# Calculating temperature change from constant decompression

I have a pipe, inside of wich there is a constant amount pressure supplied from one end (like a compressor). On the other, the gas (air), is free to leave and experiences decompression.

Knowing the temperature of the gas inside the pipe, as well as the temperature and pressure (1 atmosphere) when it leaves, how would one go about calculating the initial pressure? Would this apply to liquid (water) as well?

I tried to have a go at this question, and found this handy dandy Gay-Lussac's Law. Moved around a bit to my favor it reads: $$P_1 = P_2 \frac{T_1}{T_2}$$

If for example,

\begin{align} P2 &= 101325\rm\ Pa = \text{1 atmosphere} \\ T2 &= 173.15\rm\ K = -100°C \text{ (My goal is to cool to this temp via decompression.)} \\ T1 &= 298.15\rm\ K = 25°C \end{align}

The result is about $174473.28\rm \ Pa$. This is just over 1.7 atmospheres. So apparently I can make dry ice with my bike pump? That's not right.

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