  Motivated by a long-standing debate concerning the nature and interrelations
of surface-tension variables in fluid membranes, we reformulate the
thermodynamics of a membrane vesicle as a generic two-dimensional finite system
enclosing a three-dimensional volume. The formulation is shown to require two
tension variables, conjugate to the intensive constraints of area per molecule
and volume-to-area ratio. We obtain the relation between these two variables in
various scenarios, as well as their correspondence to other definitions of
tension variables for membranes. Several controversies related to membrane
tension are thereby resolved on a model-free thermodynamic level. The
thermodynamic formulation may be useful also for treating large-scale
properties of vesicles that are insensitive to the membrane's detailed
statistical mechanics and interactions.
