Free Physics NEET Video Lectures for Behavior Of Perfect Gas And Kinetic Theory

00:00:01 [Music] so in these set of lectures I will be give a brief introduction to thermal physics so essentially I will be talking about thermal physics first thing will be heat and temperature I'll be doing heat and temperature I will tell you what is heat and what is temperature secondly kinetic theory of gas thirdly thermodynamics and forth thermal properties or matter so as we know materials have properties elasticity and other properties which are mechanical properties here I will be talking about

00:02:25 thermal properties of matter which means if I increase the temperature how does a material respond to that change in temperature that is what is reflected in the thermal properties of matter then what is kinetic theory I will elaborate on that and what is thermodynamics I will also elaborate on that now let me first ask the question what is heat so first question which I am going to ask is what is heat well in physics we know everything we deal with has to do with energy so we talk about mechanical

00:03:09 energy which we learn at length in our mechanics course these had two pieces one is kinetic energy which we talk about and then for interacting systems which we talk about also potential energy if there is a force a particle is subjected to a force we can talk about a potential energy so it is nothing but a form of energy a form of energy okay well how do I know it is a form of energy we know that anything which is related to energy can be obtained from other forms of energy so what do we know for example if I have a rough surface

00:04:16 we're off surface with a friction and then I Drive an object on the surface I push it then I know I generate heat that means the mechanical energy is converted to heat energy okay so energy flows from one end to the other we know heat energy goes from hotter to cooler body this is how it propagates so it is the form of energy and what is temperature temperature is a measure which measure which dictates which direction it flows okay it always flows from higher to lower temperature so when I talk about temperature I know that

00:05:40 temperature should be measured with something called thermometer okay we know of mercury thermometers in our day to day use and we can think of various other kinds of thermometer but here let us try to give some different description of temperature temperature in the sense of kinetic theory of gas kinetic theory of gas I can say temperature capital T I will soon tell you what is capital T capital T is the temperature measured in some scale which I will call absolute scale of temperature this I am going to define

00:06:39 soon so kinetic theory of gas there is a definition of temperature which says that T is proportional to average this word is very very important this word average okay average translational that is also very very important and I will be talking about ideal gaseous mostly it's a kinetic energy so it will be proportional to kinetic energy of the molecules and that is a more well-defined definition of temperature you know when we increase temperature there is always Y bration in the molecules molecules you can go from

00:07:32 solid to liquid and liquid to gaseous in the process they gain more more energy because temperature comes an average kinetic energy of the molecules keep on increasing okay but now comes the question if I define a temperature I can define a thermometer and we all know from our basic ninth 10th standard education that thermometer is a scale where we always talk about ice point which is the lowest ice point which is the rest and steam point let us say steam point which is the highest skill which we know zero degree and 100 degree

00:08:29 Celsius of water these can help me in calibrating my thermometer and this set be scale so we can measure anything with respect to these two temperature in our Celsius scale one is zero degree and other is 100 degree Celsius now coming back to the point what do I mean by average why I am calling it's an average distribution average well firstly I have written down already two things one is kinetic theory of gas and other is thermodynamics of gas now what do I mean by average okay firstly this is first time I will be

00:09:25 dealing with a huge number of particles huge number of particles what I mean that I will be dealing with a system of particles which 10 to the power 23 molecules let's say which is typically the Avogadro number we all know a huge number of particles now 10 to the power 23 what we have learnt over mechanics each molecule described by Newton's laws which is mass of the molecules let's say m and i can write d 2 x dt 2 this force acting on the molecule so this is my Newton's law now

00:10:24 you see this is a vector equation a position vector of the particle has three components now I will be totaling these type number of equations which is impossible for me to handle it I cannot handle with 10 to the power 23 particles I cannot handle with this many second order differential equations as you know we cannot solve this problem by any means even using the most advanced computer what we have today so we must have an average description here comes the advantage and also the difference between kinetic theory so I need an

00:11:17 average description and here comes absolutely the necessity of doing kinetic theory and thermodynamics ok so let me first ask equation what is kinetic theory and what is thermodynamics ok then I will go to the details of other things I will talk about firstly kinetic theory here I will talk about distributions distributions okay I will look at distribution I will look at gas molecules and probe the distribution for example of their velocity or speed though I talk about a molecule here are not really interested in an individual

00:12:34 molecule and its behavior rather I will be interested in the distribution of their velocities from that I can talk about average of velocity from average of velocity which will give me some average of kinetic energy okay and from that I will try to find out what is the temperature I will relate it to the temperature I will relate it to the pressure and volume of the container so here I will be talking about molecular level but please note I talked in terms of molecular level in an average sense

00:13:26 so I need a distribution a distribution of velocity and that distribution of velocity will give me average properties which will be related to kinetic energy for example I will be talking about average kinetic energy these average kinetic energy will be letter related to temperature pressure and volume okay this is the approach of kinetic theory what is the approach of thermodynamic plane well in thermodynamics we really don't care about individual molecules okay I don't care about molecules their velocity distribution

00:14:11 nothing so kinetic theory if I use more formal language it's a microscopic theory microscopic theory at the microscopic level I am trying to see what is happening to the molecules thermodynamics in some sense more formal language is a coarse-grained description coarse-grained description what do I mean by cadre coarse graining Mills I don't look at molecular level rather I look at macroscopic level well I look at macroscopic level okay and the macroscopic level let me erase this part in the macroscopic level I will just

00:15:10 consider the measurable quantities for example measurable quantities all the physics all the equations I am going to write will involve only the measurable quantities namely pressure volume temperature and so you see pressure volume temperature these are the quantities with measure in experiments but there is a difference volume is a quantity which certainly different from pressure and temperature which I am going to say later there are quantities of two types one is called extensive quantities and other called

00:15:56 intensive quantity I just throw the nomenclature at the moment pressure and temperature are intensive variables so they are intensive variables whereas volume is an extensive variable okay now does these two have do these two approaches which I mentioned here do they lead to different set of results No when I talk about these two approaches maybe they are two different approaches but if I am in equilibrium whatever I will talk about is related to equilibrium okay and in equilibrium whatever I measure okay in physics it is

00:16:45 important that whatever we measure should be independent of the theory I construct so therefore in the equilibrium in equilibrium results arrive from kinetic theory or result arrive from thermodynamics they should be same identical realized okay now before I proceed further I should tell you what is equilibrium I will give a very informal definition in the beginning that equilibrium means nothing depends on time okay nothing depends on time I measure pressure I'm is a temperature for example with a

00:17:37 thermometer maybe it is my just medical thermometer clinical thermometer whatever I measure measurable quantities not which depends on time so you can for example you can take liquid in a container and stir it wait sufficiently long okay now everything settles down and you say that system has reached an equilibrium configuration which means no longer my measurable quantities depend on time so in equilibrium if I study kinetic theory will show a different approach for equilibrium maybe thermodynamics will give you

00:18:21 another approach to equilibrium but I will be constantly talking about equilibrium and in equilibrium whichever approach I take kinetic theory or thermodynamics my ideal gas equation PV is equal to NRT if you like PV is equal to NRT whatever n number of moles okay it doesn't change so this is very very important that equilibrium results do not depend upon whatever approach I am going to take now since I have come to ideal gas now question is what is an ideal gas okay first question is what is

00:19:09 an ideal gas this is something which with which we have to spend plenty of time and then it idea should be clear name itself suggest it's not real not real - not real - ok because you soon see the assumptions and it will be clear to you why I am saying that it's not real first it's I will assume there is a point particle so all the ideal gases I assume to be point particles so it's very very tiny or rather more formally one can say that I will assume that size of the particle is very very small

00:20:05 compared to the intermolecular distance okay so that is what I mean by point particles that inter particle distance is very large in compared to the size of the molecule if you like but remember this is an approximation ideal gas approximation second thing is I will assume there is no interaction this is not true we know that whenever we have molecules they are occupying a container there must be an electrostatic force and typical interaction between two molecules if you like it is attractive when they're far apart and they are

00:20:58 repulsive when these two molecules approach each other so you see I cannot assume that they are not at all interacting with each other okay so this is also an approximation that they have no interaction and the energy is then entirely kinetic entirely entirely kinetic okay now force condition this energy is entirely kinetic and they have collisions okay collisions but these are all elastic collisions this is very very important these are elastic collisions no interaction and collisions collisions

00:22:17 conditions which are completely elastic in nature so no energy is dissipated energy is internal it concept now question is that whether I can have this type of ideal gas in nature well point is that I cannot get an ideal gas in nature because these two conditions is never satisfied but if this is a situation now question I am going to pose to you that if there a situation when I can approximate a real gas as an ideal gas well often you will find the answer in the books but I will try to explain to you why in some limiting

00:23:08 situation okay ideal gas can be approximated ideal gas can be approximated as a as a limiting situation okay now this can be very easily thought in terms of comparing different length scales of the problem well i am using a georgian length Kel you will soon see what do I mean by length counts okay firstly inter-atomic distance I told you or intermolecular distance in a gas average intermolecular distance that I can say is a length scale of the problem okay that usually can be found it is related to density if I have a container

00:24:20 of volume V and in that volume I put n molecules so I know V by n typically this to the power is typically on an average this is my intermolecular distance and now I should be able to compare this distance with other length scales of the problem let me bring in something which you will learn later or already you know that if I have a particle of momentum P okay roughly momenta P I know from wave particle duality picture that there is a lambda associated with this which H over P okay now roughly in a very hard waving sense

00:25:31 let us assume that the average kinetic energy of a molecule which I know is of the order of KT then you know P is of the order of root over 2m 80 so you see there is a wavelength or there is a length scale associated with this which is to m80 okay now look at these two length scales this if you do not know you have to just look for de Broglie wavelength what is the bromley wavelength and what i have said that there is a length H over P and typically kinetic theory tells us which I will elaborate on later

00:26:19 that P squared over 2m is of the order of KT so P is of the order of root over 2m KT so this is the other length in the problem so I remind you two lengths one is a other is lambda these two are to be compared if a is much much greater than lambda let's say somehow or the other a is much much greater than lambda all I can say particles interact on existence or intermolicular distance which is an important role okay is very very high in compared to whatever the de Broglie length curve this de Broglie wavelength

00:27:07 tells you that particles are interacting with each other maybe those interaction lengths are interactions may be of quantum mechanical nature let's not go into that I am just constructing two length scales in this one is de Broglie wavelength which is given by H over root over 2m Katie and other is a which is inter atomic distance now you will see the question I have I need to have a much much greater than lambda how do you achieve that you can achieve that in two ways firstly increase the temperature if

00:27:48 you keep on increasing the temperature very high temperature you see your lambda becomes smaller and smaller this quantity out here becomes smaller and smaller okay if this quantity becomes caller this is automatically satisfied a is much much greater than lambda you can say I am just having some isolated particles who do not interact with each other and this is a limiting situation where ideal gas condition could be valid now I leave it for you to think about it that vyn okay you can look at the vyn

00:28:27 quantity this is inverse of density because what is density density typically given by n over V so if I make n by V is my density if density is lower and lower is automatically increasing so once again I can say a is much much greater than lambda if you don't like the concept of lambda you just like how I can make a comparable e large very very large in compared to the other length scales of the problem first increase the temperature or reduce the density that is why in books you often see this statement that ideal gas

00:29:19 approximation ideal gas is a good approximation good approximation when T is high and density is okay this is the region where we can talk about ideal gas systems so I told you what is an ideal gas ideal gas is a limiting situation of a real gas if I talk about very high temperature gas and low density gas and secondly we all know the following things for ideal gas for example keep T constant temperature constant PV is equal to a constant these we know famous as Boyle's lock then if I constitute or have a given substance or

00:30:37 given quantity of gas I can write V is proportional to temperature which is known as Charles law and then this is Boyle's law this is Charles law and all together taken together for any substance okay or any amount of gas I can have this equation PV is equal to n RT R is the ideal gas constant T is temperature and L is the number of moles of the sample I do have okay so now this also helps us in defining the absolute scale of temperature if you keep V constant let's keep V constant P is proportional

00:31:42 to absolute temperature and we all know if I approach T is equal to 0 or in Celsius scale if I call it in Celsius scale roughly minus 273 degrees Celsius this is how my absolute temperature is for example absolute temperature zero is my Celsius scale temperature minus 273 degree Celsius and then if I have 10 which means minus to 63 degree of Celsius and if this is absolute zero that means I have reached in the Celsius scale minus 273 degrees Celsius I will know that pressure of an ideal gas will

00:32:32 go to you so if I take a very high temperature or whatever you like a very low density gas so that it is roughly an ideal gas if I plot P as a function of T at T is equal to 0 if we could reach T is equal to 0 this is very very important we cannot reach T is equal to 0 then pressure would be equal to 0 if I keep volume fixed we know this is unphysical T is equal to 0 is unattainable but why do you need the absolute scale of temperature given in terms of this ideal gas equations because of this fact that it is

00:33:17 universal I am not using mercury I am not using any other particular substance rather I am working on something which is universal does not depend of the material I put in the thermometer so I have a scale which is universal and which is independent of scale I am choosing okay secondly it is always positive temperature now you may ask me question can I have negative absolute temperature well that is not an equilibrium situation in some books you will find they mention very adult book mention negative absolute temperature

00:33:58 but remember here we are dealing only with absolute temperature which is equilibrium situation in equilibrium situation it's always positive okay having this preamble and assume that we know bit of absolute scale of temperature now let me proceed to what I know as kinetic theory of gas okay so kinetic theory of gas definitely as I mentioned to you before it's a microscopic description okay but in an average sense I will explain to you what does it mean that I talk about in an average sense so I wrote few minutes

00:34:49 back what is called a distribution and there is some distribution of velocities so I don't actually talk about an individual molecule rather I talk about a distribution okay so ever it is important again I repeat the charm I write the term average okay so what are the properties I will be talking about molecules are moving in all directions this is the first assumption I will make all directions okay and second assumption as I told you point particles I absolutely ignored any size of the molecules okay which is an

00:35:46 approximation but it is good approximation as long as size is smaller than interact on each distance okay third I will not assume any interaction no electrostatic interaction between the molecules I will not assume any interaction only two interaction I will talk about is collisions so these molecules are caped gas molecules are kept in a container there is a container in which these gas molecules are moving in all random directions okay and only they can exchange energy through collisions collisions between two molecules this

00:36:53 molecule colliding with this molecule or they can collide with the wall of the container this is something very very important wall of the container gives me the pressure which I measure I told you whatever approach I take I take the approach of thermodynamics or I take the approach of kinetic theory I will always be ending up same equation of state same measurable quantities which I get from either of the approach yes the system which I am dealing with is in equilibrium okay so these are main assumptions secondly I will consider

00:37:38 classical classical motion though when I talked about de Broglie wavelength I brought in a little bit of quantum because all of you know that in the microscopic world in the world of electrons if we talk about the world is actually quantum mechanical that's why to tell you to give you an inkling about that I mentioned a bit of de Broglie wavelength that helped us also to tell us the role of temperature in finding an ideal gas that temperature I can define for an ideal gas if it is temperature very very high I can assume

00:38:21 that it's typically an ideal gas so now I will make two assumptions more I will completely talk about classical motion so all these molecules satisfy Newton's laws of motion nothing more than that Newton's laws of motion okay so these elastic collisions which I mentioned here collisions they are all elastic collisions so these elastic collisions are all dictated by the classical mechanics the mechanics we have learnt conservation of energy momentum that is what I am just going to use here and in

00:39:14 the form of Newton's laws of motion okay so classical mechanics on who is the subject on which will build our this kinetic theory cause this is first second thing it's homogeneous that means I will have you density is uniform everywhere I will assume the density wherever I have a container if I am very much inside the container wherever I am the density is independent this is same everywhere if I take same amount of volume number of particles will on an average will be the same in this volume wherever I place it inside

00:40:00 the container this is homogeneity thirdly which is very very important I have a three dimensional system in a three dimensional system I know if I talk about velocities velocities will be having three components react V Y and V that I will consider complete isotropic in today the this is our first thing first discussion our I'm throwing some jargons which will be familiar to you as I proceed complete ISO trophy means that three directions if you like three directions V X V Y and B red they are identical

00:40:46 okay if I measure something in the X direction for velocity okay I will be having same for y and z there is no way I can distinguish that this is in the V X Direction this is V Y direction this is V Z direction now there will be a distribution of velocities which I will call velocity distribution an important point I would like to make that velocity distribution is same and as I am talking about equilibrium is independent of time I told you my definition of equilibrium is nothing depends on time so it same

00:41:38 everywhere like I said density same everywhere the distribution is also same everywhere now since I have been talking without explaining what do I mean by distribution and what is probabilistic about it because I am talking about average you know if I have a dice dices six phases so I can play the dice I can get 1/3 3 1 2 all with equal probability 160 okay so you know there is a probability of getting six out of six in one throw is one-sixth because these events are not correlated 6% doing it six person can get six different results

00:42:28 this can be generalized to a continuous system I will here when you talk about dive you throw a dice you can get one value out of six possible values now question is I have a continuous variable okay which is you can think of if I could construct it dies which had many many faces this probability of getting these things becomes smaller and smaller is a 50 faces I can have 50 values but if it is unbiased it can fall on any side okay you have to extend your imagination to see that if I have a 50 dies with 50

00:43:14 faces so what do we have it can go from 1 to 50 what is the value we get 25th it it it is still 150th but you see probability has become smaller the possibilities had become larger so if I keep on increasing this eventually things become continuous 125th 151st it was 1/6 now if dice somehow has 50 faces its 150 if I could construct a dice with 100 faces okay in real world we cannot imagine because we are living in 3-dimensional world ok if I could then probability would have been 100 so if I increase the possibilities you see the

00:44:00 probability also goes down okay it can take a continuous value let us take such a random variable X ok simple random variable X which can take any value X ok from minus infinity to plus infinity you may ask me that well what is the average value I see this information that X goes from minus infinity to plus infinity it is not sufficient to tell what is the average value we need something what is the probability of having it between X to X plus DX that is very very that is called the distribution okay so

00:44:53 now I want to give you what is the meaning of distribution okay let's say I know P X D s ok what is the probability probability is P X DX it is the probability that X lies between X to X plus D X so you can think of I am doing many many experiments which in the standard example I have given mins I am throwing a dice okay in the standard example I have given which means throwing a dice so now I am asked many many experiments and I am asking the question as you will now X is a continuous variable which can take any

00:45:46 value from minus infinity to plus infinity I am asking the question what is the average value of x and that is given in terms of this probability distribution let us take a very simple example let's say px is of the form e to the power minus Alpha X square okay well and then there will be some constant which is called normalization constant because you know px DX minus infinity to plus infinity minus infinity to plus infinity you are integrating what does it tell you it simply tells you the

00:46:31 total probability and I know that total probability is 1 if you recall your dice problem okay then every probability is 1/6 of any faith I am getting a single throw I am getting 1 1 okay for 2 all with probability 1/6 but total probability you won 6 times 6 is 1 this is all written in a mathematical language okay so these gives me a constant end we can forget about this end for this our time being but what we have to remember what is pxdx okay if you plot this function okay this is called a typical Gaussian

00:47:18 function which goes like this okay now look at this function you ask me what is the probability at X is equal to zero that is given by this number n what is the probability of getting X plus infinity you can see it is exponentially small it is actually 0 alpha is something positive let me tell you alpha is something positive greater than zero okay so if alpha is greater than zero X the Infinity value is zero it's negative infinity value is zero I am plotting P X as a function of X so you can see X

00:47:55 going to positive infinity or of X going to negative infinity these probabilities are basically zero you can ask me what is the probability it lies between X to X plus VX if this is my small interval DX this is the probability okay what is the probability of X is this probability is X plus DX is this this is my x-axis probability of having anything X and X plus DX lies within this region now you may ask me what are you doing because it is after all thermodynamic course why are you talking about this

00:48:34 probability distributions I am just trying to give you the feeling of average why do I need average you have to understand I have 10 to the power 23 particles at my disposal and given 10 to the power 23 particles I cannot write Newton's laws I cannot solve them I have to go to a probabilistic distribution for that probability distribution I need to tell you a bit about probability if you ask me what is X then average value of X now you cannot tell X is this if you write like your dice experiment if

00:49:12 you are done it can take any possible value from minus infinity to plus infinity and you are asking what is the mean value if I do is many many many times okay this will be given by integration X px DX so you are not talking about a value of x now please remember you are not talking about a value of X rather you are talking about an average value of x okay this is a brief introduction to probability theory now I will tell you how it is connected to very brief introduction to probability theory now I

00:49:52 will tell you how it is connected to kinetic theory of gas kinetic theory of gas tells you that I told you that average sense things that to be del to it okay in an average sense so this brings in the concept of Maxwell's velocity distribution okay Maxwell's velocity distribution I just briefly mentioned what is Maxwell's velocity distribution what I have at hand I have at hand ideal gas molecules molecules in a container okay which is a volume V and these molecules are doing random motion velocity components are V

00:50:38 X V Y V said I told you they're independent of each other okay this is far beyond your limit so I just want to mention few things this can be argued this is Maxwell velocity distribution so if you have vxvy visit this can be shown PV x pv y pv z DV X DV y DV read I read out this mathematical expression for you remember container volume is V these are the components of velocity let's say distribution of this velocity if I track a single particle what is the probability of that velocity of that

00:51:49 particle lies between VX plus DV X now this is the deviation from mechanics mechanics I give you a force given initial condition on ask you the question ok I ask you the question what is the velocity after a time T you solve Newton's equation you tell me the answer ok given the initial condition and if it is a solvable force you can do that but here I have already gotten rid of force there is no force ok there is no force in the problem but what I am talking about is a probability distribution of

00:52:26 velocities three components of velocity this is the PV X DV X it tells you what is the probability that velocity lies between VX and VX plus DV X ok so this is a probability distribution this has a form which one can write as up to some constant some constant a VX square V Y square V Z square okay I am not telling you at the moment what is this constant a but you can guess what should be its dimension because you know from dimensional analysis if I write something in the exponential we already have some dimension okay

00:53:24 this a should be something that the whole quantity is dimensionless okay I am not telling you what is this a okay but this should be such that the whole quantity as a whole is a dimensionless quantity and there is also a speed distribution now I do not care about I don't care about the component rather I ask the question what is the probability that speed of the particle lies between okay V - V Plus D V what is V speed V is root over VX square V y square VZ square that is my speed okay now speed distribution similar form P V

00:54:22 DV or you can ask me how many particles are on average having velocity NV whose speed lies between V - V Plus DV I put another constant a e to the power let's say I call it be this time b v square V V I am not telling you what is a what is B but again B can be argued out remembering that exponential minus b v square should be dimensionless okay so this is the distribution this is far advanced but I just wanted to tell you that what is average we often see in books that they people talk about I will

00:55:10 talk about average over some distribution whenever there is a probability there is a probability distribution and if you ask me what is the average speed average speed you can calculate taking the integral of this form here given here so this is the average so we talk about average velocities average kinetic energy but let me ask you a question now I have random motion of the particles okay I am inside a container okay and here I have two boundaries let's say this is my boundary one and this opposite phase is my boundary - now

00:56:00 if it goes randomly with equal probability with velocity VX in the positive direction equal velocity minus VX okay in the negative direction if this is my x axis let's say this is my V x axis if you like it goes like this with VX and comes back again with minus VX what do you expect to the average to be I will tell this answer later I just don't want to tell you but it's obvious since particles are moving in the equal and opposite direction with equal probability there is nothing that

00:56:44 distinguishes plus VX and minus VX if you look at this expression it's given in terms of VX square so there is a equal probability of having VX and minus VX because it is given in terms of VX square probability is given in terms of VX square so average you would expect to be 0 and that is indeed decay okay now we have next what I will do in the next set of lectures I will put ideal gas in a container and try to form an equation state which is PV is equal to one-third MN let's say v square average I will

00:57:39 explain the meaning in the next class this v square average is actually averaging in this sense so what I told you today that there are two approaches to study the thermal properties or equation of state of a gas or for that matter any system one is kinetic theory approach where you go microscopic other is thermodynamic approach in equilibrium the results would be similar I have started with the kinetic theory approach I have tried to tell you what is a probability distribution how do you talk about averages okay and the next lecture

00:58:23 which I am going to say click I will go to the details today's were more like preamble and introduction tomorrow's or whenever our next meeting is I will tell you about the equation of state already it is very close to what we know about PV is equal to NRT can we see that from here I told you in the beginning that temperature is related to the average kinetic energy can we see it from here how do I get this form at all what is pressure pressure whose pressure particles are hitting the walls of the

00:59:00 container and that gives me a force distribution there is a momentum transfer this momentum transfer will be related to pressure and I will arrive at this form in the next set of lectures thank you for [Music]