Halliday-Resnick makes the following argument: Suppose that a room of volume $V$ with diatomic ideal gas (air) at temperature $T_1$ and pressure $p_0$ is heated by a furnace to $T_2$ with constant pressure (since the walls are not airtight). Assume that the walls do not experience a significant change in temperature.

Because $E_{int} = \frac{5}{2}nRT = \frac{5}{2} p_0 V = \textrm{constant}$, the internal energy remains the same despite the different temperatures, and all the energy supplied by the furnace has gone to heat the air outside the room.

But, if all the heat escapes in the room, why bother lighting the furnace at all?

  • 2
    $\begingroup$ Why bother having a room at all then either? $\endgroup$
    – Jon Custer
    Commented Aug 21, 2021 at 21:19
  • $\begingroup$ This problem is also discussed in Kreuzer and Payne's "Thermodynamics of heating a room." $\endgroup$ Commented Aug 22, 2021 at 0:09

1 Answer 1


The total energy is the same and the total pressure is the same, but the temperature is higher and the number density of atoms is lower. The higher temperature makes humans comfortable, because temperature, not pressure or energy density, affects how heat moves from your skin to the air. I guess you could think of this situation as spending some energy to remove some atoms to increase the average energy per atom.

To put it another way, either your heat is going to heat the air outside, or the furnace heat is going to heat the air outside. You will enjoy having the furnace help you.


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