The temperature of a rapidly expanding gas decreases. Explain why in terms of the first law of thermodynamics. (Hint: Consider whether the gas does work and whether heat transfer occurs rapidly into the gas through conduction.)

According to the first law of thermodynamics, the internal energy change is equal to the difference between the heat transfer and work done

\({U_2} - {U_1} = Q - W\)

Where \({U_2}\) stands for the initial internal energy of the system, \({U_1}\) stands for the final internal energy of the system and \(Q\) stands for the heat transfer and \(W\) stands for the work done.

As the gas expands quickly, the machine doesn’t have much time to interact with the surrounding. Hence the heat transfer is \(0\) that is \(Q = 0\). When \(Q = 0\), automatically work done is in negative.

So, the final internal energy is less than that of initial internal energy. This shows that work is done by the system and the work is done at the expense of internal energy of the system. As internal energy is directly related to temperature, the temperature of the system falls.