 Hello and welcome to lecture number 21 of this lecture series on Introduction to Aerospace Propulsion. We have been discussing over the last several lectures on basic aspects of thermodynamics including some of the basic cycles which are which form very important aspects of engineering analysis and our day to day use as well. And we have also had several tutorial sessions during which we have solved some problems related to basic thermodynamics as well as in the last lecture we have solved some problems related to cycle analysis of some of the cycles like the auto cycle, the diesel cycle and the Erikson and the Brayton cycles. And what we shall be discussing today is a very different topic in some sense in as compared to what we have been discussing in the last few lectures. And but of course, they will the discussions of today's lecture will also be an important aspect of thermodynamic analysis of certain systems which involve mixtures of gases or in processes which involve vapor water vapor for example. Like if you have to use Rankine cycle for a steam power plant how do you carry out a thermodynamic cycle analysis of such a cycle because Rankine cycle involves water and steam and therefore, we need to calculate properties associated with water and steam. And so today's lecture is aimed at understanding of properties of water known as pure substances as well as mixture of ideal gases and real gases and so on. So, what we shall be discussing in today's lecture of the following we shall be talking about properties of pure substances what are meant by pure substances. We shall then be discussing about what is meant by compressed liquid saturated liquid saturated vapor superheated vapor and so on. We shall be also discussing about what is meant by saturation temperature and pressure then we shall be talking about property diagrams of pure substances property tables composition of a gas mixture then we shall talk about PBT behavior of gas mixtures ideal gas and real gas mixtures and properties of gas mixtures. So, these are some of the aspects that we shall be discussing in today's lecture which is primarily aimed at understanding of properties of pure substances. So, what do we primarily mean by a pure substance because that is going to be our topic of discussion for today's lecture. Now, a pure substance is one which has a fixed composition throughout the process and so such a such a substance is referred to as a pure substance. Examples of pure substances can be water or gases like nitrogen oxygen helium and so on. So, these are certain substances which will have the same composition throughout the process and that is why they are called pure substances. And a mixture of one or more phases of a pure substance will still remain to be pure substance as long as the chemical composition of individual phases of the mixture does not change. That is a mixture of water and vapor that is water vapor will still be qualified as a pure substance because the chemical composition of neither water nor water vapor changes during the process. So, as long as that is maintained then even a mixture of different phases of the same substance which is a pure substance will still be called as a pure substance itself. And so a substance which has a fixed chemical composition throughout is called a pure substance. Examples are water nitrogen helium and so on. A mixture of two or more phases of a pure substance will still be a pure substance as long as the chemical composition of all the phases is the same mixture of ice and water and so on. Properties of water and its phase and its different phases form a very important part of analysis because in different systems engineering systems one would come across use of water and its different phases like ice or steam. And therefore, analyzing different properties of water and its different phases will be a very significant part of an analysis of any system which involves water and its phases. And therefore, it is important for us to understand the significance or importance of water and its different phases which is why we are basically trying to understand properties associated with water and its different phases. So, before we try and understand those properties we need to understand certain terms associated with how water behaves at different states or phases. Now, if we consider water which is at a state which is far away from being vaporization that is water which is or a liquid which is at a state which is not about to vaporize is known as a compressed liquid or a sub cooled liquid. So, water or any other liquid where it exists in a state wherein it is not going to vaporize at all such a state of water or that liquid is called compressed liquid or sub cooled liquid. For example, water at a temperature of 20 degree Celsius and 1 atmosphere is a sub cooled state because water is not at a state of evaporation at a 1 atmosphere because we all know that at 1 atmosphere water boils at 100 degree Celsius. A liquid that is about to vaporize is called a saturated liquid that is liquid which is in a state that it is just about to vaporize is known as a saturated liquid. And a vapor on the other hand which is about to condense is called a saturated vapor. So, vapor which has reached a state wherein it is about to condense is known as saturated vapor. And a vapor which is not about to condense that is it is not a saturated vapor is called a super heated vapor. That is the vapor happens to be in a state where even some amount of cooling does not cause it to vaporize or condense is known as super heated vapor. That is when you have vapor which is at a temperature and pressure which is not close to the saturation temperatures then that vapor is known as super heated vapor. So, what we have discussed are compressed liquid or sub cooled liquid wherein the liquid is not about to vaporize. A liquid which is about to vaporize is called a saturated liquid. And a vapor which is about to condense is called a saturated vapor. And a vapor which is not about to condense is known as super heated vapor. So, what I will do now is to illustrate all these different states by some example we will use water as our example because that is the most common liquid that we come across in engineering systems. So, if you look at different states of water let us look at state 1 for example, at state 1 water is at a pressure of 1 atmosphere and 20 degree Celsius which means that the water is not about to vaporize. It is not at a state that it will begin to vaporize. Therefore, this particular state that is state 1 is known as the compressed liquid state for water. Now, we consider state 2 that is we continuously as you can see we are adding heat to the system. As we add heat at constant pressure that is the pressure remaining constant now the temperature increases. At the state 2 temperature of water is now 100 degree Celsius pressure remains 1 atmosphere. So, this state is known as saturated liquid state for water that is water is at a state now that it would start vaporizing any minute now or that is at any moment the water would start vaporizing which is why it is called saturated liquid. State 3 is a mixture of saturated vapor and saturated liquid because as we continue to add heat at 100 degree Celsius water will start vaporizing provided pressure is 1 atmosphere and once it starts vaporizing you can see there will be one part of the system which consists of a vapor of water and second part consists of saturated liquid and so we have a mixture of saturated liquid vapor. So, this is saturated liquid vapor mixture that is state 3. Looking at state 4 we again add heat continuously till a point where there is no more water left in the system. And so at 1 atmosphere and 100 degree Celsius if you have water vapor then that is known as saturated vapor because if you start removing heat water will on that is vapor will immediately begin to condense. So, this is known as saturated vapor state. State 5 is if we again add continuously heat at constant pressure that is 1 atmosphere then we have higher temperature of the vapor which is not close to its condensation temperature and this is known as the super heated state of water or vapor. So, state 5 is super heated vapor. So, what we have seen are 5 different states of the same substance the same pure substance that is water. State 1 was of water at temperature and pressure which is far away from its vaporization temperature that is known as sub cooled or compressed water state. Second state is if you heat water up to 100 degree Celsius at 1 atmosphere water is at a point that it may begin to vaporize any moment that is. So, this is known as the saturated liquid state. The third state is as you again add heat water will begin to evaporate. So, you will have a mixture of water and vapor at 100 degree Celsius and 1 atmosphere. So, this is known as the saturated liquid and vapor mixture. Fourth state is that we have only vapor which is at 100 degree Celsius for water and so that state corresponds to saturated vapor state because the vapor will begin to condense at any moment. And last state is at a temperature which is far away from the condensation temperature and that is known as the super heated vapor state wherein the temperature is far away from the condensation temperatures. So, for water temperatures much beyond 100 degree Celsius at 1 atmosphere would be super heated state of water. So, what we shall do now is we know that we have seen 5 different states of water. Let us plot the temperature and volume. Let us plot this particular process that we have seen of adding heat to water. So, as to get different states or phases of water and plot them on a constant pressure line. So, what we have seen is in all the 5 different states that we have identified pressure remains a constant that is 1 atmosphere. So, on a constant pressure line we will plot this property on temperature and specific volume coordinates. So, on T V coordinates or T V diagram temperature and specific volume if you were to plot this particular process that we have seen which is basically from state 1 to state 5. This is a constant pressure process it is constant pressure at pressure equal to 1 atmosphere. Water was initially at state 1 at 20 degree Celsius and as you heat water at constant pressure this is how the process proceeds up to state 2. So, this is the compressed liquid state because at these states water is not in a position that it will begin to vaporize. At state 2 is when you have saturated liquid that is 100 degree Celsius. So, water is in a position that it will begin to vaporize. State 2 corresponds to saturated liquid state and state 2 to state 3 is saturated mixture where you have liquid as well as the vapor and after it reaches state 3 and it goes to state 4. So, any state in between 2 and 4 which is equal to state 3 that is saturated mixture state after it reaches state 4 we have the saturated vapor state that is we have only vapor which is saturated that is it may condense any moment. Further addition of heat at constant pressure increases the temperature it was 100 from all the way from state 2 to state 4 and as we increase the temperature we have process going from state 4 to state 5 and this is the super heated state that is we have now temperatures which are much higher than the condensation temperature. So, the vapor reaches a temperature of 300 degree Celsius let us say that was at state 5. So, state 5 is the super heated vapor state. So, any point in between 4 and 5 corresponds to super heated vapor. So, 4 5 is the super heated vapor state. So, this is how you would plot this process which is basically heating of water from sub cooled state that is compressed liquid all the way up to super heated vapor state. So, as you heat from state 1 to state 5 water passes through a series of different states which involves sub cooled state saturated liquid state saturated liquid vapor mixture saturated vapor and the super heated vapor. So, as we continuously heat at constant pressure water passes through these different phases and we have discussed 5 different states of water which correspond to different states of water as well. And so, we can plot the property diagram on a constant pressure scale between temperature and specific volume and we see the how we can we can actually see the variation of the different properties of water as it passes through these different states. So, property basically the temperature in this case was changing, but the pressure was kept constant at 1 atmosphere. So, it is also possible that you can change it the other way around that is we can also change pressure that is as you keep changing pressures the characteristic curve that we have just seen will also change because this is valid only for pressure is equal to 1 atmosphere. For any other pressure we know that the boiling point of water the condensation point of vapor will all be different which means that as you plot this for a different sets of pressures and temperatures we should be able to get on TV scale this temperature volume scale a series of these constant pressures lines along which water would continuously change its phase. And so, this is what we shall try and do and see different property diagrams or scales in which we can see how the properties of water would change. Now, in order to do that that is because temperature of water that is temperature at which water would start boiling also depends upon pressure it is a strong function of pressure. And so, if we fix pressure the boiling temperature is also fixed. So, at a given pressure the temperature at which a pure substance changes its phase is called the saturation temperature or T sat as it is represented. So, once you fix the pressure what is the temperature at which the temperature of the phase changes that is basically known as the saturation temperature. Similarly, the at a given temperature the pressure at which a pure substance changes its phase is called the saturation pressure. So, saturation pressure and temperature are both functions of each other in some sense that is if you fix the pressure then the temperature at which a substance pure substance changes its phase is known as the saturation temperature. And as you fix pressure the temperature at which the property of the pure substance changes its phase is known as the saturation temperature. So, saturation pressure and temperature will form a very important aspect of analysis of pure substances which continuously evolve and change their phase. So, in the previous picture or diagram which had shown variation of the properties along a constant pressure line. If you were to plot multiple number of these pressures what we get is a series of points that is if you join all the state 2's and state 4's we get a saturation line on one side we have a saturation liquid line on the other side we have saturation vapor line. And eventually this will meet at a particular point which is known as the critical temperature or critical point. So, let us plot a series of these lines on T V diagram that is temperature and volume diagram in the previous slide or 2 slides from here we had shown I had shown you a line which is looking like this constant pressure line compressed liquid saturation liquid mixture of saturation liquid and vapor saturation vapor here and superheated vapor after this. So, what if I plot this for multiple pressures. So, if I increase the pressure P 2 which is greater than P 1, but is equal to a constant we get a line like this that is for higher pressures the temperatures will also now be different. And so, as you increase the pressure the boiling temperature of water also changes you must have learnt in your school days earlier on that if you go to a high altitude place water boils at a lower temperature that is it boils at temperature lower than 100 degree Celsius. At higher altitudes we know the pressure is lower and therefore, water will also boil at lower temperatures that is if you were to plot below this let us say this is atmospheric pressure one atmosphere which is what we did last time where in water boils at 100 degree Celsius. So, if you were to plot this for let us say less than atmospheric sub atmospheric pressure which will occur at high altitudes water will boil at lower temperatures. Similarly, at higher pressures water boils at higher temperatures. So, you can see as you if you plot multiple number of these pressure constant pressure lines what we can see is and we join all these points locus of all these lines points which join state 2 which was the saturated liquid state of water. We get a saturated liquid line and on the other hand if you join all state force which was saturated vapor state we get a saturated vapor line. And both these lines meet at a point which is known as the critical point and all points on the left of the saturation line correspond to the compressed liquid state all those points on the right hand side of the saturated vapor line corresponds to the super heated vapor state. And those which fall within this curve corresponds to a mixture of saturated liquid and vapor. So, this is basically a critical diagram which also shows how you can achieve a critical point when you were to plot different states of water for different pressures. And so in general if I were to plot this for pressures which are higher than the critical pressure which means that if you have a constant pressure process which has a pressure higher than let us say the critical point that is p greater than p critical then what basically happens is that super critical pressures where pressure is greater than the critical pressure there is no distinct phase change. Because phase change occurs only during the process which is below this curve this is basically indicating the phase change process. So, after the critical point there is basically no distinct phase change process that takes place. So, at pressures which exceed the critical pressure one cannot see a distinct phase change or a boiling process. Whereas, at pressures which are below the critical pressure there would be a very distinct phase change process or a boiling process during which liquid becomes vapor and so on. So, there is a distinct phase change that occurs for processes which occur at pressures which are below the critical pressure. And for pressures which are higher than the critical pressure there is no distinct phase change or boiling process that we can see during change of phase of a pure substance. So, we have now plotted the phase change process in terms of temperature and specific volume coordinates where we have seen a constant pressure on a constant pressure line how the different properties can change and how phase water changes its phase from or water how it water changes its state from sub cooled state to saturated liquid state to mixture of saturated liquid and vapor then saturated vapor and the super heated vapor. So, the locus of all the saturation lines which include the saturation liquid is known as the saturated liquid line. And that was if you join all the state 2's of a constant pressure lines we get saturated liquid line if you join all the state 4 lines we get the saturated vapor line eventually both these lines meet at a point which is known as the critical point. And so for pressures which exceed the critical pressure one cannot really see a distinct phase change or a boiling process. So, what we will do next is to plot phase change process on a P T diagram that is pressure and temperature coordinates and see how the variation looks like. On pressure and temperature coordinates if you were to plot the phase change process. So, here we have the solid state, the liquid state and the vapor state. So, if you were to look at a process where the pressure and temperatures are changing for pressures and temperatures below the triple point triple point is by definition the point at which all the three phases coexist in equilibrium that is solid liquid and vapor. And for water the triple point is 0.01 degree Celsius. So, a line which basically demarcates the solid from the vapor that is at pressures which are lower than what corresponds to the triple point or temperature as well as pressure the process which will occur would basically be a sublimation process that is solid would directly become a vapor without becoming a liquid that is if you have pressures and temperatures which are below that of the triple point. Then if you take up ice for example, ice will directly vaporize into water vapor without actually melting. Melting would occur only at temperatures and pressure which correspond to what is shown here. And there are two melting lines shown there are substances that expand on freezing and there are substances that contract on freezing. And for example, water is a substance which expands on freezing. And so the melting line for water is which is basically has a negative slope whereas, those substances which contract on freezing have a positive slope of this melting line. So, water will follow this melting line and if because it is temperature and pressure is above the critical point above the triple point. So, solid ice would melt and become liquid and as we have seen this is basically the vaporization line where on further addition of heat water would become vapor. And you also have the critical point shown here. So, critical point was also defined in the previous diagram here. And this is that was of course, on temperature and volume coordinates this is on pressure and temperature coordinates. So, this is basically a pressure temperature diagram of any pure substance. And difference between different pure substances will come in the location of the triple point as well as the critical point because that is different for different pure substances. For example, for water the melting line is slightly different from the melting line for substances that contract on freezing and so on. So, there are three distinct phases that you can see here the solid phase which is between the sublimation line and the melting line. The liquid phase is between the melting line and the vaporization line. And the vapor phase is between the sublimation line and the vaporization line. And the point at which all these three phases coexist is known as the triple point. So, we have plotted the properties on different scales on temperature volume pressure temperature and so on. And it is also possible to plot these diagrams on a three dimensional plot which is known as the PVT surface pressure volume temperature surface. And so, which is basically a combination of pressure volume pressure temperature and temperature volume coordinates and so on. Now, the properties of different substances of pure substances are either plotted in terms of these coordinates or you can also see them in terms of a tabular format. So, there are tables associated with properties of pure substances because the thermodynamic relations for different substances can be quite complicated. And so, calculating each of the properties like enthalpy, entropy, etcetera at different pressures and temperatures can be quite involved. And to avoid that, there are standard property tables available wherein for given pressure and temperature one can determine the different properties like energy, the entropy and enthalpy and also specific volume of different pure substances. And so, if you look at the property table which is something we shall be discussing now, basically the properties are in the form of tables. So, there are different property tables available for different substances and properties of water are usually referred to as the steam table where we get different properties of water, liquid water vapor as well as superheated steam. In all these tables, subscript f is used for denoting properties of saturated liquid and subscript g denotes properties of saturated vapor. And also f g denotes the difference between saturated vapor and saturated liquid values of the same property. So, in these tables you would find subscript f which corresponds to properties of liquids that is saturated liquid, subscript g denotes properties of saturated vapor and subscript f g denotes the difference between these two of the same properties. So, for example, h f that is h subscript f basically corresponds to specific enthalpy of the saturated liquid. H subscript g is the specific enthalpy of the saturated vapor. And therefore, h f g is the difference between h g and h f that is h f g is equal to h g minus h f that is basically the difference between specific enthalpy of the saturated vapor and specific enthalpy of the saturated liquid. If you recall in the property diagram I have shown that, but process from 2 to 4 corresponds to the phase change process. And therefore, the difference between enthalpies of the vapor and that of the saturated liquid is basically the enthalpy of vaporization or latent heat of vaporization. That is the difference between h g and h f is the difference between the specific enthalpy of the saturated vapor and specific enthalpy of the saturated liquid, which means the difference between the two basically corresponds to the amount of energy required for carrying out this phase change from liquid to vapor. Therefore, h f g is basically the latent heat of vaporization or enthalpy of vaporization as it is also sometimes called. Now, so within this saturation curve that we have drawn where we have a mixture of saturation liquid as well as vapor. We know that there is some fraction of liquid water present there is some fraction of vapor present. So, we will now represent this fraction by what is known as quality that is what is the quality associated with this particular vapor and liquid mixture. So, quality which is usually denoted by symbol x quality x is basically the ratio of mass of water mass of what vapor to the mass of liquid that is quality x is basically the ratio of mass of the vapor present to the mass of liquid present. Therefore, quality has a value which ranges from 0 to 1 which means if quality has a value of 0 it means that the mass of vapor present is 0 and liquid. So, the entire mass is basically the mass of liquid. So, its quality of 0 corresponds to saturated liquid state and quality of 1 corresponds to saturated vapor state because it is at that state where the entire mass corresponds to the mass of the vapor itself. And so any value of quality which is between 0 and 1 corresponds to mixture of saturated liquid and vapor state. So, quality x is basically defined as the mass of vapor ratio of the mass of vapor to the mass of liquid and it has a value obviously ranging from 0 to 1. So, x equal to 0 means a saturated liquid state x is equal to 1 means a saturated vapor state. Now, we can actually show that in general any property which is denoted by y, y average should be equal to the sum of y f plus x times y f g that is sum of the property of the saturated liquid plus quality multiplied by the difference in the property at saturated vapor and saturated liquid state. So, here y can take any value it could be either specific volume or it could be energy specific entropy or specific enthalpy. For example, we can write h average is equal to h f which is the enthalpy corresponding to the saturated liquid state plus the x which is basically the quality multiplied by h f g which is h g minus h f. So, most of the times we would drop this subscript average because it is known that we are just taking an average of the property. So, it is basically h is equal to h f plus x times h f g similarly we can write u is equal to u f plus x times u f g s is equal to s f plus x s f g and so on. And it is also known that the value of the property at liquid state that is h f or u f or s f should be less than or equal to the average value which will in turn be less than or equal to the value at the vapor state because at the vapor state it also has additional energy because of the latent heat. So, y f will be less than or equal to y average less than or equal to y g and so this is true for any of these properties whether it is specific volume or specific internal energy or entropy or enthalpy and so on. So, what we have seen is that for from a property table if you look at properties of steam for a particular pressure and temperature all these properties will be listed as different columns you would have h f h f g and h g similarly you would have properties for internal energy entropy and specific volume. So, from the table it is possible for us to determine these values rather than calculate them though it is possible to calculate them, but it can be quite complicated especially if you look at mixture of liquid and mixtures and so on. So, to keep thing simple especially when you are trying to solve problems property tables come in handy because based on the pressure and temperature one can determine all these properties corresponding to that particular pressure and temperature. Now, we have seen that on pressure temperature diagram as well as we have also shown the temperature and volume diagram. Let us look at the temperature and volume diagram of a pure substance again. Now, here we can see that on the left hand side of the saturated liquid line it is the compressed liquid state on the right hand side of the saturated vapor line is the super heated vapor state. So, let us look at some more properties of the super heated vapor state. So, all those properties which correspond to the right hand side of this vapor line saturated vapor line correspond to super heated vapor. So, region to the right of the vapor line and at temperatures above the critical temperature we have the super heated vapor region and in the super heated region which is basically a single phase region pressure and temperature are no longer dependent properties that is you can change pressure and temperature independent of each other unlike at other phases where they are basically dependent. So, compared to a saturated vapor a super heated vapor is characterized by the following that is if you compare super heated vapor and a saturated vapor super heated vapor is characterized by lower pressures that is p will be less than the saturated pressure pressure for a given temperature and temperature would be greater than saturated temperatures at a given pressure that is if you look at the pressures and temperature for a given temperature the pressure will be higher than that of the saturation pressure and if you look at the same pressure the temperature will be higher than that of the saturation temperature for super heated state and also in super heated state the specific volume enthalpy etcetera are higher than that at a given pressure or temperature that is for super heated steam for example, volume would be higher than that of the volume of the saturated vapor. Similarly, enthalpy of the super heated vapor will be greater than enthalpy of this saturated vapor internal energy of the super heated vapor is greater than the enthalpy of saturated vapor for a given pressure or temperature. So, these are some of the properties that are associated with super heated vapor similarly, we can also identify properties that are that correspond to that of sub cooled liquid which will be basically those lines those points which are on the left hand side of the saturated liquid line. So, that those all those points correspond to the compressed liquid state of the particular pure substance. Now, what we shall discuss next is on composition of a gas mixture that is if you were to look at mixture of different gases how do you analyze different properties of that particular mixture and like if you have to calculate enthalpy of the mixture how do you calculate them and. So, in order to do that we will define what are known as mass fractions and mole fractions of a particular mixture and then apply those for calculating different properties. So, let us consider a gas mixture which has let us say k components and let m subscript m be the mass of the mixture and n subscript m is the number of moles and therefore, m subscript m is equal to summation i is equal to 1 to k m i and n m is equal to summation i is equal to 1 to k n i. So, here let us also define mass fraction m f which is ratio of the mass of a component to the mass of the mixture and mole fraction y which is basically the ratio of the mole number of a component to the mole number of the mixture. So, m f i is equal to m i by m m that is mass of the component divided by mass of the mixture and mole fraction is mole number of the component to the mole number of the mixture. So, mass of a substance of mole number n and molar mass that of m will basically be equal to product of the two that is mole number multiplied by the molar mass and average mass that average molar mass and gas constant can also be calculated as m subscript m is equal to the mass of the mixture divided by the molar number of that particular mixture which is basically equal to sigma m i divided by n m which is sigma n i m i divided by n m and since n i by n m is equal to the mole fraction this is equal to sigma summation i is equal to 1 to k y i m i and therefore, the average gas constant for the mixture would be equal to universal gas constant divided by average molar mass of that particular mixture. So, which means that we can also now relate the mass and mole fractions of the mixture using this mole fraction sorry mass fraction which is m f i is equal to m i divided by the mass of the mixture which is finally, be equal to the mole fraction y i multiplied by m i by m m. So, the mass and mole fractions are basically related through the mass fraction m f i is equal to the mole fraction multiplied by the molar mass m i by m m. Now, so this is basically to calculate if you were to calculate different properties of a gas mixture if you can basically use some of these properties to identify and calculate average properties of a mixture of gases like average gas constant and so on and we shall also see how you can calculate average properties in terms of enthalpies and specific heats and so on. Now, ideal gas of we have also already discussed in two three lectures earlier about ideal gas equation and compressibility factor. So, ideal gas equation with a compressibility factor can be used for real gases this is something we have discussed earlier. So, prediction of the behavior of a gas mixtures can basically be whether it is real gas or ideal gas can be predicted by based on two laws one is known as the Dalton's law of additive pressures and the other is known as the Amagat law of additive volumes. So, Dalton's law of additive pressure probably you would have learnt this earlier is basically stating that the pressure of a gas mixture is equal to the sum of the pressures of each gas that if would exert if it existed alone at the mixture volume and temperature. Similarly, the Amagat's law of additive volume states that volume of a gas mixture is equal to the sum of the volumes each gas would occupy if it existed alone at the mixture temperature and pressure. So, both these laws are can be used as it is for ideal gases, but these are kind of approximations for real gases that is these laws can be used in an approximate way for real gases one of the ways we could do that is by using the compressibility factor as well. Now, for ideal gases both these laws that is whether it is Dalton's law or the Amagat's law both of them give identical results whereas, it could be different it could be different results for real gases. So, let us look at the Amagat's and the Dalton's law. Dalton's law basically states that the mixture pressure is equal to summation i is equal to 1 to k p i which is the component pressure at the temperature and volume of the mixture. Amagat's law states that v m volume of the mixture is equal to summation i is equal to 1 to k v i at the mixture temperature and volume. So, both these equations are exact for ideal gases, but obviously they are only approximate for real gases. So, for ideal gases the component pressures and volume can be related to the mold fraction from the ideal gas equation that is p i we will use ideal gas equation for this p i that is component pressure at the temperature and volume t m v m divided by mixture pressure is equal to n i r u t m by v m which is for the component divided by n m r u t m by v m which is for the mixture. So, this reduces to n i by v n m that is y i that is the mold fraction. Similarly, we can also using the Amagat's law we can still write or equate that being equal to the mold fraction that is the ratio of the component pressure to the mixture pressure and component volume to the mixture volume is basically the mold fraction that is p i by p m is equal to v i by v m which is equal to number of mold ratio which is basically the mold fraction. So, this quantity which is product of y i times p m is called the partial pressure and the product y i by y i times the mixture volume is called the partial volume. For an ideal gas mixture all these that is the mold fraction the pressure fraction the volume fraction all of them are identical which is what I had mentioned here for an ideal gas this pressure ratio the pressure fraction the volume fraction mold fraction they are all basically the same. But they are not necessarily the same for real gases because if you were to use if you were to see a real gas because of real gas effects all these parameters need not necessarily be the same. So, let us look at real gases how do we apply Dalton's and Amagat's law for real gases what we could either do is use one of the advanced equations of states like Wanderweil's equation or the B.T. Bridgeman equation or so on or we could use the ideal gas equation itself with the compressibility factor. So, Dalton's law and Amagat's law can be used for real gases with approximations. So, either we use advanced equations which can be little complicated or we use the compressibility factor where P v is equal to Z n R U T where Z is the compressibility factor. So, Z compressibility factor for the mixture can be calculated if you know the compressibility factor for individual components. So, Z m is equal to summation i is equal to 1 to k y i Z i where y i is the mole fraction for each of the components. So, Z i can be determined either at T m and V m which is based on Dalton's law or at T m and P m which is based on the Amagat's law for individual gases. But again because these are not ideal gases if you determine it either is either of these ways they are not going to give you the same results. But what has been seen is that the Dalton's law seems to give more accurate results if you were to determine the compressibility factor at the mixture temperature and mixture volume. So, for gas mixtures if you are required to calculate extensive properties like energy, enthalpy, entropy and so on how do you calculate that? We can calculate them if you know the contribution of each of the individual components. For example, the internal energy enthalpy entropy can be expressed as of the mixture u m energy of the mixture is equal to summation i is equal to 1 to k u i which is equal to summation i is equal to 1 to k mi u i where mi is mass of individual component u i is in the energy associated with the individual component. Similarly, enthalpy and entropy can be expressed as summation of the products of the mass times the specific enthalpies and specific entropies of the individual components. Now, internal energy enthalpy entropy per unit mass can be basically determined by dividing the equations by the mass of the mixture that is u m that is specific internal energy of the mixture will be equal to summation i is equal to 1 to k mass fraction multiplied by the internal energy for each of the component. Similarly, enthalpy of the mixture will be equal to summation i is equal to 1 to k mass fraction multiplied by enthalpy of individual components and also for entropy, entropy of the mixture will be equal to summation i is equal to 1 to k m f i times s i where m f i is the mass fraction s i corresponds to entropy of individual components. We can also use similar properties for calculating the specific heat at constant volume and specific heat at constant pressure for the mixture that is c v m specific at constant volume for the mixture will be equal to summation i is equal to 1 to k m f i times c v i where c v i is the specific heat at constant volume for individual components. Similarly, c p m that is specific heat at constant pressure for the mixture will be equal to summation i is equal to 1 to k m f i times c p i. And so for gas mixtures, if you were to look at mixture comprising of several gas components, we can calculate the specific heats, we can calculate the internal energy enthalpy entropy etcetera for the mixture by summation of the mass fractions times the specific properties of the individual components that is mass fraction multiplied by let us say enthalpy of one component and sum it up for all of these individual components, we get the enthalpy of specific enthalpy of the whole mixture. We can also use the same properties for calculating the specific heats at constant pressure and constant volume for the mixture. Now, if you look at changes in these properties that is changes in enthalpy, changes in internal energy and entropy, we use the same principle that we have used for calculating the entropy enthalpy internal energy of the mixture itself. So, for calculating changes in internal energy enthalpy and entropy of a gas mixture during a process, we can calculate them as the following way delta u m which is the change in energy of the mixture is equal to sigma that is summation i is equal to 1 to k delta u i which is equal to summation i is equal to 1 to k m i delta u i. Similarly, delta h m is equal to summation i is equal to 1 to k m i delta h i and summation and for entropy delta s m is equal to summation i equal to 1 to k m i delta s i. So, that brings us to the end of this lecture, where we have been discussing about properties of pure substances. We have defined what is a pure substance, we have also defined what is meant by compressed liquid, saturated liquid, saturated vapor and superheated vapor. Basically, these are different states of water as they change phase from liquid to vapor. We have defined what is meant by saturation, temperature and pressure. We have different, we have discussed different ways of expressing these phase change processes in terms of property diagrams of pure diagrams, pure substances p v and p t diagrams and temperature volume diagrams. And property tables are those which basically help us in estimating many of these properties that is temperature and enthalpy, energy and so on. If we know the pressure and temperature of a particular process, then we have also looked at pressure p v t behavior of gas mixtures and ideal gas and real gas mixtures and properties associated with gas mixtures on how you can calculate the different properties like enthalpy and entropy and internal energy of a gas mixture, if we know the mass fraction of each of the components and the corresponding specific properties of that particular component. That is what we had, these are the topics we had discussed during this particular lecture. In the next lecture, what we shall be discussing are certain aspects which are quite different from what we have been discussing so far. We shall talk about one-dimensional compressible flows and we shall define what are known as stagnation properties and then we will define Mach number and the speed of sound and we shall take up analysis of one-dimensional isentropic flow. We shall understand the variation of fluid velocity with flow area and also look at isentropic flow through nozzles that is convergent nozzles as well as converging and diverging nozzles and so isentropic flow through these different types of nozzles is something we shall be discussing in next lecture as well as one-dimensional flow, compressible flow and properties of compressible flows is something that we shall be discussing in the next lecture which will be lecture number 22.