Thermodynamics - A Level Physics

hello today we are continuing in our a level physics revision series this time looking at thermodynamics and thermal physics if you take an ideal gas and we'll consider what one of those is in a moment and for a given temperature at constant temperature you plot the pressure of that gas against the volume you will get a curve like that which satisfies the equation that the pressure times the volume is a constant which for the moment will simply call C if we were to plot pressure against one divided by the volume then we would get a straight line which suggests that pressure is proportional to 1 over volume remember that this is at constant temperature that's called Boyle's law now there's another law called Charles law Charles law relates volume to temperature and if you plot volume against temperature you get a straight line which means that volume is directly proportional to temperature but if we extend the axis this way and then plot backwards we find that this line crosses at this point here which is approximately minus 273 degrees Celsius and at that point you can see that the volume of the gas is now 0 and it clearly can't go any further it cannot become a negative volume so that's as far down as it can go and we call that point absolute zero and that is zero in the Kelvin scale so zero degrees Kelvin equals minus 273 degrees Celsius but every degree Kelvin is equal to one degree centigrade in other words the divisions on the scale are the same but zero Kelvin is minus 273 degrees Celsius you might just notice a convention here when we're talking about degrees Kelvin we don't put the little degree signal in here whereas when we're talking about degrees Celsius we do put that little degree symbol in there now finally we have a pressure law and the pressure law relates pressure to temperature and once again it's a straight line and once again if you project it backwards you find that the pressure becomes zero at minus 273 degrees Celsius which is zero degrees Kelvin and that tells us that pressure is proportional to temperature so we have that pressure is proportional to one over volume volume is proportional to temperature and pressure is proportional to temperature and if we put all of those together and combine them we get the pressure times volume over temperature is a constant which I'll call C for the time being but that constant is obviously going to depend on how much gas you actually have so let's consider one mole of the gas now a mole is simply the atomic weight of the gas expressed in grams so for example hydrogen the atomic weight is one so one mole of hydrogen is 1 gram of gas then we would say that PV over T equals R for one mole of gas and R is the molar gas constant and its value is 8.3 one gosh 8.31 joules per mole per degree Kelvin if you have n moles of the gas in PV over T will equal n which is the number of moles times R which is the molar gas constant and if you rearrange that formula you get that PV equals n R T and that as I've said before applies to an ideal gas so what is an ideal gas well it has several qualities first there must be a large number of particles or molecules in the gas second they must be moving rapidly and randomly third their motion follows Newton's laws of motion fourth all the collisions between the molecules are perfectly elastic no energy is lost fifth there's no attractive forces between the particles 6 all the forces apply instantaneously and 7th the particles have negligible volume compared with the volume of the gas that's quite a hefty set of constraints but in practice most gases behave like an ideal gas if two things happen one the pressure is not too high and 2 the temperature is much higher than the boiling point of the gas temperature scales tend to have fixed points such as the centigrade or Celsius scale has zero degrees for the freezing point of water and 100 degrees for the boiling point but these aren't actually temperature dependent is there a way to come up with a fixed value that doesn't depend on pressure and the answer is that there is and the Kelvin scale has two fixed points zero degrees Kelvin is the point we established before it's when volume or pressure is reduced to zero the second fixed point is what's called the triple point of water which is 273 point one six degrees Kelvin it is the one only place where ice and steam or sit together harmoniously they are in equilibrium now clearly that doesn't happen at room atmospheric pressure because ice and water are at naught degrees and water and steam are at a hundred degrees but if you reduce the pressure sufficiently there comes a point where ice water and steam all coexist and that's called the triple point of water and it's 273 point one six degrees Kelvin now the speed of a particle the velocity of a particle which of course is also determines the energy because the energy is 1/2 MV squared that's the kinetic energy of particle is proportional to the temperature the higher the temperature the higher the velocity that doesn't mean of course that all the molecules in a gas are traveling at exactly the same velocity if you were to plot the number of particles in the gas compared with their velocity then for one temperature let's say this is 200 degrees Kelvin you would find that there would be a distribution of velocities but this would be the average velocity for 200 degrees Kelvin if the temperature goes up to say 500 degrees Kelvin then you would find that the distribution spreads out like this and the average has gone up you will notice that as the average temperature increases so the average speed increases the maximum particle speed increases and the distribution curve spreads out this is more spread out at 500 degrees than this is a 200 degrees we must remember the particles or molecules in the gas are colliding all the time and therefore they're transferring energy from one to another but the total energy of all the particles will remain the same for a given temperature now let's consider a box offside it's a cube of side L and it has n capital n molecules not moles but molecules of a gas inside it what is the pressure on let's say this face of the cube well pressure we know is force divided by area and force is the rate of change of momentum P is momentum well momentum is mass times velocity so we can say that that is d MV by DT and since the mass doesn't change that is M DV by DT and DV by DT of course is simply the acceleration so force is mass times acceleration which is Newton's second law now let's make an assumption which is not as daft as it sounds that the molecules in the Box are moving in only three directions a third of them are going left to right a third of them are going up and down and a third of them are going back and forth and although you will say that the molecules will be traveling in all directions and that is perfectly true the fact is that you can always resolve a velocity into one of these three directions they are the coordinates of a velocity you can resolve them into those coordinates and let's further assume that all of those molecules are traveling at an average speed V in fact to be technically true what we should say is that it is the root mean squared speed to get the root mean square speed you take all the sum at all the speeds and you square them and then you add them all together so that's the sum of the squares and then you take the square root and that is the root mean square but I shall use the general term average speed or average velocity now let's consider one molecule which is traveling from left to right and it hits this wall and as it does so it is traveling in this direction at V with mass M so it has momentum MV and when it hits the wall it bounces back still with same mass but now with velocity in the other direction what is the change in momentum the change in momentum DP is 2 MV because it was going in MV in this direction now it's doing MV in that direction the total change is 2mv how often will that molecule hit that wall well it hits the wall it rebounds it then travels all the way over to this wall it rebounds again and it comes back and when it does that it has travelled a distance of 2l there and back so its distance is 2l but it's traveling at speed V and therefore the time to travel there and back the distance to L is 2l divided by V and now we can write down that DP by DT which is the rate of change of momentum is DP which is 2 MV divided by DT which is 2 L sorry 2 L over V and that comes up there and becomes V squared and that equals MV squared divided by L per molecule so the force which is DP by DT is equal to MV squared divided by L per molecule but what we said was that there were total of n molecules in the box and a third of them were traveling from - right and that means that the total force exerted by all the molecules on this surface is going to be equal to the N divided by three which is the total number of molecules times the force of any given molecule and that is the force on this surface but pressure is force over area and the force we know is n over three MV squared over L divided by the area well what's the area the area of this wall is L times L l squared and that gives you n MV squared divided by three L cubed but what is L cubed L cubed is simply the volume of this box so we can replace L cubed by volume and that means that P equals n MV squared divided by 3 V capital V now where V stands for volume and rearranging that we can say that PV by taking the V up here is equal to n M V squared divided by 3 but we've already shown that PV is n RT where n is the number of moles R is the molar gas constant and T is the temperature and if we rearrange this equation here we get that 1/2 MV squared which of course is the energy equals 3 over 2 n over n R T where n is the number of moles n is the number of molecules and R is the molar gas constant now it was a begad row whose that one mole of gas has the same number of particles it doesn't really matter what gas it is if you take one mole of it you will have the same number of particles and that's Avogadro's number and that is 6.02 times 10 to the 23 particles per mole and that's often given n a so now we can write that the total number of molecules in the box is equal to the total number of moles times the Avogadro number n ay and that means that n divided by n this term here is equal to 1 divided by the Avogadro number in a so now we can write that the energy which is 1/2 MV squared here equals 3 over 2 n over N is now 1 over na R T and since R is a constant and in MA is a constant we can define R over na as a new constant which we shall call K and K is called the Boltzmann constant and its value is one point three eight times ten to the minus twenty-three joules per Kelvin and what we can say is that R is the gas constant for one mole of gas whereas k is the gas constant for one particle or one molecule of gas we're now going to look at thermodynamics there are four laws of thermodynamics which can be expressed in varying ways I'm going to keep them simple today the four laws are zero one two three the reason there is a zeroth law is that the first three were already formulated when someone realized that there was a necessary law before those three which had to be called the zeroth law the reason was that the three laws of thermodynamics makes refer to temperature and then somebody realized that no one had defined temperature so consequently the first law which is called the zeroth law defines temperature the first law says that heat is work and work is heat the second law says that heat cannot of itself flow from one body to a hotter body you can't get heat to move to a hotter body it doesn't flow upstairs and the third law essentially says that you can never get to absolute zero you can never get to zero degrees Kelvin the first law which defines temperature is often described in this way let us take a body which we'll call body a and that is at a certain temperature now we'll take another body B which is another temperature and what we say is that if we bring these into thermal contact such that heat can flow he will always flow from the hotter body to the cooler body never the other way around and it will continue to do that until those two bodies are what are called in thermal equilibrium which is kind of another way of saying they're at the same temperature what the first law says sorry the zeroth law says is that if two bodies are in thermal equilibrium that isn't heat no longer flows between a and B and if there's another body C which is also in thermal equilibrium with a that is to say a and C at the same temperature and a and B are at the same temperature then it must follow that B and C are at the same temperature and hence in thermal equilibrium there was a very well-known song in the 1960s by a popular singing duo called Flanders and Swann they wrote songs about the most unlikeliest of subjects and one of them was about thermodynamics and their great song was that heat is work and work is heat heat is work and work is heat heat cannot of itself flow from one body to a hotter body and those are simply just the two expressions of the first and second of thermodynamics let's look at the first law according to Flanders and so on but actually to physicists before that the total amount of energy or rather we talk about energy change since you can never be sure precisely how much energy you've got so we call that Delta U which is the energy change in a gas is equal to the total amount of heat put in plus the total amount of work done we always use positive terms when we're putting heat or work into a system and we use negative terms if we're getting heat or work out of a system so just to recap Delta U is the change in total energy of a system Delta Q is the change in the heat going into the system and Delta W is the change in work being done to the system what does this mean well let's take a container into which we will put a piston or a plunger that's free to move up and down but nothing can escape outside the plunger and we're going to have a gas inside that plunger now we might press down on the plunger and squash the gas what we are doing is we are doing work and consequently there's no heat going in but the energy of that gas will increase because we're putting work in alternatively we could take exactly the same arrangement here's the plunger here's the gas but instead of pushing it down or pulling it up we put a flame underneath to heat it up we're now putting heat in Delta Q we're not doing any work but heat in if that plunger doesn't move then what will happen is that the gas will heat up and his energy will increase so Delta U will equal Delta Q the heat in now there are different ways in which all this can happen different ways in which energy can change there's a process called adiabatic which means that there is no heat in and no heat out but the temperature inside the system can still change the way this usually works is that you take a system here and you surround it by some kind of insulating material such that no heat can get in and no heat can get out and that's called a dive attic the second type of change is what's called isothermal the temperature remains constant and the way this is done is that the system is the system whatever it may be is placed in what's called a heat bath this is usually a fluid of constant temperature that you maintain a constant temperature and it's got so much heat in that heat bath that it can ensure that the temperature of this system remains absolutely constant and that's called isothermal and the third type of change is what's called constant volume or in some cases constant pressure where you make the changes but you ensure either that the volume doesn't change that is to say that stays the same volume or the pressure acting on that system doesn't change let's just consider an example of three changes we're gonna plot pressure against volume and what we're going to do in these three examples is we're going to take a gas that has a volume v1 and we're going to take it down to a lower volume v2 and we're going to start with the pressure at that point and the volume v1 so our starting point is a pressure with a volume v1 now there are three ways that we can get down to volume v1 that's where we've got to get to the first is the adiabatic and that will look like a curve like that the second is isothermal and that will look like a curve like that and the third is constant pressure and if constant pressure means that pressure doesn't change so it will simply go like that now what does that actually mean how can we achieve this if we're going to reduce the volume but not change the pressure the only way we can do that is to reduce the temperature so for path three the temperature goes down for path two isothermal that's a constant temperature T is constant for path that's path to path one is the adiabatic change and in that case the temperature is going to go up so essentially you achieve these three different paths depending on what happens to the temperature in this case the temperature must go down you actually physically have to reduce the temperature in order to get the volume to decrease here this is just a straight volume change with a consequent pressure increase but all at constant temperature and here you've got a volume decrease a pressure increase but also a temperature increase and that's a diabetic we can now consider what's called a heat engine and it's sometimes also called a Carnot cycle once again we're going to plot pressure against volume and we're going to go through four points each of which as you will see has a distinctive pressure and volume pressure and volume pressure and volume pressure and volume and the path we're going to take goes something like that this path is isothermal which means that there is no temperature change this path is adiabatic which means that there is no heat in no heat out this path is isothermal again constant temperature and this path is a diabetic which means no heat in no heat out so you can by the same process that we achieved these changes you can achieve these changes isothermal constant temperature a diabetic no heat in no heat out and you can go right the way around until you come back to where you've started and the area within this curve is equal to the work done and that's either by the system you might get work out or to the system you might have to put work in now let's go back and look at this term that we derived earlier PV equals n K T where n is the total number of molecules K is the Boltzmann constant and T is the temperature this term PV is in fact an energy term we can demonstrate that fairly easily what is pressure pressure is force over area but what is area well area is just a distance squared what is volume well volume is a distance cubed and so pressure times volume is Force Times distance but what is force times distance that equals work or energy so PV is an energy term and what we can derive from this is that PV which is energy equals KT for a single molecule because it's MKT for n molecules so it's KT for a single molecule and this now relates the energy of the gas to its temperature it shows why as the temperature increases the energy increases and it also means that we don't really need a temperature scale we could in fact describe all temperatures in terms of energy simply by multiplying the temperature which of course has to be in degrees Kelvin by the Boltzmann that does not of course mean that every single molecule in a gas will have an energy equal to KT if you look at the number of molecules in a gas and you look at their energy then you're probably going to find that there will be a distribution some molecules will have a much higher energy some will have a much lower energy because they're constantly colliding with one another and transferring energy but the average is going to be KT now let's think about particles in matter they are held together by some kind of bonds we don't need to work out exactly what kind of bonds at this stage simply to say that particles that are in a solid are obviously very tightly held together particles that are in liquid are much more loosely held together and particles that are in a gas are barely held together at all so in other words the bonds are much tougher and tighter in a solid than they are in a liquid and they are tighter and liquid than they are in a gas so what do you need to do if you want to change the state of a solid to a liquid let's say you want to take ice and you want to melt it the ice will have very tough bonds between the particles and molecules and they need to be broken and how do you break bonds you have to put in energy and we've just shown that the energy equals K T so you might think that you can break these bonds by having an energy which equals KT and that energy will be the energy that's capable of breaking the bonds let's say that the energy needed to break a bond is epsilon then what we would be saying is epsilon equals KT the bonds get broken but actually that's not quite true for the reason that I just showed in the graph that if you pop the number of molecules against their energy you get a distribution and the average distribution I said was KT but way up here somewhere right towards the our end there might be molecules that had the energy epsilon so the average doesn't necessarily have to be epsilon if some of the molecules have an energy epsilon then they will have the capade capacity of breaking those bonds so in fact the bonds can start to be broken when the average energy is very much lower than epsilon as long as there are some fast moving molecules that have sufficient energy to break the bonds and what typically happens is that if epsilon is equal to about 15 times KT or if epsilon divided by kt is equal to 15 then the process of bond breaking will begin and that enables us to talk about what's called the Boltzmann factor which is the exponential of minus epsilon over KT and what we said was that if epsilon over KT is 15 then the process will begin to begin to start which means that if e to the minus 15 because epsilon over KT is 15 equals approximately 10 to the minus 7 and what that means is that if 1 in 10 to the 7 particles has enough energy to overcome the what's called activation energy or the bond energy to break the bonds then that process will begin there are of course millions of particles millions of molecules and consequently if just some of them have enough energy in those bonds can be broken broken and the rate of the reaction is given by this formula here finally I want to look at the subject of specific heat specific heat is energy and it's the energy required to make the temperature increase or indeed decrease you put energy in to get the temperature to go up you take energy out to get the temperature to go down and specifically the specific heat is the energy or the heat needed to raise the temperature by one degree Celsius or Kelvin is the same per kilogram of the material so if you've got one kilogram and you want to raise the materials temperature by one degree then you have to put in that amount of energy and that's the specific heat so essentially the specific heat or energy is equal to the mass in kilograms times the specific heat times the change in temperature that you want to achieve the specific heat of water for example is four one eight zero joules per kilogram per degree Kelvin so if you want to raise one kilogram of water by ten degrees you need to put in ten times this amount in other words the total energy or the total heat that you need to put in is the mass which is one kilogram times the specific heat which is four one eight zero times the temperature change which is ten so you need to put in forty one thousand eight hundred joules of energy or heat in order to raise one kilogram of water through ten degrees now that's the specific heat needed to take a product and increase its temperature without changing its state suppose you have ice and you want to convert it into water then that's called the specific heat of fusion and to take one kilogram of ice and to convert it into one kilogram of water both at the same temperature both at zero degrees Celsius you need 80 times that specific heat you need 80 times to get one kilogram of ice changed to water no temperature increase at all it's just that's the amount of any to break the bonds to turn ice into water and to turn water into steam you need approximately 500 times this amount now remember this is the energy you need to take one kilogram through one degree Celsius or one degree Kelvin if you took water from naught degrees to a hundred degrees Celsius that is from freezing point to boiling point but without it changing state in other words it remains water all the time you'd need to use a hundred times this amount of energy once you got it to a hundred degrees centigrade if you want the water to boil you have to put in 500 times this amount of energy which means you're putting in five times as much energy just to get the water to change from water to steam as you had to put in to get the water to go from freezing point to boiling point

Channel: DrPhysicsA

Views: 192,336

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Keywords: Level, Thermal, Physics, Boyle's, Charles', Pressure, Laws, Thermodynamics, Specific, Heat

Id: JOs8UQSWmos

Length: 36min 33sec (2193 seconds)

Published: Thu Mar 15 2012