Show that the internal energy of the air (treated as an ideal gas) contained in a room remains constant as the temperature changes between day and night. Assume that the atmospheric pressure around remains constant and the air in the room maintains this pressure by communicating with the surrounding through the windows, doors, etc.

We know that internal energy at a temperature


U=nCvT


Where U=internal energy


n=number of moles


Cv=molar specific heat at constant volume


T=temperature


Air in home is chiefly diatomic molecules, so


Cv for diatomic moles is given as



So,



We know ideal gas equation


PV=nRT


Where V= volume of gas


R=gas constant=8.31Jmol-1K-1


T=temperature


n=number of moles of gas


P=pressure of gas


Substituting the value of nRT from ideal gas equation to U



Now, it is given in question that pressure remains constant throughout the day and volume V of room is also constant. So, PV is a constant.


Hence U is also constant.


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