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# Heat capacity at constant volume and pressure

## Video transcript

imagine you had a monatomic ideal gas in the cylinder here and there was this tightly fitted piston above it that prevented any gas from getting out well we know that the total internal energy for a monatomic ideal gas is just three-halves P times V or three-halves and Katie or three-halves little n RT and we know that saying you internal the internal energy is really just code for the total kinetic energy of the monatomic ideal gas that these are the same thing when we talk about internal energy we're talking about how fast are these particles moving in other words what's the total kinetic energy of all of these particles added up my question is how do we go about changing the internal energy let's say we wanted to increase the total kinetic energy what could we do well you could say increase the pressure or the volume or the temperature yeah but I mean physically actually in the lab what do we do and there's basically two ways to change the internal energy if you want to add internal energy ie get these particles moving faster we can heat it up so put this above a flame or on a hot plate and heat will flow into the gas which will cause these particles to move faster and faster that's one way to do it to add heat the other way to do it is to do work on the gas I could take this piston and push it down and if you push this down hard enough it'll squash this gas together and those impacts with this piston while it's moving down will cause them to start moving faster and faster that will also add internal energy to the gas so if we wanted to write down a formula that told us how you could get a change in the internal energy if I want to change the internal energy Delta U which is really just saying changing the kinetic energy well there's two ways to do it I can add heat so if I added 10 joules of heat I'd add 10 joules to the internal energy but I've also got to take into account this work being done and so I could do plus the work done on the gas and that's it this is actually the first law of thermodynamics it's a law of conservation of energy it says there's only two ways to add internal energy to a gas let me talk a little bit more about this work done though because getting the sign rights important if you're doing work on the gas compressing it you're adding energy to the gas but if you let the gas push up on the piston and this gas expands pushing the piston up then the gas is doing work that's energy leaving the system so if the gas does work you have to subtract work done by the gas if the outside force does work on the gas you add that to the internal energy so you've got to pay attention to which way the energy is flowing work done on gas energy goes in work done by the gas energy goes out and you'd have to subtract that over here let's say the gas did expand let's say the gas in here was under so much pressure that the force it exerted on this piston was enough to push that piston upward by a certain amount so let's say that started right here and it went up to here so that piston went from here to there no gas is escaping because this is tightly fitting but the gas was able to push it up a certain distance D how much work was done you know the definition of work work is defined to be the force times the distance through which that force was applied so the work that the gas did was F times D but we want this to be in terms of thermal quantities like pressure and volume and temperature so what could we do we can say that this volume not only did the piston raise up that there was an extra volume generated within here that I'm going to call Delta V and I know that this Delta V has got to equal the area of the piston times the distance through which that piston moved because this height times that area gives me this volume right in here why am i doing this because look I can write D as equal to Delta V over a and I can take this and I can substitute this formula for D into here and something magical happens I'll get work equals F times Delta V over a but look F over a we know what F over a is that's pressure so I get that the work done by the gas is the pressure times Delta V this is an equation that I like because it's in terms of quantities that we're already dealing with so work done you can figure out by taking P times Delta V but strictly speaking this is only true if this pressure remained constant right if the pressure was changing then what am I supposed to plug in here the initial pressure the final pressure if the pressure staying constant this gives you an exact way to find the work done you might object and say wait how's it possible for a gas to expand and remain at the same pressure well you basically have to heat it up while the gas expands that allows the pressure to remain constant as the gas expands and now we're finally ready to talk about heat capacity so let's get rid of this and heat capacity is defined to be imagine you had a certain amount of heat being added so a certain amount of heat gets added to your gas how much does the temperature increase that's what the heat capacity tells you so capital C is heat capacity and it's a defined B the amount of heat that you've added to the gas divided by the amount of change in the temperature of that gas and actually something you'll hear about often is the molar heat capacity which is actually divided by an extra n here so instead of Q over delta T it's Q over and the number of moles times delta T pretty simple but think about it if we had a piston in here are we going to allow that piston to move while we add the heat or are we not going to allow the piston to move there's different ways that this can happen and because of that there's different heat capacities if we don't allow this piston to move if we weld this thing shut so it can't move we've got heat capacity at constant volume and if we do allow this piston to move freely while we add the heat so that the pressure inside of here remains constant we'd have the heat capacity at constant pressure and these are similar but different and they're related and we can figure them out so let's clear this away let's get up nice here we go two Pistons inside of cylinders we'll put a piston in here but I'm going to weld this one shut this one can't move we'll have another one over here they can move freely over on this side we'll have the definition of heat capacity regular heat capacity is the amount heat you add / the change in temperature that you get so on this side we're adding heat let's say heat goes in but the piston does not move and so the gas in here is stuck it can't move no work can be done since this piston can't move external forces can't do work on the gas and the gas can't do work and allow energy to leave q is the only thing adding energy into the system or in other words we've got heat capacity at constant volume is going to equal well remember the first law of thermodynamics said that Delta U the only way to add internal energy or take it away is that you can add or subtract heat and you can do work on the gas in this case Q if I subtract W from both sides I'd get Delta U minus W over delta T but since we're not allowing this piston to move the work done has got to be zero so there's no work done at all so the heat capacity at constant volume is going to be Delta U over delta T what's Delta U let's just assume this is a monatomic ideal gas if it's monatomic we've got a formula for this Delta U is just three-halves P times V over delta T that's not the only way I can write it remember I can also write it as 3 halves + K delta T over delta T and something magical happens check it out the Delta T's go away and you get that this is a constant that the heat capacity for any monatomic ideal gas is just going to be three-halves Capital n K Boltzmann constant and is the total number of molecules or you could have rewrote this as little an R delta T the T's would still have canceled and you would have got three-halves little n the number of moles times R the gas constant so the heat capacity at constant volume for any monatomic ideal gas is just three-halves NR and if you wanted the molar heat capacity remember that's just divided by an extra mole here so everything gets divided by moles everywhere divided by moles that just cancels the Seau and the molar heat capacity at constant volume is just three-halves r so that's heat capacity at constant volume what about heat capacity at constant pressure now we're going to look at this side again we're going to allow this gas to have heat enter the cylinder but we're going to allow this piston to move up while it does that so that the pressure inside of here remains constant and this is going to be the heat capacity at constant pressure well again we're going to get that it's Q over delta T and just like the first law said Q has to equal Delta u minus W so we get Delta u minus W the work done over delta T this time W is not zero what's W going to be remember W is P times Delta V so this is a way we can find the work done by the gas P times Delta V so this is going to equal Delta U we know that is if this is again a monatomic ideal gas this is going to equal three halves n our delta T plus this is P times Delta V but we have to be careful in this formula this work is referring to work done on the gas but in this case work is being done by the gas so I need another negative technically the work done on the gas would be a negative amount of this since energy is leaving the system so that negative cancels this negative and I get plus P times Delta V all of that over delta T so what do we get three halves n R delta T plus I want to rewrite P times Delta V but I know how to do that the ideal gas law says PV equals NRT well if that's true then P times Delta V is going to equal an R delta T so I can rewrite this as an R delta T divided by delta T almost there all the Delta T's go away and the go to I'm left with I'm left with C heat capacity at constant pressure is going to be equal to three-halves NR plus NR that's just five halves and our and if I wanted the molar heat capacity again I could divide everything everything around here by little n and that would just give me the molar heat capacity constant pressure would be five halves are and notice they're almost the same the heat capacity at constant volume is three halves n R and the heat capacity at constant pressure is five halves n R they just differ by n R so the difference between the heat capacity at constant volume which is three halves and R and the heat capacity at constant pressure which is five halves n R is just CP minus CV which is n R just n R and if you wanted to take the difference between the molar heat capacities at constant volume and pressure it would just be R the difference would just be R because everything would get divided by the number of moles so there's a relationship an important relationship that tells you the difference between the heat capacity at constant pressure and the heat capacity at constant volume