Internal resistance

This is the one thing the ideal battery leaves out, and the CED puts it back in exactly that way: the internal resistance of a nonideal battery may be treated as the resistance of a resistor in series with an ideal battery and the remainder of the circuit (11.5.B.2).

The consequence is measurable. When there is current in a nonideal battery with internal resistance $r$, the potential difference across the terminals is reduced relative to the potential difference when there is no current (11.5.B.3):

Both CEDs label that a derived equation, and it is printed on neither equation sheet. Check the appendices and you will not find it: you are expected to produce it from the loop rule, not look it up.

Reading the equation. At $I = 0$ the terminal potential difference equals the emf, which is exactly how the CED defines emf in the first place. Every ampere you draw costs another $r$ volts, spent inside the source itself and dissipated there. A tired battery is usually one whose $r$ has grown, not one whose emf has collapsed.

That also gives the standard lab: plot terminal potential difference against current for a real cell, and the vertical intercept is the emf while the magnitude of the slope is the internal resistance.

AP Physics 2 and AP Physics C: E&M carry this block word for word, so nothing here is a Physics C extra.