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When a firework explodes, the powder in the firework burns rapidly to produce a small volume of gas at high temperature and pressure. Can anyone explain to me why the temperature of the gas falls rapidly? Thank you :)

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You want to know why something hot embedded in something cold cools down? I don't understand what kind of answer you are seeking here, firstly it is an experimental fact, secondly one can model this behavior using thermodynamics/statistical mechanics. –  Alexander Nov 9 '13 at 19:41

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The energetic (i.e. hot) gas particles quickly diffuse to the surrounding environment. Because the number of particles in the air greatly outnumbers that of the gas from the fireworks, the two systems quickly reach thermal equilibrium.

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I dont know much about this topic but in a laymen way what i know is that because all the heat energy is lost in expanding the gas the same reason why it feels cold when you spray a deodorant on you.

It will not be totally accurate but here it is,

When you light the firecracker its chemical burns and produces heat and gas which is used to raise the pressure and temp inside the casing of the firecracker. But when the pressure exceeds the strength of the casing and the case blows(which what it was intended to do) due to the pressure inside it the gas expands really fast and all the energy is used in doing this so the gas cools down

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