We can figure out the initial volume backwards from the final conditions using the ideal gas law (P*V=n*R*T).

We want to use the ideal gas constant that corresponds to the given units. R=0.08206 L atm /(mol K).

First we should solve for n, the number of moles using the formula n= P*V / (R*T)

Using the final conditions of P_{F} = 1.2 atm, V_{F} = 48 Liters, and T_{F} = 320 K, we get n = 2.19 moles.

It's important to know that for a closed, non-reacting volume of a gas, the number of moles, n, is a constant no matter how the conditions change. So, now that we know n, we can solve for the initial volume by rearranging the ideal gas law again. V= (n*R*T)/P.

Using the initial conditions of T_{I} = 325 K, P_{I} = 0.5 atm, and n= 2.19 moles, we get V_{I} = 117 L.

Another easier and quicker method of solving this uses the fact that both the ideal gas constant and n are constants. From the ideal gas law this means P*V / T = n*R = constant. Therefore P_{I}*V_{I} / T_{I} = P_{F}*V_{F} / T_{F}.

We rearrange to solve for initial volume: V_{I }= P_{F}*V_{F}*T_{I }/ (P_{I}*T_{F})

Using this method we get the same answer V_{I}=117 L, without having to solve for n, the number of moles or using the ideal gas constant. Hope this helps!