Why does the compressed gas leaving the propane tank cause the temperature to drop around the valve?
Look up the Joule-Thomson effect.
http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Joule%E2%80%93Thomson_effect.html
http://www.princeton.edu/~achaney/tmve/wiki100k/docs/Joule%E2%80%93Thomson_effect.html