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 PF = 1.2 atm, VF = 48 Liters, and TF = 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 TI = 325 K, PI = 0.5 atm, and n= 2.19 moles, we get VI = 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 PI*VI / TI = PF*VF / TF.
We rearrange to solve for initial volume: VI = PF*VF*TI / (PI*TF)
Using this method we get the same answer VI=117 L, without having to solve for n, the number of moles or using the ideal gas constant. Hope this helps!