Similar questions have been asked here, but there's one part of this that I don't understand; in a resistor connected in a circuit, as the energy of the electrons decreases when they pass through the resistor, shouldn't the difference in potential across a resistor increase?
And, therefore, shouldn't the terminal potential difference across a cell that is connected to the circuit increase?
You can compare your battery with two water reservoirs at different heights. When water flows down through a pipe - the resistor- waterparticles lose energy , but the difference in height does not. You always have water or e- at higher levels or higher potential than others, but the difference in height does not change, till the upper reservoir is almost empty. so you loos energy, (chemical energy in your battery, potential energy for the Water, but it surely will not increase
You are correct if there were only a finite number of electrons in the cell. However we assume there to be an infinite number of electrons so that after a couple have dropped in potential energy after passing though, and heating up, a resistor, the total potential energies on each side of the resistor barely change due to the vast number of electrons present. This change is so small that we can take it to be negligible and ignore it.
Also note that the drift velocity of electrons moving down a wire is normally very small (the Wikipedia article gives an example speed of $2.3\times10^{-5}\,\text{ms}^{-1}$), so the number of electrons dropping over the resistor per second is not as large as you may be envisaging.
