Gas diffusion from an encapsulated microbubble is modeled using an explicit linear relation for gas permeation through the encapsulation. Both the cases of single gas (air) and multiple gases (perfluorocarbon inside the bubble and air dissolved in surrounding liquid) are considered. An analytical expression for the dissolution time for an encapsulated air bubble is obtained; it showed that for small permeability the dissolution time increases linearly with decreasing permeability. A perfluorocarbon-filled contrast microbubble such as Definity was predicted to experience a transient growth due to air infusion before it dissolves in conformity with previous experimental findings. The growth phase occurs only for bubbles with a critical value of initial partial mole fraction of perfluorocarbon relative to air. With empirically obtained property values, the dissolution time of a 2.5 micron diameter (same as that of Definity) lipid coated octafluoropropane bubble with surface tension 25 mN/m predicts a lifetime of 42 minutes in an air saturated medium. The properties such as shell permeability, surface tension, relative mole fraction of octafluoropropane are varied to investigate their effects on the time scales of bubble growth and dissolution including their asymptotic scalings where appropriate. The dissolution dynamics scales with permeability, in that when the time is nondimensioanlized with permeability, curves for different permeabilities collapse on a single curve. Investigation of bubbles filled with other gases (non-octafluoropropane perfluorocarbon and sulfur hexafluoride) indicates longer dissolution time due to lower solubility and lower diffusivity for larger gas molecules. For such micron size encapsulated bubbles, lifetime of hours is possible only at extremely low surface tension (<1mN/m) or at extreme oversaturation.