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Hello All, I have a problem I've been trying to solve for some time and can't quite seem to figure it out. I need the help of some experts out there. I am no physics expert but I do have a basic understanding of math, algebra, and geometry. I am a beginner with calculus.

I am trying to find out the TOTAL amount of heat in Joules that will be conducted into a gas per unit of time(like 1 second). I understand fouriers law and how you can determine the local heat flow rate when you have 1. Thermal conductivity 2. Surface Area 3.Temperature difference 4. Length between temperatures. shown as:

Q(per unit time)= k*A*T(difference)/L

However I don't understand how to use this equation(or any equation) to solve what I am trying to figure out.

Let me explain this theoretical problem:

This is in one dimension, keep in mind. Say I have a cylindrical jar with a metal cap and I fill this jar with an ideal gas(like helium) at room temp. There is a hole in the bottom for venting. The jar is sitting with cap on top like a jar normally sits. Say I theorectically can INSTANTLY heat the metal cap to a higher temperature(like 500 kelvin)(this is the start of the time unit). The cap always maintains its temperature and it has unlimited amounts of heat available for transfer. I want to know how much heat will conduct to the gas per unit of time. Because the gas is heated from the top there is no convection taking place, the hotter portion of the gas is always on top so no hot gas rises, no flow, no convection. I realize that as soon as the gas heats it will expand thus changing the amount of substance in contact with the metal also potentially changing its thermal conductivity. If this is too complicated, we can leave it out, I just want something close. There will also be no pressure building because of the vent hole in the bottom. I am not concerned with the walls of the jar or any conduction through it, this is in 1 theorectical dimension only from the top.

This may be an easy problem for all you experts out there, maybe I am over analizing it, but I just can't come to understand it. This is one way I tried to figure it out:

I tried to use fouriers law and just made the length 0. Because there is no length between the molecules of the gas touching the molecules of the metal, the length would be zero. I also tried used a very small length of like 20 microns(.000020 metres). But then I realized that during this unit of time the temperature of the gas touching the metal will change thus changing the temperature difference between these two points(which are basically touching) and thus changing the heat flow rate. So I guess the heat flow rate varies in time? I am confused.

Anyone who can help me I would greatly appreciate it. If I have left anything out, please let me know and I will correct it. I tried to explain my problem as best I can. I am beginner with thermodynamics and I have a desire to understand it better. Thanks in advance.

-Mystic

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# Help Heat Conduction into a gas

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