Serway & Jewett — The First Law of Thermodynamics: Problem 32.

You are working in a condensed-matter laboratory for your senior project. Several of the ongoing projects use liquid helium, which is contained in a thermally insulated vessel that can hold up to a maximum of \(V_{\text{max}} = 240\text{ L}\) of the liquid at \(T_c = 4.20\text{ K}\). Because some of the liquid helium has already been used, someone asks you to check to see if there is enough for the next day, on which four different experimental groups will need liquid helium. You are not sure how to measure the amount of liquid remaining, so you insert an aluminum rod of length \(L = 2.00\text{ m}\) and with a cross-sectional area \(A = 2.50\text{ cm}^2\) into the vessel. By seeing how much of the lower end of the rod is frosted when you pull it out, you can estimate the depth of the liquid helium. After inserting the rod, however, one of the experimenters calls you over to perform a task and you forget about the rod, leaving it in the liquid helium until the next morning. How much liquid helium is available for the next day’s experiments? (Aluminum has thermal conductivity of \(3100\text{ W/m}\cdot\text{K}\) at \(4.20\text{ K}\); ignore its temperature variation. The density of liquid helium is \(125\text{ kg/m}^3\).) Assume that gaseous helium can escape from the top of the vessel.