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When the mass oscillates up and down until it stops and the water plus 400-g mass have their temperature raised, we say that "heat is added to them". What does that really mean? It means that the water molecules and the molecules in the mass have, on average, more kinetic energy than before which appears as an increase in temperature.
Say you have a gas in a piston. If you compress the gas adiabatically (no heat added), you do mechanical work W on the gas and the temperature rises meaning that, on average, the gas molecules move faster than before. If you then lock the piston in place and put it over a flame adding heat Q at constant volume, the gas molecules will move, on average even faster resulting in an even higher temperature. What have you really done during this last step? The much faster moving molecules in the flame collided with the walls of the piston molecules making them move faster which in turn made the gas molecules colliding with the piston walls move faster than before. So this addition of "heat" is really transfer of kinetic energy from the outside into the system. However, the complicated microscopic processes of kinetic energy transfer between a prodigious number of molecules is significantly facilitated if you lump the total energy transferred as "heat entering the system."
Then you have a generalized form of the work energy theorem. ##ΔK = W## becomes ##ΔK = Q + W## to account for the intractable amount of kinetic energy that goes into the system when "heat" is added. Remembering that the average kinetic energy of an atom in an ideal monatomic gas at temperature ##T## is ##\bar K=\frac{3}{2}kT##, the total kinetic energy content of ##N## atoms is ##K=\frac{3}{2}NkT.## Thus, in an ideal monatomic gas the total kinetic energy is the same as the internal energy ##U.## We use a different symbol because in polyatomic gases energy coming in from the outside does not only go into translational kinetic energy but also into rotations and vibrations of the molecules.
Say you have a gas in a piston. If you compress the gas adiabatically (no heat added), you do mechanical work W on the gas and the temperature rises meaning that, on average, the gas molecules move faster than before. If you then lock the piston in place and put it over a flame adding heat Q at constant volume, the gas molecules will move, on average even faster resulting in an even higher temperature. What have you really done during this last step? The much faster moving molecules in the flame collided with the walls of the piston molecules making them move faster which in turn made the gas molecules colliding with the piston walls move faster than before. So this addition of "heat" is really transfer of kinetic energy from the outside into the system. However, the complicated microscopic processes of kinetic energy transfer between a prodigious number of molecules is significantly facilitated if you lump the total energy transferred as "heat entering the system."
Then you have a generalized form of the work energy theorem. ##ΔK = W## becomes ##ΔK = Q + W## to account for the intractable amount of kinetic energy that goes into the system when "heat" is added. Remembering that the average kinetic energy of an atom in an ideal monatomic gas at temperature ##T## is ##\bar K=\frac{3}{2}kT##, the total kinetic energy content of ##N## atoms is ##K=\frac{3}{2}NkT.## Thus, in an ideal monatomic gas the total kinetic energy is the same as the internal energy ##U.## We use a different symbol because in polyatomic gases energy coming in from the outside does not only go into translational kinetic energy but also into rotations and vibrations of the molecules.