 Next class, welcome to the session last class we discussed the application of one- hospital of thermodynamics with an open system or a flow process earlier we discussed the application of first lord to a closed system, so today we will be discussing first some important thermodynamic relations, as an outcome of both first and second laws, so let me first you the first law now if you see that first law applied to any process it can be written like that d q is equal to d u plus d w so we have already recognize this that the first law when applied to a stationary system or a closed system where internal energy is the only intermolecular energy u then the first law for an infinite small process can be written as d q d u plus d w with a cut in d represents infinite small amount of it infinite small amount of work well now for a special restriction of an irreversible process and for an irreversible process and only displacement work displacement work only displacement work and only displacement work we can write this equation as this d q is d u where d w can be represented as p d v that means the work done infinite small amount of work done can be written as p d v or w for any finite process connecting the state points one and two as you know from basic thermodynamics can be written like that so until and unless we know pressure as a function of volume we cannot integrate this so for a reversible process and only displacement work that means the work transfer between the system and the surrounding takes place to the displacement of the system boundary under reversible conditions that means always there will be an equilibrium of pressure that means if this is the system so pressure exerted in the boundary will be same p uniform pressure and that has to be same with the pressure of the surrounding that means there will be always a pressure equilibrium that is the concept and the process is infinite small quasi static process very long process very slow process which is a reversible process where all dissipative effects are absent I am not going to tell you again about all the requirements for a reversible process so under the conditions for a reversible process where only the work transfer is confined with the displacement of the system boundary system boundary either expands or collapses then the type of work transfer takes place in either directions that means either work may come out or work may go in under reversible condition this work transfer can be expressed as p dv so therefore under a reversible condition and only displacement mode of displacement work mode is there for a close stationary system we can write the first law as dq is du plus p dv now you see that before going to further for thermodynamic relations we must describe the second law what is second law of thermodynamics is very important what is second law of thermodynamics can anybody tell in a very simple manner what is second law of thermodynamics while the first law of thermodynamics is the quantitative conservation of energy what is second law which is another independent law of thermodynamics what is second law second law of thermodynamics puts yes conversion of heat to work this  inevitability six  inevitability action  inevitability run but second law of thermodynamics as a whole is same, its uses or it can be looked from different perspective. So, accordingly the definitions related to a specific field appears to be different, but the basic second law is a law relating to the direction of processes, while the first law puts the constraint on the quantitative conservation of energy. That means for any process dealing with the transformation of energy or transfer of energy, energy quantities have to remain same. If mass to energy or energy to mass conversion phenomena is kept aside. This is simply the first law of thermodynamics. So, what are the types of energy with which will be dealing the depends upon process to process. Mechanical engineers may be interested only with heat and mechanical work, chemical engineers may be interested with the chemical energy with the heat energy and other energy, but the total conservation has to be there, the total amount of energy will remain same. Similarly, second law puts a restriction on the direction of a process. That is why it is known as directional law. That means all the processes are not and do not occur in nature in all directions means there is there is there is a restriction for the direction of a process. For an example, I am telling second law is as simple as that the energy flows from a high energy gradient the sorry the flux flows from high energy gradient to low energy gradient. For example, heat flows from high temperature to low temperature mass diffuses from high concentration to low concentration. The fluid flows from high energy to low energy. So, low energy to high energy this does not flow. So, this is also can be considered the second law of thermodynamics which puts a restriction on the direction of the processes. We see that there are certain processes which takes place in a particular direction. Similarly, just you told that conversion if mechanical engineers are much more interested with the conversion of heat and work continuous conversion of heat and work. And it has been found that if you want to convert continuously heat into work it is not possible to convert the equal amount of heat into work. So, in the conversion some amount of heat has to be always left as a residual. That means heat cannot be completely converted into work continuously while if you want to convert work continuously into heat you can do it. That means one way you cannot convert the entire amount of heat into work continuously in a cyclic process. Because for a continuous conversion you require a cyclic process while you can do it for work to heat conversion. That means in a cyclic process you can continuously convert the equal amount of work into heat. So, this is a second this is also second law. So, second law gives us the restriction or puts an imposition or constraint imposed on the direction of processes. Though the conservation of energy is valid conservation of energy may be valid for both these directions, but one direction will not appear at all in nature. That means for example, if heat if heat could have been converted entirely into work continuously the same amount there was no violation of conservation of energy. Because nothing is destroyed nothing is created, but it cannot be done. Similarly, some processes cannot be made to occur in certain directions nobody has seen in nature precisely this is the second law heat flows from high temperature to low temperature. So, same amount of heat can flow from low temperature to high temperature. There is no violation of first law of thermodynamics, but this is the directional law that heat always flows from high to low temperature. So, these are the directions these are the outcome of the natural processes which follow certain directions. Now, this can be put into a more concise or compact in a compact statement in the form of the definition of an entropy. That means it is very difficult to say that some there are many processes which can occur in both the directions. Some processes may occur in both the directions. For example, you heat a body heating of a body the body can be cool the cooling is also possible. Some process may not occur in reverse direction. For example, if you convert work into heat adiabatic conversion of work into heat or intermolecular energy. For example, there is a moving body for example, there is a fluid you stir it. That means you make the work transfer the fluid becomes hot. That means the work is being converted into intermolecular energy. We will not tell heat better in a strict sense is intermolecular energy insulated vessel. Heat is the energy in transit that appears because of the temperature gradient. That are rather we will call we will tell in a strict sense that work is being converted adiabatically intermolecular. But the reverse is not happening. That means if you cool the liquid or we cool the fluid the work may not come out. So, there are some processes that distinctly occur in one direction reverse is not possible. Understand, but there are processes which can occur in both the directions. So, how do you put a directional constraint in those processes? For example, you can heat a body, you can cool a body, you can compress a gas, you can expand a gas. That means you can make work transfer to a closed system in and you can take work out from a closed system. So, to make a general restriction or constraint on the direction of all the processes it is made in a more compact and concise statement through the definition of entropy through the definition of entropy. So, entropy in a very limited sense of definition for our use we will define like that entropy is a property of a system and which is defined like that a change in entropy differential form is defined as d q reversible by t. That means if for an infinite small process executed by a system in a reversible manner if d q reversible is the heat transfer in the process then this divided by the temperature at which the heat was transferred represents the change in the entropy. This is the first line of definition of entropy in the classical thermodynamics through the definition of heat d q reversible by t. That means if you want to define the change of entropy between two points then we have to integrate it one by one to two d q reversible by t and entropy being a point function can be written as s 2 minus s 1. So, we see that one interesting thing that this heat transfer d q reversible it is a path function it does not depend on point on state points, but if it is divided by t it becomes an exact differential which is equal to d s which can be integrable from one to two. So, therefore one by t acts as an integrating factor type of thing which is being multiplied with d q gives you the definition of a point function which is precisely the definition of entropy well. So, therefore we get the definition of entropy like this and the second law tells the directional law with this definition of entropy that all process takes place in such a way that entropy change for example, if I write a finite entropy change entropy change of system and surrounding together and surrounding together algebraic sum algebraic sum should be greater than is equal to say that the very important principle delta s system plus surrounding that means whenever a system interacts with other systems the other system becomes surrounded what is the definition of system and surrounding system we define that certain mass certain quantity of mass within certain boundary the definition of system and surrounding is everything external to the system that is surrounding, but in proper sense we mean surrounding those systems or those parts external to that system on which we are concentrating our attention with which the system is interacting. That means if there are two systems interacting with each other one is system other is surrounding if there are four five systems a number of systems interacting with each other if we consider one as a system others with which a particular system which we have identified interacting is unknown as surrounding. So, therefore, if we identify one system and all other interacting systems as a surrounding this is an arbitrary definition then we tell any change delta s of this system and the delta s of the surroundings with which it is interacting that means this algebraic sum of the entropy change of all interacting systems you understand that is known as delta s system plus delta s surrounding. So, there is no restriction which one will be system which one will be surrounding for example system a interact with system b if system a you define as surround system b is the surrounding for system a. Similarly, if you consider system b as the system system a will be surrounding a system b that means delta s system plus delta s surroundings means the algebraic sum of the change will be always greater than is equal to 0 that is the directional law. That means process should take place that means a particular system will execute a process with interacting with another system such that the algebraic sum of the entropy change between the two systems will be always greater than is equal to 0 it can never be less than 0. This is the pure directional constraint and the second law of thermodynamics defined from the principle of entropy and this statement is known as the principle of increase of entropy. That means the change of entropy algebraic sum of the change of entropy of all the interacting system has to be greater than is equal to 0. That means if a body is heated that means another body is cooled where from the heat is coming. So, therefore, when the body is heated if you call this as the system the body from where it has taken the heat you call as the surrounding. So, if you calculate the entropy rise of the body which is heated and at the same time you calculate the entropy decrease of the body which has been cooled and algebraically at the rise of entropy for this and the decrease of entropy for this you will see the sum of the two will be either greater than or is equal to 0 in which case it will be 0 when the interactions will be reversible, but when the interactions will be reversible all natural processes are irreversible processes then this will be always greater than 0. That means all natural interactions all natural processes will take place in such a way that the consequence of the process is a net change of entropy to have a non-zero value which is known as the production of entropy because if the algebraic sum of the entropy change is greater than 0. That means before the start of the process if we consider entropy reservoir some absolute value of entropy was there and because of the execution of any natural process the algebraic sum of the change of the entropy for all systems interacting with each other that means for system and surrounding greater than 0 that means it is adding to the entropy reservoir which means a production of entropy that means any natural process in the universe always increases the entropy the entropy bank increases its storage that entropy is a monotonically increasing function of time. So, whenever there is a process entropy is always increasing, but for a reversible process the limiting case is hypothetical case theoretical case a reversible process quasi static process free from all dissipative effects and for all potential lack of potential gradients the entropy change is 0 reversible process hypothetical process as you know from your thermodynamics the basic definition of a reversible process is a process where the basic potential gradient which causes the process is 0 that means a reversible heat transfer is a heat transfer at 0 temperature gradient a heat transfer at 0 temperature gradient cannot occur understand. So, therefore, all reversible processes are processes where the potential gradient is 0. So, therefore, reversible process is not a process. So, it is purely a static condition. So, therefore, a limiting case of reversible process is conceived in practice by creating infinite small potential gradient. So, that a very slow process will take place. So, for a finite amount of energy interactions it takes a long time that. So, infinitely long process or a quasi quasi means almost that a quasi static process is a reversible process. So, if the interactions are reversible then the entropy change algebraic sum of the entropy change of all the interacting system that is system plus surrounding will be 0. Otherwise for all natural process a reversible process which are finite processes which takes finite time to complete a process these are all irreversible processes these are associated with dissipative effects these are associated with the potential gradients causing the process. So, for them the entropy change of the interacting systems the algebraic sum of the entropy change of the interacting system will be greater than 0 which increases the entropy. So, therefore, this is the basic second law second law which can be written that d s in differential form d s system d s system plus d s surrounding surrounding means if I identify one as the system all other interacting system is the surrounding is greater than 0. And some of these two as a whole is known as d s isolated that means isolated system d s isolated system is greater than is equal to 0 d s isolated system is greater than is equal to 0 what is this isolated system that means a system and the surroundings for example a system a interacts with system b only system b it is surrounding. So, system and surrounding combinedly make an isolated system isolated system is system as you know which is totally isolated from other part of the surrounding that means the interaction is taking place between system a and system b. So, no other body is coming in to picture as far as the interactions are considered. So, therefore, system a and system b consider to be an isolated system. So, therefore, system and surrounding surrounding means all other interacting system comprises an isolated system. So, therefore, this is known as d s of an isolated system is greater than is equal to 0 this is the second law of thermodynamics in brief I tell you now with this definition I tell you one very important thing you must remember at this juncture that again I start that d q is equal to d u plus d w this is the first law this is the first law there is no doubt first law applied to a close system close system if I write it d q well d e plus d w this is for any system for any system when I write this this is a stationary or close system this is because for a close system we neglect the kinetic energy and other potential energies only the intermolecular energies as the internal energy. Now, when we consider the interactions in a close system as reversible and we consider the displacement work is the only form of work then we can write in place of d q is t d s and this is d d now you see I take the help of a reversible process to modify the first law or rewrite the first law in the form that d q is replaced by t d s because you know if we consider a reversible process we know the definition that change of entropy is defined like that for a reversible process d q reverse that means the d q that is the amount of heat transfer in a reversible process is t d s. So, therefore, we can write t d s is d u and we know that if we consider only the displacement work as the only mode of work transfer by the close system then under reversible condition this d w the work transfer is p d v now this equation if we look you see that this contains all the property values. So, therefore, from the viewpoint of mathematics it is seen that this relationship should be valid for any process related to these properties because the property entropy is valid for any process whether it is reversible or irreversible, but this definition is made through the heat transfer of a reversible process for an example let me tell you that if there is a process which is irreversible and there is a value of d q for a small process infinite small process where the value of heat transfer is d q and at a temperature t for example then this d q by t is never equal to d s of the process d s is equal to d q reversible by t that means if that process could have been made in a reversible manner then the value of d q reversible by t could have been d s. So, in that case we know that d q by t is greater than d s what is this principle this is inequality of cross s this is known as inequality of cross s this is inequality of cross s d q by t is greater than d s. So, usually this is a this is expressed like this d s is equal to for any process d q by t plus d s irreversible that means this is the component for irreversibility of the process which is 0 for a reversible process that means for any reversible process d s is always greater than the d q by t that means for any any process there must be a value of d q by t because d q is defined there is a heat transfer to a process to understand, but the d q by t is not the entropy change for that process. So, entropy change is defined by d q by t is greater than d s sorry sorry I am extremely sorry d q by t less than d s I am sorry d q by t all right all right d q by t is less than d s or d s I am sorry this is a mistake d q by t less than this is the inequality of cross s that means d s is more than d q by t clear now it is clear now it is very important concept that though here we have taken the help of a reversible process to derive this from the first law, but this law is valid for any process so far is the property values are concerned which means that t d s is equal to d u plus p d v is valid for any process, but difference is that for any process t d s is not equal to d q and p d v is not equal to d w p d v is p d v t d s is p d s that means only for a reversible process this equation and first law are synonymous they are not independent equation, but for all other processes this and first law of thermodynamics are not synonymous first law of thermodynamics is like this, but this is a property relation this means that t d s not is equal to d q and p d v not is equal to d w this you must bear in your mind that t d s is d u plus p d v is a property relation for all processes whereas t d s is equal to d u plus p d v is a first law of thermodynamics for only reversible processes because t d s can be substituted by d q and p d v can be substituted by d w which cannot be done for all other processes so therefore this will remain as a property relations for any process, but for a reversible process this will be the first law of thermodynamics clear? I think you have understood this because here lies the major confusions that though we derive t d s is d u plus p d v from the first law with the help of reversible process executed by a closed system with displacement mode of work, but when we derive this process this constant is dealing then it acts as a property relationship for all processes that means t d s of all processes is equal to d u plus p d v of all processes but whether p d v of all processes will be equal to d w or t d s of all processes will be d q that have to be examine whether the process is the reversible or irreversible realize the concept that means this equation property relations equation and the first law of thermodynamics are synonymous for reversible processes but for only reversible process they are not synonymous well so therefore this is a very important property relations t d s is d u plus p d v another important property relations comes from the definition of h you know that h is u plus p v well so we can write d h a simple differentiation d u plus p d v plus v d v well now we know that d u plus p d v is t d s so t d s plus v d v so therefore we can write t d s is equal to d h minus v d p so these two are very important property relations we can write it in terms of intensive property that means specific values that means t small s that means specific entropy d small u plus p small v similarly t d s is d h minus v d p that means specific entropy d h minus v d v so these two equations are very important property relations always we will use t d s is d u plus p d v t d s is d h minus v d v all right so with this property relations this will come afterwards at several locations in reducing many formulae for compressible flow now we will start the propagation speed of sound in a propagation of infinite small infinite small disturbance in a compressible medium in a compressible that is speed of sound why that what sound comes I will tell you let us consider a duct where there is a flow of a compressible fluid with any velocity v now at any point here on the downstream now if you create an infinite small infinite a small disturbance here then what will happen this disturbance will propagate will propagate through this medium now usually what happens if it is a rigid body and infinite small disturbance if you create here that disturbance will be immediately sensed at the point on the upstream that means all the points in the medium will be simultaneously picking up this displacement or this disturbance that if this disturbance causes the displacement of a particle adjacent to this point it will instantaneously propagate the disturbance to all other particles this happens also in an incompressible fluid or a liquid so that it is almost an instantaneous process that means if a particle is displaced here this displacement will be sensed instantaneously almost instantaneously to all points that means if you consider in the direction of the upstream it will be propagated instantaneously but if the medium is compressible whose coefficient of elasticity is very high it does not happen so that means if you create a small disturbance what we will do it will displace the immediate adjacent points and we will increase its density reduce its volume and it will do in turn the next neighboring points and their next neighboring points and this way this disturbance will propagate during a finite time within a finite time in the upstream directions that means it will not be sensed instantaneously at any point in the upstream so this will be gradually propagating through the displacement and increasing the density of adjacent points consecutive neighboring points in a case of a compressible medium so therefore we see that if we create a disturbance this disturbance ultimately what happens it propagates in the form of a pressure wave in the form of a wave form of an elastic wave that means a elastic wave as it moves which changes the condition here after it moves through certain distance in the upstream direction then slowly slowly propagates in the upstream direction and changes the condition so it moves in the form of an elastic wave or pressure wave elastic wave or pressure wave the sound energy exactly moves through a compressible medium in by the same mechanism so that we consider it as the propagation of the sound that means you create a sound energy here or which can be thought of an infinite small disturbance how does it propagate let us consider at any instant the wave this elastic wave or pressure wave as come up to this point with a velocity c let us consider the velocity that is the pressure wave or the elastic wave is moving with a velocity c in the upstream direction let us consider the velocity of the compressible fluid or the fluid which is flowing with a velocity v its pressure is p and the density is rho now when the pressure or elastic wave as generated in just come through this part of the fluid it has imparted some velocity to the fluid in the direction of its motion how can we conceive it if the fluid could have been stagnant then if a pressure pulse is generated here or a disturbance is generated here then this disturbance while it is moving with some velocity c after it passes through this portion it has it will create by the action of impulse some small velocity in the direction in this direction to the fluid element so therefore it is in the case of a stagnant fluid so for a flowing medium we can consider the velocity of the fluid may be in this direction if this velocity is very high then this will this velocity will be very small which will be generated due to the passing of this pressure wave or elastic wave so therefore the velocity will be reduced by an amount v minus d and let because of this the pressure is changed from p plus d p less the density changes from rho plus d well now you see how to analyze this situation now this situation is an unsteady situation with respect to the flow of the fluid why because at any point with time the things are not steady because depends upon is arrival for example at this instant when the pressure wave or elastic wave is at this position the velocity is v pressure is p and rho is run disturbed but when it will cross this position then the pressure density will change velocity will change so therefore this is a situation where the flow is termed to be or appears to be unsteady so therefore if we consider a situation where we give an opposite direction velocity that means we choose a frame of reference where this elastic wave appears to be steady then the situation will be and steady flow situation let us make it here itself you can understand that means let us consider a situation where we consider this to be steady the elastic pressure wave to be steady that means this frame of reference that means we fix the coordinate axis with this pressure wave with respect to that coordinate axis the pressure wave remains to be stationary that means it is synonymous to this situation that we add a velocity c in this direction that means with respect to this frame of reference the wave will appears to be stationary while the flow will be in this direction with the velocity v plus p and rho density and when it comes here its velocity is reduced v plus minus dv and pressure will be p plus dv and rho plus dv so you see with respect to a stationary frame of reference this wave this pressure wave acts as a compression wave why the velocity the fluid is coming with an uniform steady velocity approaches this stationary pressure wave and across this wave that when it passes through this wave it is velocity is reduced and pressure and density is increased now this is a steady state situation and we can analyze this problem we can analyze this problem with the help of continue conservation of mass and momentum what we do we consider a control volume circumscribing this pressure wave let this is a control volume now if I write the equation of conservation of mass then we can write the mass flow rate m dot at this section let the cross sectional area of the duct or the pressure wave they are same a then we can write rho area into the velocity v plus c and under steady situation this will be same that means it crosses a wave but there is no generation or no depletion of mass within this control volume so the control volume represent the steady state control volume so mass flow in is equal to mass flow out so we can write this with respect to this situation where we consider the change in density as rho plus d rho and a sorry a v plus c minus dv well so now if we equate these two we get rho a v plus c is equal to a rather we can cancel so is equal to rho into v plus c rho into v plus c minus rho dv plus d rho you see v plus c minus d rho dv now we consider this wave to be infinite small and the strength is also very small so that this dp and d rho are very small in a way that their product can be neglected as compared to the value of d rho and dv so this second order terms are neglected now rho v plus c rho v plus c cancels so this minus this is 0 from which we can write dv is equal to v plus c into d rho by rho well now if we apply the momentum theory this is the continuity that is conservation of mass continuity now if we write the momentum theory momentum theory if we write the momentum theory then what do we get the momentum theorem applied to the control volume at steady state you know the momentum theorem at steady state tells that is the net force acting in any direction let us example we consider this direction the direction of flow of the fluid is equal to the net rate of momentum a flux from the control volume in that direction that means a flux minus influx since there is no generation that is there is no change in the momentum within the control volume so if you apply the momentum theorem let us write the net force acting in this direction in the control volume so net force at the forces which is due to the pressure p and p plus dv we discard the frictional forces this is because that this pressure wave is very thin so the control volume circumscribing this pressure wave enveloping this pressure wave will be very thin so that these areas are very small so the surface area so small the frictional forces are negligible compared to pressure forces so therefore the forces in this direction direction of flow will be p into a minus this side p plus dp p plus dp into a and that is the net force acting in this direction of flow all right that must be equal to the net rate of momentum a flux in this direction what is momentum a flux in this direction that is the mass flux rho a v plus c times the a flux velocity that is v plus c minus dv that means the fluid is coming out from the control volume with this velocity v plus c minus dv what is the influx of momentum which has to be subtracted that is the mass mass remains the same because this and these are same and this is multiplied with this inflow velocity to the control volume that is v plus so if you calculate it you will get dp into a minus dp into a so rho a v plus c so v plus c dv minus v plus c that means simply minus dv that means we can write dv is equal to minus minus cancels so dv is equal to what is that dp by rho into v plus so now we equate dv from these two dv from these two this is the final step for the continuity equation and this is the expression of dv as the final step from the momentum equation if we equate these two then you will get dv if we equate dv that means dv is eliminated you will get v plus c is equal to root over dp by d rho you see that root over dp by d rho so v plus c we get as root over dp by d rho that is the increase in pressure divided by the increase in density change of pressure with respect to density dp d rho all right so therefore we see that we can write v plus c as root over dp by d rho what is v plus c what is v plus c c was the speed of the elastic wave or pressure wave and v was the velocity of the fluid in this direction and this is moving in the upstream direction because it was created this pressure pulse of the disturbance was created at downstream so it is moving in the upstream direction so v plus c is the relative velocity that means it is the you write it as a which is the velocity of pressure pulse pressure wave relative to the relative to the fluid medium relative to the flow medium relative to the flow medium that means flow medium as a velocity is equal to root over dp that means if it is a stagnant fluid so the velocity of the pressure pulse in the direction that means there is no question of upstream downstream it is a stagnant that means where from it was originated from there it flows in this direction with a velocity root over dp by d rho so in case of a flowing medium we can tell that the velocity we represent it by a now the velocity of pressure wave relative to the flow medium is root over dp by d rho since the sound energy moves in a compressible medium in the similar fashion this is the speed or velocity of sound velocity of sound or acoustic velocity so therefore we see the acoustic sorry acoustic velocity in a compressible medium relative to the flow medium relative to the flow medium that is relative to the velocity of that medium is given by root over dp by d rho as you know that definition of e that elasticity of modulus what is the definition of e is equal to rho dp by d rho we have seen it earlier so therefore we can replace it and we can write root over e by rho now it is clear that all medium are compressible medium but liquids are incompressible that means whose change of pressure with respect to density is very high that means the value of elasticity is very high that means it requires a very high pressure for a change in density well so therefore the speed of sound in a medium which is more or less incompressible is very high that means the speed of sound is related to the modulus of elasticity in this way that the modulus of elasticity is more that means fluid is more in less compressible then the speed of sound is very high e by rho now next is this can be represented in a different manner that now you see this dp by d rho that is the change of pressure with the change of density the value of this depends upon how you make the change of pressure because we know that the change of pressure can be made in with various process constants for example an isothermal change of pressure may be made where the volume may change because out of PVT if you think this three properties we can keep t constant and we can vary p and v that means we can change pressure and we can see the variation in v for example p rho t similarly other process constants may be kept fixed where the temperature may be allowed to vary along with the variation of p so therefore the values of dp by d rho the exact values of dp by d rho can be found out provided we put a process constant until and unless it is very difficult to be described even if we define in terms of the bulk modulus of elasticity the bulk modulus of elasticity from its basic definition cannot be evaluated until and unless we put a process constant to evaluate the dp by d rho because it is very difficult to know the dp by d rho because p can be varied to change the rho under various process constant we can keep the temperature constant we cannot may not keep the temperature constant if you do not keep the temperature constant we may keep some other thing some other restriction there may be a number of restrictions where temperature may vary so specifically we will have to know the restrictions and we will have to know the explicit relationship between pressure density and temperature which is known as the equation of state for that medium or for that substance or for that system only when we can find out or evaluate the value of dp by rho or ascribe even the value of the bulk modulus of elasticity do not think that either it is root over e by rho or root over dp by d rho its value is known until and unless we put the process constant so therefore we have to find out from our physical understanding what should be the constraint process constant or what situation the mechanism the pressure pulse is propagating so that we can put from there a process constant to evaluate the value of dp by d rho or the value of p so that we can finally find out the numerical value of the speed of the sound in a compressible flow that I will discuss in the next class well please any question this you have to understand there is a lot of concept rather than mathematics well thank you thermodynamics ok