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A great question, Julian. Let's just consider Na+. If you open a Na+ channel in a neuron, why does Na+ go into the neuron in the first place? Well...TWO reasons: FIRST, Na+ is a cation (it is positively charged) and the inside of the neuron is negatively charged. So all those positive charges are drawn into the negatively charged inside of the neuron.
At rest, the concentrations are as follows: high intracellular K, high extracellular Na. When the cell fires, that changes, K flows out, Na flows in. When the cell is done, it has to revert to its original state so that it can fire again which is why a pump is necessary. Since we're dealing in concentrations, the cell tries to get equal amounts of sodium and potassium inside and out, once it gets close, you need the pump, powered by ATP, to pump the ions against the direction that they are … (more) trying to flow to revert the cell to rest.