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I'm trying to find out how to determine how a pressure of gas changes in time, but I can't find a way.

I have a system with constant volume $V$ (closed system, exchanges heat with environment through walls), the system is heated to temperature $T_0$ and then left to cool down on its own (heat source is removed after reaching $T_0$) until steady temperature $T_1$ is reached.

Is a way to determine the time dependent function for change in gas pressure inside the system (for example gas in some container)?

Assuming an ideal gas is okay.

At first I thought maybe differentiating $pV = nRT$ as $dp/dt = nR/V * dT/dt$, but I don't know where to go next.

Then I thought that maybe finding how heat is transferred through the walls to the environment and through the gas and using that to find $dT/dt$ (and eventually $dp/dt$) but I don't know how to go about approaching the problem that way.

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I am not sure how much it will help, but normally temperature drops in proportion to the difference in temperature - and as you point out pressure is directly proportional to temperature - so the temperature will drop by exponential decay and the pressure will also drop in this way as well.

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