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we already know that by expension the kinetic energy of molecules decreases hence causes a little decrease in temperature but when suuden expension takes place then fast moving molecules takes energy from that area and rapidly decrases temperature.

Now let's prove it mathematically we know that

PV=nRT

when expension occurs pressure decreases more than volume increses hence PV product decreases on the right hand R & n are same hence temperature decreases.

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