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The following question is here Bidon, Energy of room. Ideal gas law : URL (version: 2018-10-27)

Although my question is a little bit different.

I’m currently studying Thermal Physics and I am reading “Concepts in Thermal Physics”, by Blundell. And when I was trying to solve some problems, I found this one:

Mr Fourier sits in his living room at 18ºC. He decides he is rather cold and turns the heating up so that the temperature is 25ºC. What happens to the total energy of the air in his living room?

I made some calculations and I just got the expression Bidon got in his question:

$$ E = \frac{3}{2}NKT $$

Where $E$ is the total energy of the room, $N$ is the number of particles, $K$ the Boltzmann constant and $T$ the temperature.

I read that the pressure is constant, because it’s atmospheric pressure, the volume of the room is assumed constant (and closed!) and what changes is the distribution of the particles in the volume (density), $E$ would be constant with $T$ increasing.

This doesn’t make any sense to me, I know that when $T$ increases, the mean kinetic energy will increase, but then the total energy kinetic of the room will increase too, right? Also I think this is wrong because the pressure will increase if the volume is constant.

What should be the answer to this situation?

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I read that the pressure is constant, because it’s atmospheric pressure, the volume of the room is assumed constant (and closed!)

Here is the problem. The room is not closed. If it were closed then you could not assume that the pressure was atmospheric. The pressure is constant and the volume is constant, but gas can enter and leave the room. It is not closed to gas even if it is closed to humans.

As the room heats up from 18 C to 25 C the number of atoms in the room decreases. By $PV=nRT$ since $P$, $V$, and $R$ are constant as $T$ increases $n$ must decrease proportionally.

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  • $\begingroup$ @cell you can certainly apply the ideal gas law to the (remaining) gas in the room. The IGL is an equation of state. But you also need other things (ie being told the new T) to fully characterize that gas. $\endgroup$ – Bob Jacobsen Dec 22 '19 at 3:36
  • $\begingroup$ @BobJacobsen You can apply the ideal gas law to a smaller volume but the phrase "as T increases n must decrease proportionally." is not true. Mass carries enthalpy with it and as it leaves a control volume P and T will change as shown in this answer physics.stackexchange.com/questions/487012/… $\endgroup$ – Cell Dec 22 '19 at 15:12
  • $\begingroup$ The 3rd paragraph of the top answer you reference says exactly what you’re denying here. P in a room is fixed by the atmosphere. $\endgroup$ – Bob Jacobsen Dec 22 '19 at 15:14
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    $\begingroup$ @cell You am to have missed that some external system is adding heat to raise the temperate to a given temperature T. Yes, exiting mass carries energy and enthalpy, but that’s just replaced as needed to meet the final P V T state. $\endgroup$ – Bob Jacobsen Dec 22 '19 at 15:21
  • $\begingroup$ @Dale the pressure is constant ant it is equal to the atmospheric pressure because the room is open to the gas, ok i got that. But how does the volume of the gas remains constante? If the pressure is constante, by heating the gas, the molecules will start wizzing around more and more and they will tend to occupy more space, right? $\endgroup$ – user249212 Dec 22 '19 at 19:05

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