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# Intuition behind formula for thermal conductivity

## Video transcript

so I have an interesting system over here I have two compartments on the Left compartment I have a gas that is at a temperature of T sub a and on the right side of this I have gas it is a temperature of T sub B and they are separated by a wall of depth D or I guess you say if thickness D and the contact area between the wall or the contact area of the gas onto the wall that area is a and I'm destroying a section of it we're assuming that these two compartments are completely separated now what I am curious about and we're going to assume that the temperature on the Left is higher than the temperature on the right and so because of that you're going to have a transfer of thermal energy from the left to the right and that that thermal energy that gets transferred we call that heat and we'll denote that with the letter Q I'm curious about how does the rate at which heat is transferred or so how much heat is transferred per unit time that's the rate at which heat is transferred how would that change depending on how we change these different variables so for example if our if our area if our contact area were to go up what would that do for Q over T well then Q over T would also increase per unit time because I have that I have more area for these hot Maalik these hot air particles or hot air molecules to bump into and they'll heat that wall and then there'll be more heated wall to heat up the colder air MARTA particles so in that case our rate of heat transfer would also go up well what if we what if we and obviously if I made my area smaller maybe I should just write that explicitly if I made my contact area smaller then my rate of heat transfer my rate of heat transfer would go down and that feels like common sense now what about what about the thickness if I were to make it if I were to make the thickness larger if I were to make this a thicker wall what would that do to my rate of heat transfer well then I would have I would have more things that I would have to heat up to get it to a certain temperature too before I can or to which then can heat up the particles on the right and obviously this is a continuous process it'll always be happening but be more stuff to heat up and it's going to take longer and more of that and more of that more of that kinetic energy that average kinetic energy is going to get dissipated in this wall so if this wall becomes thicker if the wall becomes thicker then the rate of heat transfer is going to go down or if you because if the wall became thinner if this depth decreased then the rate of heat transfer then the rate of heat transfer would go up so you can say the rate of heat transfer is going to be inversely proportional to the thickness of this wall now what else could we think about well we could think about the temperature differential the temperature differential that's T sub a minus minus T sub B minus T sub B well if this temperature differential if this temperature differential were to go up well what's going to happen well it's common sense then well this is super hot if this is super hot over here this is way hotter than what we have on the right well we're going to have more heat transferred so the rate of heat transfer you're a more heat transferred per unit time your rate of heat transfer is going to go up and likewise if this if this differential were to go down and you could take the extreme case if there was no differential if T sub a was the same as T sub B then you would have no net heat transfer frankly in any unit of time so it makes sense that the rate of heat transfer is going to be proportional to the temperature differential so how can we encapsulate all of this intuition into maybe a formula for describing thermal conductivity for the fur for thinking about how quickly some how quickly this heat will be transferred the rate of heat transfer well we could say the read the rate of heat transfer and this is really hopefully you know comes out of a little bit of common sense or an intuition of what would happen here the rate of heat transfer I could say it's going to be proportional to well what are the things that's going to be proportional to it's going to be proportional to the area the more surface area we have on this wall more contact area the more heat we're going to have transferred per unit time so it's going to be proportional to that contact area it's also going to be proportional to the temperature differential so let's multiply this times the temperature differential so T sub a minus T sub B minus T sub B and it's inversely proportional to the thickness of the wall so all of that over the thickness of the wall and now another thing that you might be saying okay I have this proportionality constant but wouldn't this be different for different materials for example if this was a metal wall wouldn't this conduct the heat quicker then if this was a wood wall and you would be correct a metal wall would and so this K this K right over here this is dependent this is dependent on the material so what is the wall made of so material material of of the wall and you can actually you can actually measure this thing and different different materials will have different thermal conductivities which this which this variable right over here would actually represent so going through a little bit of intuition we were able to come up with what looks like a a fancy formula and you will sometimes see this formula formula for thermal conductivity through a solid barrier but it really comes out of hopefully common sense the rate of the amount of heat transferred per time is going to be proportional to and the proportionality constant can be dependent on the material styrofoam for example would be very low here that's why coolers are made out of styrofoam and it's going to be dependent on the area it's going to be proportional to the area the temperature differential and then inversely proportional to the thickness so if you wanted to really insulate something you would want to minimize surface area and and you would want to and you would want to maximize the thickness and you would want to have something with a very low thermal conductivity so a thick styrofoam wall that's maybe shaped in a sphere might be a pretty good container for for keeping something hot or for keeping something cool