constitutes a different equation of state the equation of state will vary the structure is same functional relation is same because they belong to a particular class van der waals state equation of State but the particular equation will go on varying so if you have got 10 gases following just for an example van der Waals gas with different values of a and B so we will have to provide compressible chart compressibility chart for 10 gases plane compressibility charts try to understand jet as a function of P and T okay but now if we want to make that for a particular class of gases following a particular type of equation of state by a single equation of state by a single equation of state there will not be any parameter which is distinguishing from curved gas to gas then in that case so we can express the compressibility factor in some normalized variables in terms of some normalized dimensionless pressure and dimensionless temperature which will be unique just like following the principle of similarity same thing for an example you come to the fluid mechanics a recapitulate we know that pressure drop is a function of flow rate for a given fluid okay for a given diameter of the pipe pressure drop per unit length for example in a parallel flow as you know the pressure drop per unit length is constant so pressure drop per unit length is a function of flow rate for a given fluid that means for a given value of Rho mu and for a given diameter that means if I have to provide a relay a figure pressure drop versus flow rate then you will have to mention this is valid for a pipe of this diameter for a liquid of this density and this viscosity if you go if you see if you have to show this similar relationship Q versus Delta P versus Q for different pipe diameters and this for 10 diameters I will have to produce 10 cups again for each diameter if I have to show the influence of density then I left to produce so many curves for variation of density this is the relationship Q but it is not done then what happened from the principle of similarity we have shown also from the theoretical equations the navier stokes equations you have read probably has been found that this phenomena that pressure drop depends pressure drop per unit length depends upon the flow rate depends upon the density depends upon the viscosity depends upon the diameter so all these things clubbed together we define a implicit functional relations of all these variables then apply the buckingham's pi-theorem and show ultimately this entire physical phenomena is guided by two governing parameters in non-dimensional forms they are known as five terms one is the friction factor and at the Reynolds number so density viscosity diameter velocity that means the flow rate in terms of the velocity of flow average velocity of flow is clubbed in one dimensionless term Reynolds number and the pressure drop is normalized with half Rho v square is one parameter as the friction factor so we give the value of friction factor versus Reynolds number which takes care of pressure drop flow rate curve for all diameters and all fluids in a particular group what is that group for what is that group anybody Newtonian fluid yes for Newtonian fluid non Newtonian fluid there is the different relations Newtonian fluid gives the group homologous series in fluid machines also you will read that thing that a machines of the similar kind similar geometrical shapes are falling on a particular class similar these are known as homologous series and a particular group very good similarly here also a group a group of number a number of gases following the van der Waals equation can be expressed in terms of these dimensionless parameters so that one can express one single equation of state and the single compressibility chart to work with all gases of this particular time with this genesis and theory the concept of law of corresponding States came now law of corresponding States is very important corresponding States law in thermodynamics we tell that this corresponding odd is same to the similar points similarity that these similar parameters similarly this calab corresponding States now to do that the basic physical concept is taken that PV diagram because there has to be some sort of similarity otherwise the similarity parameters that PI parameters cannot be taken for example when I compare the two pipe flow problems with different diameters or different fluids then I see that there is some similarity in the physics physics of flow that two flows are governed by the pressure force and viscous forces similarly when the flow in a surface waves or in a free surface flow we know the physics are same that two flows different situations different liquids different geometry are governed by the gravity force and inertia forces similarly some common similarity should be there what is that it has been found that this vapour dome is similar for all substances that means there is a similarity in the structure of this vapour dome that means if you draw the PV diagram you see their structure is same quantitative value may be different there is a critical point and the slope and all these things are similar that means they reveal a geometrical similarity on this PV diagram so therefore from this it was thought that probably a similarity similarity parameters can be resorted to make the similarity principle valid there so that we can do this what we have explode and for that what we do this critical point which is different for different gases even the van der Waals gases there may be number of gases following the same equation of state but having different AV values different a critical point similarly for ideal gases also all gases are not same because they are values of R the characteristic gas constant will be different but ideal gases one thing is a unique that if you express in terms of the molar volume they reduce to a single equations R Bar is same for all gases which is not so for the axial gases even if we expressed in terms of the molar volume AV parameters for an example van der Waal gas all gases will be having such parameters they are not same their critical points are also different but we can use these values PCV CTC to normalize the corresponding damaged dimensional parameter PV T and we define that dimensionless pressure as P R P by PC that means these are used as the reference variable for normalizing the actual dimensional variables we define TR is T by TC and V R is V by V and this non-dimensional terms in thermodynamic science of thermodynamics are known as reduced properties well reduced properties that means P R is the reduced pressure TR is the reduced temperature and V R is the reduced volume and if you do so you will see that these equations take a unique form just I give an example let us do it for van der Waals gas now van der Waals equation is P plus a by v square into V minus B is equal to R Bar T now along with that one thing you have to take care of one another physical fact that if we draw this curve sorry this PV diagram for example well the PV diagram is our liquid vapour as you know I am NOT drawing all these things is unknown so one thing is that at the critical point you know these are the isotherms so the isotherms at the critical point has a zero slope and zero curvature this is an information - you have to accept this information this is a physical fact which means mathematically del P del V this is for all gases del P del V at the critical point okay at T C that means the slope of the isotherm that is T is equal to constant which is TC here it is the isotherm that at TC this slope is 0 and del square P del v square at TC 0 that means this is almost flat here which has got both a zero slope and the zero curvature so if you use this you write this P is equal to R Bar T by V bar minus B minus a by square then if you find out del P del V at alright at TC is equal to what it will be minus R Bar T by V Bar C minus B whole square then it will be 2 plus 2 a V Bar C whole key okay this one equation is this this is one equation another equation is the second derivative this is correct then second derivative del square P del v square at TC second derivative that means again it will be plus 2 R Bar T then VC bar minus B whole cube ok then - what it is 6 a by to the power 4 4 is 0 so that is another if you solve these two equations that I am NOT doing it here is simple algebra then you get okay please you get a B and R Bar in terms of P see this you have done at school label I understand but again it will be a recapitulation I cannot help so you get what is the value you get do you remember B is equal to one-third vc-1 our volume at the critical state and a is equal to three PC VC square and R Bar is equal to 8 by 3 PC VC by PC this is jet CHEO VC VC by TC that is compressibility at the critical point if you substitute this in the van der Waals equation of state then the equation of state becomes like this 3 by V R square into 3 now you understand small V sorry so these are the reduced volume reduced pressure that means dimensionless is equal to a TR so now this van der Waals equation of state is valid for all gases following the van der Waals equations a B is eliminated and in that case if you define j2 by jet see as we reduce here of course Jed is non-dimensional I understand PV by R Bar T but if you normalize it with scale it with jet C and if you plot J dot as a function of PR TR you get only one equation that means a series of curves one family of curves in Tjader PR plane with different values of TR is unique that means these families of curves are unique for all vander waal gases okay so this is the theory brandy law of corresponding state this is expressed like that and if law of course what is love or what is corresponding States that means a state of a system gas for example Vander Waals gas or any gas which are having the same pvap RV RT are unknown as correspondence is similar like similar situations in fluid flow that means the same Reynolds number in two flows same Froude number you know all this thing that means their actual velocity may be different they are actual gravity may be different gravity condition may be different but they are having the same Reynolds number that means the situation's are similar so here also the Pvt may be different at two states but if you for the two gases but if you normalize with PCV CTC and if you see the reduced properties are same then they are told as corresponding States that means states are similar okay corresponding States so for corresponding States you get the same value of Jayda for an example in Reynolds number is fixed friction factor is fee but if you have got a higher velocity then you get a higher pressure drop if you have got a lower velocity you get a lower pressure drop that will be decoded from the friction factor friction factor is unique because friction factor if you decode in terms of the pressure drop you will see automatically those the flow for which velocity is high you get a more pressure drop for flow with velocity is net with less pressure drop but arrows number is same that means the velocity is high for that case d may be lower or Rome you may be so I just added a knows number is same this is the concept of the similarity situations similarly here the same similarity situations are just replaced by the term in thermodynamic science that is the corresponding States so there is nothing much in the law of corresponding States so this is alright now after this what very simple I will leave you today early I do not know how much I can tell so then I come to mixture of ideal gases mixture of gases but first I will tell mixture of ideal gases the mixture of real gases excluded from this syllabus for you mixture of ideal gases now I again go back to ideal gas this is some information about the real gas and specially about the Vander Waal gas as an example how you can reduce it into in terms of the reduced properties and the concept of corresponding states how you can express the real gas in terms of a virial expansion and there of course I have forgotten to tell these coefficients BCD are known as virial coefficients forgotten this is another important thing that this coefficients are known as virial coefficients and this varial coefficients and this expansion is known as virial expansion or virial series so how can you express this then the definition of the compressibility and compressibility is one in case of a an ideal gas and the law of corresponding states now go back to ideal gases mixture of ideal gas in many situations we have to deal with a mixture of ideal gases that means is not a single component system for example here itself is a mixture of ideal gas here itself is a body's composition is throughout same that it behaves almost like a single component that so it is a pure substance but somewhere composition may change sometime but however at any state a mixture of ideal gases also behave as an ideal case this is a hypothesis this has been found experimentally this has been proved from kinetic theory of matter that means if you assume all the real gases hypothesis H sorry ideal gases hypothesis then you will see that mixture of ideal gases are also ideal gas that means if a mixture of ideal gases there have a fixed composition this gas will behave as an ideal gas the way AR does okay so mixture of ideal gases first thing is the ideal and that behave as an ideal gas now there are certain laws for example if we have in a container for example there are number of gases with must ami mb m c like that there are n number of gases with mass m in and corresponding number of moles are there in a NB like that there are number of gases and they when a mixture of gases are in equilibrium means what the pressure and temperature are same that means they the entire system having a pressure P and a temperature T okay now in this case what happens when you mix two gases what happens the ideal gas hypothesis the molecules do not have any volume that means there is an inter molecular diffusion both the gases diffuse into one another and what happens if there is a fixed container all the gases all the Constituent gases occupy the same volume so horrible each gas is under an expanded state that means his to understand that means this pressure is reduced because it has to expand to in his volume that means is gold in an inter molecular diffusion with other gases so therefore the pressure exerted by each cash individually is less than that the that of the pressure which the mixture is exhibiting and that pressure is known physically as the partial pressure which means that if any one of the gas at the same temperature the temperature of the mixture is allowed to exist alone in that particular volume because exactly it is opening the same volume then what pressure it could have exerted because other molecules are there of the other components of the other constituent constituting component so that is known as partial pressure now if you deal with the moles then it will be better that means I can write if the partial pressure of a is T and the volume is V of the container that means the mixture PA B is equal to number of moles na I know and R Bar is same for all constituting components ideal gas that is one good thing the this actually is the equation defining the partial pressure this is the mathematical equation it defines the partial pressure and which I have told just now is the physical concept how can you conceive the partial pressure of a particular constituting ideal gas in a mixture of number of ideal gases so therefore this is the basic equation for the partial pressure similarly if we define the partial pressure for the component B then we get another expressions like that and B I am sorry very very very very good and ultimately all these things I can write for the NH component is equal to n in R Bar T now if I add this thing what I will get simply school level thing again recapitulation dot dot dot p.m. V is equal to na plus NB plus dot dot dot in in now what is this these are the total number of moles of the gas because they have the amount of the thing so amounts are additive is that the extensive properties and therefore this for the entire mixture as an ideal gas I can write its pressure into his total volume of the ideal gas ideal gas as a whole and not considering as a pure substance ideal gas as a whole is a pure substance if it has got same composition which does not change PV is equal to in our body so if I compare I see that the pressure of the mixture okay that is sometimes known as total pressure is equal to the sum of the partial pressures who is the person who first derived developed it Dalton Dalton is the scientist that is why it is known as d'harans law that some of the partial pressures equals to total pressure now if you see that for a given particular component P AV is na RT now for example here now if you see for any component gate component let us write for K component that is is equal to NK R Bar T and if you now substitute R Bar T as Sigma n like this that means NK by Sigma n pv pv then you cancel P it is a very simple school level thing that means the ratio of the partial pressure to the total pressure is the mole fraction that means depending upon the mole fraction its partial pressure will be determinant that means ratio of the partial pressure to the total pressure is the ratio of its mole divided obviously it depends upon the number of moles what pressuring will exist it will depends upon his number of moles as compared to the total number of moles so this is one equation now you see that therefore the sum of the partial pressure is the total pressure now for extensive properties how do you write other properties for example what is now you before that I must tell how do you define the characteristic gas constant again refer the same figure okay I am drawing it again that number of component now I am writing the MA M be like this MC amen and P and T now if you write in from the partial pressure point of view PA v is equal to Amy RA T now I see now I write because all our ideal gases but when I write the equation of state in terms of mass the characteristic gas constant is very so I am giving a sorry capital a sorry similarly PB V is equal to that means I am writing in terms of the mass earlier I wrote na R Bar T now I am writing ma RA MBR BT like that like this and if we add it now we will get therefore P P into V sorry I am NOT writing again because PAP V PC is total pressure you know that is is equal to Sigma M R into T what is PV PV is Sigma M Sigma M R of the mixture that means I define the R of the mixture by this equation PV is equal to mixture mass our mixture T because if I tell that mixture is an ideal gas which is at a pressure P and temperature T then what is the definition of the our mixture then I have to use this equation which will define the our mixture you understand so this is the equation it defines the our mixture so if you write this then you get M R into T that means our mixture is the weighted mean of the mi4 example okay you write mi RI i is equal to 1 to n I am NOT writing in that fashion so you know this is very simple that means it is an weighted average of B so this way all our extensive properties can be defined that means internal energy of an ideal gas enthalpy of an ideal gas these are all mass basis that means if total mass is M okay Sigma M is the total mass the specific internal energies are defined total internal energy what will be ma UA plus MB you be like that dot dot dot is equal to total mass Sigma M u okay the specific internal energy that means per unit month that is CV into T so therefore U is Sigma mu by Sigma M similarly you can H Sigma M H by Sigma H so all extensive properties can be written as a weighted average very simple mass basis that means total energy is the contributed energy from all these things but specific internal energies CV into T which one oh sorry very good I am in a little alright okay so today I have to finish it yeah number of pluses now I will a solves very quick certain examples first I think this one will be very much appreciated I do first this now the change of entropy in a mixture of ideal gases that I will not describe in terms of this nomenclature these are very simple a just school level things a recapitulation but I will solve this problem and then you will see how interesting it is probably you know these things already example some simply how to calculate the change in entropy when a mixing of different gases take place which is a very important thing which defines how do you define which makes the definition that how do you define the entropy of a mixture of ideal gases let us consider this example a closed rigid cylinder is divided by a diaphragm into two equal compartments each of volume point one meter key each compartment contains air no I am sorry I will not go to this is a simple example but later eyes what I told let us first sorry I am sorry this one a closed vessel is divided into three compartments a B and C the compartments a contains 32 kg of oxygen the compartment B contains 28 kg of nitrogen and the compartment C contains 44 kg of co2 you can understand from their masses that they are one mole carbon dioxide all the components are at a pressure of 0.1 mega Pascal's same pressure that most fear equation and at a temperature of 20 degrees when the partitions are removed the gases mix obviously determine the change of entropy of the universe assume the vessel is insulated problem is clear now what is this problem problem is like this there are three compartments one contains oxygen 32 grams another contains nitrogen kg does not matter nitrogen is 28 kg and co2 is 44 kg and pressure for all these P is equal to 0.1 mega Pascal's P is equal to 0.1 mega Pascal's P is equal to 0.1 mega Pascal and temperature 20 degree Celsius well T small T I die 20 degree Celsius now you see if I rapture this diaphragm now all these gases are in mechanical and thermal equilibrium what is the definition of thermodynamic equilibrium 3 equilibriums have to be maintained mechanical thermal and chemical so what process is going on there is no work transfer process there is no heat transfer process but are they in chemical equilibrium any to Nome because there is the difference in thermal thermodynamic potential chemical potential which you can visualize there simply by the concentration gradient in this compartment nitrogen concentration is zero whereas oxygen concentration is 100 similar is the case reverse is the case here so therefore there is a concentration gradient thermodynamic potentials are different so thermodynamics of multi-component systems are not being taught here that is why you cannot understand what is thermodynamic potential but at least you can consider when there raptured so the gases will mix there will be inter molecular diffusion which is a process divided by the gradient of thermodynamic potential or concentration gradient that means if you have two gases at different chamber if you remove it there is a concentration gradient that flow just I explained last class that how water vapour flows because water vapour from a surface it flows into the surroundings because surrounding water vapour is less there is a less concentration whereas water vapour concentration is more here that is why it flows as the diffusion molecular diffusion so the diffuse and they mix so there is a process which may not incur work transfer or heat transfer but there is a process inter molecular diffusion because of this Chemical non-equilibrium to be existing there so now my problem is that how to find out the Delta is the universe that means you will have to find out in that case Delta is o2 Delta s into and Delta s co2 and all of them contain one mole one mole one mole o kilo mole very good one not kilo mole last class I told that you do not call it kilo mole many book tell kilo mole 1 kg mole ok now when the VIX total mixture is 3 kg mole 3 kg now you tell me first that there is an entropy change of the oxygen what property of the oxygen is changing for which entropy changes then there is no change in temperature volume volume of the molecule is changing from one volume but I do not know the volume divisions I do not know the volume divisions any other any other parameter you think here that is correct that is the volume is changing oxygen is expanding from this volume to the total volume very correct but can you take any other parameter pressure pressure because we have got equations of entropy change relating to both volume and pressure because we have TDS is either CP DT minus VDP or TDS is CV DT plus PDV if you use this equation you write the molar volume or the total whatever way you write then if you use this equation then you get Delta s is CP Ln t2 t2 by t1 T on is the initial State - what we get are bar into Ln P 2 by P 1 at 2 & 1 are the final and initial State otherwise if you use this you will get CB here plus R Bar Ln V 2 by V 1 but if I use this one then I can tell what this case for all gases this part is 0 but all gases starting from an initial pressure of 1 megapascal is attending a final pressure which is their respective partial pressure in the mixture when the equilibrium mixing state is there after mixing equilibrium state all gases have got their individual partial pressures and that partial pressure is nothing but 0.1 mega Pascal's into 1 by 3 so they are good this term is 1 by 3 ok so therefore this is R Bar into Ln 1 by 3 this is minus very good because this is minus so this will be plus so in both the cases there is a increase of entropy minus R Bar Ln 1 by 3 and because there is 1 mole for each so R Bar into 1 R Bar into 1 if the values are different we have to calculate dividing by the molecular rate what are the number of moles number of moles number of moles then some of this and then find out the mole fractions so for each gases the ratio of the final T initial pressures may change to make the calculation simples I have made this data so that this is easier to calculate concept is the same here are all 1 moles so everywhere it is one third one third one third so if you do it Delta s this is a very interesting problem 3 R Bar L in one bite then we are not it is the one one more one more one more pressure temperature is say molar volume is volume will be same correct obvious yes correct but you don't have to consider the volume you can find out the volume of course that is another thing that you can tell that when the moles are same pressure temperature same volume is same okay that is a good thing that vo2 oh then you can use that and three okay that is also correct okay okay alright any doubt so then another problem time is up but this problem you solve this is a given as your task homework a closed rigid cylinder I will give you the answer also a closed rigid cylinder this is a homework is not there in the tutorial sheet you take it close rigid cylinder is divided by a diaphragm into two equal compartments to equal component each of volume point one meter cube that means the volume is same each compartment contains air this is not a mixture of different components same component air at a temperature of 20 degree Celsius the pressure in one compartment is 2.5 mega Pascal's and in other compartment is 1 mega Pascal that means there is a difference of pressure so they are in mechanical disagreement disequilibrium but the partition is strong enough to sustain that pressure difference so that no work transfer is taking place the diaphragm is now ruptured so there will be on work transfer in between so that the air in both the compartments mixes to bring the pressure to uniform value throughout this cylinder obviously when the mix final state will represent a unique value of the pressure equilibrium state means what all properties will be uniform and constant which is insulated throughout this cylinder which is insulated that means entire cylinder is insulated there is a partition earlier problem also the vessel was insulated there are two partitions three compartments similarly insulated find the net change of entropy for the mixing process okay you have to find out here before calculating this what is the final pressure you use the equation of State for finding out the pressures and accordingly the same by the same procedure you can find out Delta si Delta s B and finally I tell you the Delta s universe is equal to 0.1 1/3 kilo Joule per kg k ok Delta is a you take it for your homework it will come as 0.3 here the situation is little different Delta s B will be negative yes one point minus sorry minus zero point one nine one kilo Joule per kg case so that the algebraic sum of these two gives this one ok then thank you for today Adam means I will take so next class I will start this cycle the vapor power cycles and the you you
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