CV, REVISITED

Why an ideal capacitor draws a rectangle

I = C · dV/dt = C · v

Part 1 introduced the CV loop. For an ideal capacitor, the current is the capacitance times the rate the voltage changes, which is the scan rate v. The current sits flat at +C·v on the way up and −C·v on the way down, tracing a rectangle.

Rectangle means capacitor: capacitance sets the height, and scan rate scales it.

READING THE SHAPE

What a CV's outline tells you

Near-perfect rectangle Ideal double-layer behavior

Rectangle with broad humps Pseudocapacitive redox reactions are contributing

Tilted, leaf-like parallelogram High series resistance is slowing the charging

Sharp current rise at the ends The electrolyte is breaking down at the window's edge

Shape bends at fast scans Ions can't keep up with the faster sweep

Shape is the fastest diagnostic you have, before any number is calculated.

CAPACITANCE FROM A CV

Turning a loop into farads

C = ∮ I dV / (2 v ΔV)

Integrate the current around the whole loop to find the charge exchanged. Divide by two, since the loop holds a charging half and a discharging half, then divide by the scan rate v and the window width ΔV. Divide again by mass, area, or volume to normalize.

One loop, one capacitance, but only at that scan rate.

SCAN RATE IS A DIAL

Capacitance depends on how fast you sweep

Slow scan Ions have time to reach deep pores, so capacitance looks larger

Fast scan Only outer surfaces respond, so capacitance drops and the shape bends

Rate capability How much capacitance survives as the scan rate climbs

Reporting A capacitance quoted without its scan rate is incomplete

Good papers show capacitance across a range of scan rates, not just the flattering one.

CAPACITIVE OR DIFFUSION-LIMITED?

The b-value fingerprint

i = a · vᵇ

Plot peak current against scan rate on log–log axes; the slope is b. A slope near 1 means capacitive, surface-controlled storage, since current scales with v. A slope near 0.5 means diffusion-limited storage, like the √v dependence in the Randles-Ševčík equation from Part 1. Values in between suggest a mix.

One slope separates surface storage from bulk, diffusion-limited storage.

UNIT 26 STUDY COMPLETE

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Ready for the Fab Challenge?

You've covered reading CV shapes, computing capacitance, and scan-rate fingerprints.