Place the mint leaves, limeade, and 2 cups of water into a blender. Pulse-blend the mixture until the mint leaves are in small pieces, but not completely pulverized. Strain the mixture into a 1 gallon container. Stir in the rum and remaining 2 1/2 quarts of water. To serve, fill tall glasses with ice cubes, pour in the limeade mixture, and garnish each glass with a lime wedge.