The C-rate normalizes the performance of batteries with respect to the capacity of the materials, and a current of 1C is the one that in theory discharges the battery in one hour.

Using A/g to understand the rate performance of a material, hence normalizing with the mass, can be a problem when coupling an anode material with a cathode material in a full cell configuration. Normally, anode and cathode materials have different specific capacities, i.e., the capacity per unit mass, expressed in mAh/g. For instance, NMC 622, a typical cathode material, has usually around 180 mAh/g when charged up to 4.3 V, while graphite, a typical anode material, can store up to 372 mAh/g. If I want to assemble a battery with a capacity of 1 Ah, the anode and cathode amounts should be balanced to have 1 Ah of capacity in both electrodes (assuming that the N/P ratio, i.e., the ratio between the anode and cathode capacity, is set to 1). Hence, I will need:

- (1 Ah * 1000 mAh/Ah) / (180 mAh/g) = 5.56 g of NMC 622 for the cathode

- (1 Ah * 1000 mAh/Ah) / (372 mAh/g) = 2.69 g of graphite for the anode

In a full cell, the current applied to the battery discharges (or charges) of course both the anode and the cathode with the same current. For instance, if I apply a discharge current of 1 A to the battery, I will get:

- 1 A / 5.56 g = 0.18 A/g of current rate at the cathode

- 1 A / 2.69 g = 0.37 A/g of current rate at the anode

As you see, the anode and the cathode are in this case discharged at a different current per unit mass, but the current per unit capacity (i.e., the C-rate) is the same! In fact, having both anode and cathode the same capacity:

- 1 A / 1 Ah = 1 h ⁻¹ (=1C) at the cathode

- 1 A / 1 Ah = 1 h ⁻¹ (=1C) at the anode

Hence, normalizing with the capacity using the C-rate has the advantage of understanding better what would be the performance of a material in a full cell configuration, since as you saw in almost any case the cathode and anode materials have different specific capacities.