Potassium occurs in two stable isotopes (Ar atoms trapped inside minerals.
What simplifies things is that potassium is a reactive metal and argon is an inert gas: Potassium is always tightly locked up in minerals whereas argon is not part of any minerals. So assuming that no air gets into a mineral grain when it first forms, it has zero argon content.
A second problem is that for technical reasons, the measurement of argon and the measurement of potassium have to be made on two different samples, because each measurement requires the destruction of the sample.
If the mineral composition of the two sample is different, so that the sample for measuring the potassium is richer or poorer in potassium than the sample used for measuring the argon, then this will be a source of error.
But consider what happens if the argon came from deep within the Earth, where it was formed by Ar ratio as is found in the atmosphere, and the formula that corrects for atmospheric carbon will not correct for this.
Finally, we must consider the possibility of argon loss.
The reasoning is as follows: the atmosphere does not only contain Ar as being atmospheric argon.
As a result under most circumstances we don't expect to find much argon in igneous rocks just after they've formed.
The rock sample to be dated must be chosen very carefully.
Any alteration or fracturing means that the potassium or the argon or both have been disturbed.
These steps help remove as much atmospheric Ar from the sample as possible before making the measurement.
Next, the mineral sample is heated to melting in a vacuum furnace, driving out all the gas.