= Solution
Define mass-specific differential <capacitance>, or <specific capacitance>, by
$$
\boxed{C_{\rm sp}(V)=\frac1m\frac{dQ}{dV}},\qquad [C_{\rm sp}]=\mathrm{F\,g^{-1}}.
$$
State whether $m$ is one electrode's active mass, both electrodes' active mass or the entire device mass; these conventions are not interchangeable. An average capacitance inferred from a finite voltage interval is $\Delta Q/(m\Delta V)$ and need not equal the local derivative.
\b[A nearly ideal double-layer <supercapacitor> has an approximately flat capacitance over its stable operating window], so stored charge rises nearly linearly with potential. Real electrodes need not be flat: dilute-electrolyte double-layer models can give a minimum near the potential of zero charge, while concentrated electrolytes, finite ion size and pore confinement can produce bell-shaped or double-peaked curves. <Pseudocapacitance> can add broad potential-dependent contributions near a redox transition. The sketch above therefore labels several possible behaviours instead of asserting a universal monotonic trend. Constant-potential simulations have directly found transitions between double-peaked and bell-shaped differential-capacitance curves as interfacial interactions change.<Insight into the Camel-to-Bell Transition of Differential Capacitance in Ionic Liquid-Based Supercapacitors by Zhongdong Gan and Yanlei Wang>[2022, constant-potential molecular-dynamics results]{ref}
Back to article page
