LP gases are actually quite safe in comparison with other fuels. Propane has a high ignition temperature, about 850-950 F (450-510 C), compared to about 495 F (257 C) for gasoline. This makes it less likely to ignite spontaneously.
Also, the tanks used to store propane are stronger than gasoline tanks because of the pressure needed to keep the propane in liquid form. This makes on-board propane storage safer than a typical gas tank -- it is more resistant to rupturing in the event of a collision. Special safety valves and cut-offs increase the safety factor.
To store LP gas safely, there are some guidelines. It's important to remember that a propane tank is never really empty. When the tank is filled, most of the propane is under enough pressure that it is in liquid form. But at safe pressures, not all of the propane is liquefied -- a small amount is in gas form, filling up the rest of the space in the tank. As more propane is used, the pressure decreases, leaving less propane in liquefied form and more propane gas filling up the remaining space.
Tanks should only be filled to roughly 80 percent capacity. Changes in temperature can change the pressure inside the tank. If you were to fill your tank to 100 percent on a cool, cloudy day, and then leave your car out in the sun the next day, the increase in temperature would cause in increase in pressure within the tank that could cause it to fail without that 20 percent headroom.
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