When liquid methane at -165 degrees Celsius and 1 atm is vaporized, it expands significantly, with estimates suggesting a volume increase of around 600 times. To determine the exact expansion, one must use the ideal gas law (PV = nRT) and compare the densities of liquid and gaseous methane. The density of liquid methane is crucial for calculating the number of moles present in a given volume, which can then be used to find the volume of methane gas at standard conditions. The discussion emphasizes the importance of temperature and pressure in these calculations, particularly noting that methane's behavior as an ideal gas can guide the volume estimation. Ultimately, the expansion of methane when vaporized depends on the specific conditions of the system.