The solution before revision #19158 of , by JMMA2006. This is not the current version.

Statement

5.6.3. The air in the room was heated from the temperature T_0 to T. At the same
time, the pressure did not change. Has the internal energy of the air inside
the room changed?

Solution

Assuming we're working with ideal gases
For an ideal gas, the internal energy only depends on the temperature
, where n is the number of moles and is the molar heat capacity at constant volume
Careful here, we could think that obviously if the temperature increases, the internal energy increases too, but let's think this
We have
, because P and V are both constant
but this is only possible if n changed (air can escape), because if n is constant, the product nRT would vary with T, contradicting that PV is constant
So, from , we get

and substituting this into the expression for the final energy

, which is exactly the initial energy
so
the internal energy did not change

Answer

The internal energy did not change