We study the mechanics of pressurized graphene membranes using an experimental configuration that allows the determination of the elasticity of graphene and the adhesion energy between a substrate and a graphene (or other two-dimensional solid) membrane. The test consists of a monolayer graphene membrane adhered to a substrate by surface forces. The substrate is patterned with etched microcavities of a prescribed volume and, when they are covered with the graphene monolayer, it traps a fixed number (N) of gas molecules in the microchamber. By lowering the ambient pressure and thus changing the pressure difference across the graphene membrane, the membrane can be made to bulge and delaminate in a stable manner from the substrate. This is in contrast to the more common scenario of a constant pressure membrane blister test, where membrane delamination is unstable, and so this is not an appealing test to determine adhesion energy. Here, we describe the analysis of the membrane/substrate as a thermodynamic system and explore the behavior of the system over representative experimentally accessible geometry and loading parameters. We carry out companion experiments and compare them to the theoretical predictions and then use the theory and experiments together to determine the adhesion energy of graphene/SiO2 interfaces. We find an average adhesion energy of 0.24 J/m2, which is lower but in line with our previously reported values. We assert that this test—which we call the constant N blister test—is a valuable approach to determine the adhesion energy between two-dimensional solid membranes and a substrate, which is an important but not well-understood aspect of behavior. The test also provides valuable information that can serve as the basis for subsequent research to understand the mechanisms contributing to the observed adhesion energy. Finally, we show how, in the limit of a large microcavity, the constant N test approaches the behavior observed in a constant pressure blister test, and we provide an experimental observation that suggests this behavior.