Thermodynamics concept question

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Problem: Why does an ideal gas cool when it expands adiabatically and reversibly?

Is the volume of the container expanding? so the total kinetic energy of the gas is getting smaller as it occupies more space making the temperature cool?
 
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The gas does work in order to expand. This work is done at the expense of the internal energy of the gas. (KE of its molecules)
 
Is that all? so when the gas is doing work, it will loose energy? is that because the system is loosing energy and the temperature willl decrease??
 
The gas does work.
The energy to do this comes from the internal energy of the gas.
The temperature of the gas depends on the (average) KE of the molecules.
If internal energy is lost, KE is reduced, temperature goes down.
(Because its adiabatic, no heat energy is allowed into the gas from outside.
If it was allowed, this heat energy would appear as internal energy and cause the temperature to rise.)