When the battery is put under load, the anode releases electrons to the negative terminal and ions in the electrolyte through an oxidation reaction. The cathode accepts these electrons, completing the circuit for the flow of electrons. This is the end of the story for non-rechargeable batteries. However, lithium-ion batteries can be recharged. So, when current is properly applied, the electron flow happens in reverse, recharging the battery.
The physically larger the battery, the higher capacity, and the flatter the curve, and the less pronounced the voltage drop is when the battery charge is low. Given the current battery debacle, this is why the Plus-sized iPhones are less affected by slowdowns, and why the iPad doesn't seem to be at all —physically larger batteries. The better and more regulated the charging and discharging of the cell is, the longer the battery will live.
The more temperature excursions the battery endures, like leaving it in a hot car, or outside in freezing temperatures, the shorter the life of the battery will be. The output voltage of any given battery over a discharge cycle can be plotted over percent charge. The reactants aren't eternal, though. In the case of a lithium-ion battery, "metal whiskers" can form in the cell, shorting out afflicted portions of the battery cell and cutting down on available power.
Ultimately, the whisker formation, coupled with reactant depletion lead to a completely dead battery from under-voltage and non-reversible oxidation. The voltage "curve" and slope varies a great deal based on a number of factors. Battery wear shrinks the horizontal axis. Obviously, device power demand decreases the amount of time it takes to progress along the curve on any given charge.
Damage to the cell from the environment, faulty charging gear providing more than allowable voltage, or other issues permanently increases the steepness of the voltage drop while the battery is being used. Operating temperature has a temporary increase in steepness as well, with low temperature having more of an impact on capacity than high.
Just because your battery is "dead" doesn't mean that it doesn't have any stored power. This is evidenced by Apple popping up a "plug in" graphic when you try to start a phone with a drained battery. In the case of a lithium-ion battery, the battery must have more than 2V capacity, or the electrode starts to oxidize. This happens fairly quickly and cannot be reversed by recharging.
This is why a lithium-ion battery left idle for a period of time ultimately completely dies, or has next to no capacity. Without delving too deeply into battery manufacture, device manufacturers have to make choices based on the performance curve. Battery capacity and output voltage are two different things that have to be considered, to say nothing of safety factors. Every device has a critical voltage, at which point it won't stay operational, may lose data, or crash entirely. The critical voltage is universal, varies from device to device, and is not just applicable to Apple.
Grossly simplified, manufacturers have to look at the battery performance curve, and pick a point that it is making engineering choices for. If they miss their guess, then as a device's battery chemically ages, it will shut down. Yes, this does happen in Android devices —with the Nexus 6P from Sept 2015 now seemingly affected by the problem —and no software fix to prevent the shutdown in sight.
If a lithium-ion battery is compromised by overcharge, overheating, damage, or simply age, the inner cells can "outgas" the contained flammable electrolyte mixture. Mobile processors don't consume a set amount of power. They draw more volts depending on how hard they are being worked. Benchmarking applications are designed to run a processor for as long as it can, as hard as it can to complete a calculation, or do a series of them in an set period of time.
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