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Thermal expansion in gas

How to quantify thermal expansion coefficient for gases? Gases are little different than solids and liquids because they neither have definite shape or volume. So we need to be careful while defining its expansion coefficient. In this video, we will explore this in detail :) Created by Mahesh Shenoy.

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Video transcript

gases are a little bit different compared to solids and liquids and so in this video let's focus on these key differences from the thermal expansion point of view one of the differences can be understood by taking an example so imagine you have a plastic bottle which is completely filled with say a liquid like water or you could also have this bottle completely filled with say sand so either you take some liquid or some solid and completely fill it then try to crush it what do you think is going to happen first of all great experiment if you're not try it do it but what you will see is that nothing happens the bottle will not budge it will not get squeezed you'll not get crushed nothing's going to happen all right now you take out all the water or you take out all the sand which means what remains now is air and again try to crush it again try to crush it what's going to happen this time the bottle will get crushed in other words the volume of a liquid or a solid does not change with pressure that's the key thing to understand we have applied pressure we have increased the pressure by squeezing it but the volume of the liquid has not changed so the volume does not depend on pressure and this is the way I'll write it this is dependency and it does not depend but when it comes to a gas the volume does depend on pressure does depend on pressure again this is not equal to I'm just saying it depends on pressure because notice when we increase the pressure over here when we try to squeeze it the bottle got crushed and the volume decreased now our goal here is to calculate how much the volume of a gas changes with temperature not pressure we don't want the volume to change with pressure so what should we do well we should make sure the pressure does not change and therefore for a gas for a gas to calculate change in volume to calculate volume changes and the way we Calvin volume changes is by calculating this number called as the volume expansion coefficient and volume expansion coefficient is just the change in volume per unit volume per unit change in temperature that's how we calculate how the volume changes with temperature and we discuss about this in previous videos so if you need more clarity then it's better to go back and watch that with you but anyways if you want to calculate this we need to keep pressure so we need to keep pressure constant constant now we didn't have to worry about this for liquids and solids because pressure doesn't affect the volume at all so whether you keep the pressure constant or you don't do it or you change the pressure it wouldn't matter to us at all so we didn't have to worry about pressure in the previous cases but when it comes to gas we have to make sure this remains a constant that's one big difference between gasses and solids and liquids another difference can be seen from a table this table is telling us the values of alpha V and alpha L let's focus on Alpha V of alpha V for different different materials alright now here's a question could we predict the Alpha V value of say say gold theoretically could we do that could we just figure this out that was 42 the answer is no not likely these numbers were found experimentally theoretically it would be extremely complicated I would have no idea how we would do that and the reason for that is because we have no idea how the volume of these things change with temperature we don't know about it so there is no way we can derive it however with gas the story is a little different we do know how the volume of a gas changes with temperature and since we already know that and we will talk about that don't worry but since we already know that we can figure out what would be the Alpha V value of a gas theoretically at least approximately we can figure it out alright so let's go ahead and do that let's get rid of this table so what is the connection between volume of a gas and temperature of a gas can you think of any expression connecting volume and temperature of a gas I think you already learned one it's called the ideal gas equation ideal gas equation and the equation says if you take pressure and you multiply it by volume PV equals M the number of moles R which is a constant because as the gas constant times T the absolute temperature temperature in Kelvin now in reality most real gases don't obey this equation but they go become pretty close to this almost all gases under certain conditions they come pretty close to obeying this equation and so we can approximate almost all our gas as an ideal gas and then figure out what is the alpha V value for such an ideal yes now to calculate the change in the volume we need to change the temperature by keeping the pressure constant all right so let's do that let's say initially the temperature is t1 and the volume is v1 so initially we would have an equation P v1 equals and our t1 let's say finally the temperature changes to t2 the temperature changes to t2 which makes the volume change to v2 and so our final equation would be P v2 equals n R t2 now to calculate the changes let's subtract these two equations let's do 2 minus 1 so we'll do 2 minus 1 on the left hand side P would be a constant and you would have in bracket v2 minus v1 because we're subtracting right and v2 minus v1 is Delta V that's the change in volume that would be equal to on the right hand side and are their constants they don't change but in the bracket would have t2 minus t1 that will be delta T and that will be delta T now from this equation we can calculate what Delta V is the change in volume but we want to calculate how much the volume changes per unit volume per unit temperature change so let's go ahead and divide this equation by V delta T so let's do that over here v delta T on this side you were divided by V delta T on this side the delta T cancels out over here and naught is we now have n R divided by V n R divided by V is just P by T so this can be further written as P divided by T divided by T and the P goes and notice now the left hand side is what we want change in volume per unit volume per unit change in temperature and we call that as alpha V and notice we have now calculated where alpha V is he couldn't do this for solids or liquids but we have done that for gas for solids and liquids this number was pretty much a constant pretty much but for gases not at all notice it depends on temperature at higher temperatures the the volume expansion cohesion is lower that means as the temperature increases gases expand lesser and lesser all right the second thing to note is that for soils and liquids different material expands differently but for gases since most gas behave almost ideal for almost all gases we have the pretty much the same value of alpha V we can even go ahead and calculate this at some temperature let's say we'll do that at room temperature so if we do that at room temperature at room temperature the room temperature is about 300 Kelvin so alpha V would be let's see 1 over 300 Kelvin how would that be well 1 divided by 3 is point 3 times 10 to the power minus 2 Kelvin inverse but let's convert this to 10 power minus 6 that's how it is in the table we'll get back the table in a second so to convert that 10 power minus 6 we will shift the decimal four places towards the right that'll give us three thousand times 10 power minus 6 Kelvin inverse all right now let's bring back the table sure it is we can now fill it up for gas we get about three thousand three thousand times 10 to the power minus 6 here it is 10 power minus 6 and this only works at room temperature and we have to keep the pressure constant and that's an approximate value yeah all these things are there but anyways if we compare this value with that of liquids or solids you see gases are the one that expand the most when you heat them up all right and there's one most interesting thing we can think about is that when we build a thermometer the idea is that we put a liquid like say mercury or alcohol in glass and the idea is if the temperature changes glass doesn't expand much but the liquid expands a lot but guess what we can now build something called as the gas thermometer by putting a gas inside a glass container that would be even better thermometer because gases expand much more in fact the most precise thermometers today are gas thermometers and one of them are called as the constant pressure gas thermometers so the idea is we keep the pressure over there constant and as the temperature changes its volume changes and just by calculating how much the volume has changed we can figure out very accurately how much the temperature has changed gas thermometers