Battery Facts
Capacity
We chemistry will usually think in terms of moles of this or that when talking amounts of stuff. For a redox reaction that is providing electric current, we can get at those moles with some math. We learned earlier that you can calculate the total number of moles of electrons being pumped around a circuit by using current times time divided by the faraday.
\[{I\cdot t\over F} = {\rm mol\;e^-}\]
And as much as I'd love a world that counted in moles, it just doesn't work that way. The real world only uses the top part of that equation (the numerator) and calculates coulombs of charge passed which is just
\[I\cdot t = {\rm coulombs}\]
And we don't even say coulombs either (sigh). We say amp-hours which is just straight up current units times time units in hours, an amp-hr or Ahr. And, most "little" batteries that we use (the AAA, AA, C, D) only pass a wee bit of current usually in their devices which means you'll get better numbers if you use milliamp-hours or mA·hr (commonly shown as just mAh). So multiply the current in milliamps by the time in hours and you'll get mAhr which IS the go-to standand for battery capacities in this size of application (small electronics/toys/devices). Here is a typical AA alkaline battery capacity written out for you:
AA capacity = 2800 mAh
Which is true only IF you are only pulling 25 mA or less during the operation. Yes, the drain-rate is important and it affects the actual capacity of the battery. Bump that drain-rate up to 500 mA and you'll only get 1300 mAh which is less than half the capacity at the smaller 25 mA rate. So when you are comparing bang for your buck with batteries, you need to know the current drain of your device and then check the capacity of the battery under those conditions. Alkaline batteries are NOT so great in high-drain devices like digital cameras because you'll only get about 1/2 or less of the advertised capacity.
Size Matters
Also remember that those electrons zipping around your circuit have to come from somewhere. Yes, from your reactants in the redox reaction. Do some quick chemistry math here... a 2000 mAh AA battery means you are running 25 mA for 80 hr. If we use our alkaline cell redox reaction (see last section) we know that 2 moles of electrons are passed for the overall reaction. So using our old conversion:
\[{0.025\cdot 80(3600)\over 2(96485)} = 0.037\;{\rm mol\;rxn}\]
Now we can convert that into the amount of Zn and MnO2 which is 2.4 g of Zn and 6.5 g of MnO2. So only about 9 grams of actual material is being converted to products in a AA battery. A AA battery weighs about 23 g total so the other 14 g is the materials holding it all together, plastic, stainless steel, more plastic + the electrolyte, water, and a little excess Zn and MnO2 for good measure.
Now say you want more current (capacity) in your set up... you can either add more AA batteries OR use a higher capacity battery like a C or D cell. If you go with a D alkaline, you'll get 16000 mAh. Wow, that is 8× that of a AA battery. You DO pay the cost in size and weight though, now you are at 139 grams for the weight which is about 6× more than the AA.
Max Current scales with Size
It is important to remember that ALL the electrons entering and leaving a battery HAVE to go to and from the two reactants via the electrode SURFACE. It is the surface area of a battery that ultimately limits the maximum current possible from the battery. Yes yes fine, there are other factors too but the main one is surface area of the electrode. You could have a big-ass battery with lots and lots of material, but if the electrode surface is still tiny, then you will NOT pass any high currents at all. It's like draining a bathtub full of water. The water is going to drain much faster if you have a very wide drain and pipe to drain the water. Put a tiny drain and narrow drain pipe on and you are going to wait a long time to drain the tub. The analogy is good for thinking about electrons moving in and out of the cathode and anode in an electrochemical cell. Find ways to increase the surface area of the electrode, and you will increase the maximum current possible. Let me write that really big for you.
max current \(\propto\) electrode surface area
So bigger batteries generally have larger electrodes and surface areas which means that more current can be delivered. Plus, if you still just want small currents, big batteries deliver that as well AND last a lot longer due to their much larger capacity.
Design Matters Too
The classic "can" type batteries are great a cylinder electrode centered in a can electrode. This is what alkaline cells are like. The electrode sizes in a can type housing are limited by the size of the battery (the radius and height). A very clever way to get even more surface area out of the same basic size is to roll the battery out of 2 sheets of electrode materials. NiCads, NiMH's, lithium, and some Li-ion batteries are made this way. They are called spiral bound or jelly rolled. You can get about 20× more surface this way and therefore 20× more current. This can actually be a bad thing if a battery like this gets short circuited. The high current causes a tremendous heat rise and the battery can catch on fire and even explode. All the spiral bound batteries I mentioned above have been known to do this... so don't short circiut a battery. Good idea to not short circuit a battery of any type while you're at it.
Performance vs Temperature
All batteries have much lower performance at low temperatures. As the diagram below shows, cold temperatures decrease run time considerably. Less of the battery's energy is available at low temperatures. Best performance is at room temperature or a bit warmer (not hot, just warm).
Bottomline: batteries hate the cold and they perform poorly in it. Manufacturers will promote how great their battery performs in cold weather. A car battery has a performance rating known as CCA which stands for cold cranking amps. This tells how many amps can be delivered from a battery at freezing temperature (0 °F, or –18 °C) for 30 seconds and keep the voltage above 7.2 volts.