The nature of the electronic interface between a nanotube and solvated ions in a liquid electrolyte is governed by two distinct physical phenomena: quantum and chemical. The quantum component arises from the sharply varying electronic density of states and the chemical component arises from ion screening and diffusion. Here, using an integrated on-chip shield technology, we measure the capacitance of one to a few nanotubes quantitatively as a function of both bias potential (from -0.7 V to 0.3 V) and ionic concentration (from 10 mM to 1 M KCl) at room temperature. We determine the relative contributions of the quantum and electrochemical capacitance, and confirm the measurements with theoretical models. This represents an important measurement of the quantum effects on capacitance in reduced dimensional systems in contact with liquid electrolytes, an important and emerging theme in the interface between nanotechnology, energy, and life.