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# PV diagrams - part 2: Isothermal, isometric, adiabatic processes

## Video transcript

all right so last time we talked about isobaric processes this time let's talk about isothermal processes ISO means constant thermal this is short for temperature so this is a process where the temperature remains constant or in other words T equals a constant which we could also write if temperature is constant that means the change in the temperature means there is no change in the temperature so the change in the temperature is just zero and before we move on let me show you one more important thing remember we said previously the internal energy of a gas is not equal to but it's proportional to the temperature of a gas and so if the temperature doubles the internal energy doubles if the temperature doesn't change the internal energy doesn't change so for an isothermal process not only is delta T equal to zero but more importantly in terms of the first law of thermodynamics Delta U is also equal to zero this is important this is something you have to know for an isothermal process Delta U is zero now you might be confused here you might say hold on a minute how can you have a thermal process if there's no thermal if there's no change in the temperature at all well you can stuff is going to happen here this doesn't mean nothing happens things are going to happen but they're going to happen in such a way that there's no change in the temperature and there's no change in the internal energy so what can we say well let's look at the first law of thermodynamics the first law holds for any thermal process in here whether it's an isotherm and isobar any of them so we can say that Delta U has got to equal Q which is the heat that flows into a gas plus W which is the work done on the gas and now we know for an isothermal process the Delta U is just zero what does that mean that means that Q plus W have to add up to zero so this means if you do say 300 joules of work by pushing this down you do 300 joules of work the only way the temperature's going to remain constant is for 300 joules of heat to leave the gas 300 joules of heat would have to leave that would mean the Q is a negative 300 joules that way 300 joules and negative 300 joules add up to zero you've got an isothermal process but it's not enough for just the initial temperature to equal the final template in order for this process to be truly isothermal the temperature has to remain the same at every moment during the process so every bit of energy you add has to immediately get taken away or every bit of energy you take away has to immediately get added back in there can't be a delay otherwise you'd add this 300 joules the temperature the gas would increase and then the heat would conduct out of here you know at its leisure take some time and then finally you'd reach the same temperature as before that doesn't count that's not an isothermal process the gas has to be at the same temperature at all moments so how do you do this well just make sure you push down the piston or if you're pulling it up you do so very slowly that way the heat always has time to conduct out or in accordingly if you make the process happen too fast this heat you know takes some time for heat to conduct through a container if you make the process happen too fast this heat can't conduct out of the container or into the container fast enough so you've got to make the process happen very slow add one Joule per hour per day make it as slow as possible so that this heat always has time to conduct accordingly and maintain a constant temperature constant temperature with what well just stick this whole thing in a thermal reservoir this is how you could actually do it put the whole container in a tank of water that has a temperature of say 290 Kelvin a huge tank of water water doesn't change its temperature very easily since it has such a high specific heat so if the tank is very large this water is going to maintain the same temperature it's not going to care about a little piston in here but the gas in the piston is going to try to maintain equilibrium with the temperature of the water so if you make this process happen very slow if I push down the piston very slow I'll add energy but that Energy's got to get taken out the temperature of the gas will remain the same if I do it slow enough or I can pull up on the piston very slowly then some heat has to enter into the gas so that it always maintains the same temperature with the outside environment ensuring that it's an isothermal process so what's an isothermal process look like on a PV diagram well let's look at the ideal gas law the ideal gas law says that P times V equals n KT at least the Boltzmann's constant version of it does so I wanted to know what the pressure is as a function of volume let me just solve for pressure pressure equals n K T over V now look at n number of molecules in here that's a constant we're not letting any gas molecules in or out k that's Boltzmann's constant that number doesn't change for an isothermal process temperature is also a constant everything up here in the numerator is a constant and we just have P as a function is constant over V so P just goes like 1 over X it's like having a function y equals a constant over X and we know what 1 over X looks like looks like that so on a PV diagram an isothermal process is going to look something like this it's going to curb like 1 over X and it can be an isothermal expansion if volume increases or an isothermal compression if volume decreases so the actual shape of the line drawn on a PV diagram for an isothermal process is sometimes called an isotherm and they look like that notice that we cannot find the work done by just saying work is P Delta V remember that's how we found the work done by the gas in an isobaric process but that was because we had a nice rectangle the area underneath this graph is still going to give us the work done that's true this definitely is the work done by the gas but it's not a perfect rectangle so you can't use this formula you'll have to know you have to be given the heat and then you can figure out the work or given the work you'll find the heat there's not a really good way unless you're going to do calculus to figure out the area underneath this curve one more thing that you should definitely know because the NKT is a constant right all of this stuff is not changing for an isothermal process that means P times V is also not changing that's another thing that doesn't change so T doesn't change you the internal energy doesn't change and P times V does not change as well because T isn't changing over here in the ideal gas law that means if you take the pressure times the volume at any point along this isotherm you'll get the same number so this volume here and this pressure right here if I take those two and I'm old apply those two together I'll get some number and if I take the final one this volume and this pressure and I multiply those two numbers I'll get the same number I'll get the same result for P times V I'll get the same result here if I take these two any P times V value along this line is going to be the same because that number can't change because if it did that mean the temperature had to be changing then you wouldn't have an isotherm so that's the isothermal process that's one of the four most common thermal processes we've got two more to go let's talk about the isometric process the first thing you should know is this is sometimes called isochoric and it's also sometimes called iso volumetric why does it have three names I don't know but they all mean the same thing ISO means constant volumetric & korek and metric all refer to size or volume this means constant volume how do you make sure that happens we'll just don't let the piston move the piston is the thing that regulates the volume well that thing shut I don't know keep that thing from moving and you'll have no matter what else happens an isometric isochoric isovolumetric process which all mean the same thing now since the piston can't move that means no work can be done the gas can't do any work the outside forces can't do any work you can't do any work on the gas no work can be done if this piston cannot be moved up or down so W is always going to equal zero for one of these isometric processes that means if we write the first law the first law of thermodynamics is true for every process says that Delta U equals Q the amount of heat that flows into or out of the gas plus W except W is zero so we have no work done this is zero and our first law just becomes Delta U equals Q R in other words for an isometric process the only way to change the internal energy would be to add heat or to take heat away so these isometric processes are actually pretty simple what do they look like on a PV diagram well the volume is staying constant pressure staying constant as a horizontal line so volume staying constant is a vertical line and if I add Heat I'll increase the pressure and if I take heat away I'll decrease the pressure and this volume will remain the same because this piston is not allowed to move now remember that work is the area underneath the curve does that make sense over here how much area is underneath this curve there's no area underneath this curve there's no air there's you've got this line here that's not an area that's infinitesimally thin and so that means there's no area no area means no work is done and that agrees with what we know about an isometric process all right one more of the big four processes to go let's talk about the adiabatic process this is one in which no heat is exchanged so sometimes people hear that and they think oh that means that there's no change in the temperature right no that is not right this is definitely not what we're saying no heat exchanged means that Q our letter that we use to represent the heat is zero it means that no heat is allowed into the gas no heat is allowed to flow out of the gas these do not happen for an adiabatic process and that does not mean that the temperature can't change the temperature can change here because the piston can do work or the work can be done by the gas but no heat can flow in or out so you've got to get good at delineating between the temperature and the heat these are not the same thing temperature is kind of a measure of how much energy a gas has at a given moment Q the heat is how much thermal energy is flowing into that gas or out of that gas it doesn't represent how much energy the gas actually has it's how much thermal energy you're adding taking away and for an adiabatic process there is no thermal energy conducted in or out what does that mean for the first law first law is true for every process Delta U equals Q plus W the work done on the gas but for an adiabatic process there is no heat so that just means Delta U equals the work done on the gas that's the only way you're going to add energy to the gas is by doing work on the gas or allowing the gas to do work then energy can be removed but you can't add or take away energy thermally conductive Li through the walls of the container it has to be done by the piston so how do you do this how do you make sure no heat gets conducted one thing you should do is insulate this so that the heat does not easily conduct through the walls of this container that's not really good enough you've got to make sure no heat is exchanged so you take this piston and you shove it down as fast as you can or you lift it up as fast as you can it's the opposite of an isothermal process there we wanted the process to happen slow so that the heat always had time to flow in or out here we want the process to happen so fast that the heat has no time to flow in or out that way we ensure that it's an adiabatic process and the queue is actually zero so what does an adiabatic process look like on a PV diagram it looks kind of like an isothermal process it's just steeper so this would be an adiabatic expansion and these lines are sometimes called adiabat and if you have an adiabatic compression it would look like that if you compare that to an isothermal process say that started here it would not get as far down you can tell that that's an isothermal process because it's not as steep so those are the four most common thermal processes you'll hear about when talking about PV diagrams and each of them had something unique and special about them the isobaric process had constant pressure and you could find the work easily because it was a nice rectangle which meant you could just do height times width to get the work done by the gas there's the isothermal process where temperature is constant internal energy is constant and the quantity P times V pressure times volume is also constant there's the isometric process also known as isochoric or isovolumetric where the change in volume is zero which meant remember that means no work can be done the work was also zero for an isometric process and then there's the adiabatic process where no heat is allowed to flow into or out of the system
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