If you have kids, you probably know this already, but lots of stuff needs batteries. Remote control toys, Wii remotes, laser pointers (well, that is for me), flash lights, even Nerf guns. For me, I have found the best place to pick up batteries is at one of these "dollar" stores. Sure the batteries are cheaper, but are they any good? Who knows. Let's find out.

Energy ——

The first way to look at the quality of a battery is to see how much stored energy is in it. How could you measure this? Well, here is how I did it. I took a battery and connected to a light bulb and let it run for as long as it could. Like this:

With this setup, I can measure both the current ( I ) from the battery and the electric potential ( ΔV ) across the battery. At any given instant in time, the power from the battery will be:

Power tells me how fast the energy is changing, but not the total energy in the battery. In order to find the total energy, I can write the power like this:

If the current and the change in potential were constant for the whole time interval (Δt), this would be a fairly straightforward calculation. Alas, these are not constant. So what do I do? I cheat. If I instead look at a very short time interval, the current and potential do not really change too much. This means that I can reasonably calculate the energy during this short time. Then I just need to do this a whole bunch of times to get the total energy.

Adding up a whole bunch of small pieces is called "an integral". In this case, I won't use calculus to evaluate an integral since I don't know a mathematical function for the power. Instead I will do it numerically with the following formula (by "I will do" I really mean "make a computer do"):

And that is it. The total energy that the battery produced.

Measuring Energy —————-

Vernier makes both a current and voltage probe for the LabQuest system. Collecting data was fairly simple (even though each battery would take quite some time). Here is the data from LoggerPro (Vernier's software):

This software can calculate both the power as well as integrate to find the total energy. But I am not going to do this. Why? Because I prefer to do things myself with python and matplotlib. That's just me.

Here is a plot of the voltage vs. time for three different AA batteries. I used an Energizer, Duracell, and DG (the one from the Dollar Store).

I will talk about the significance of this voltage curve in a bit, but for now I am concerned with the power. Here is a plot of the current for these three batteries.

It looks like something happened to the data for the current with the Energizer battery. I tried re-running the experiment with this battery, but again the current data didn't look pretty. The current probably shouldn't jump up like that. I guess it could have been a loose connection or something. Either way, the Duracell and the Energizer batteries seem to have similar curves, but the DG is significantly lower.

Now, I can multiply the current and potential to get the power. Here is that plot for all three batteries.

I can already see that I was wrong. Clearly, these cheaper batteries are not nearly as good as the more expensive ones. Ok, but what about the total energy? If I integrate these three curves, I get the following stored energies:

DG (Dollar Store) = 2983 Joules (0.829 Watt*hours).

Energizer = 10,798 Joules (3.00 Watt*hours)

Duracell = 9,398 Joules (2.61 Watt*hours)

It seems that maybe the Duracell and Energizer are essentially the same. Yes, the Energizer has a higher stored energy, but it also has that small jump in the current that may or may not be real. Other than that current hiccup, the curves for those two batteries look quite similar.

Are They Worth It? ——————

Yes, the better batteries have more energy. But how much do they cost? First, for the DG batteries. I thought these were a great deal since they cost $4 for a pack of twenty. That is 20 cents per battery. You can't beat that with a stick. What about the energy per dollar? Really, you are paying for energy - right? Let me call this the money-energy density. I will use the symbol u to represent this. So, for DG:

What about the Energizer batteries? Well, since I am cheap I only purchased two of these. Really, to be fair I should look at the price of a twenty-pack. Here is a 16-pack from Walmart. This sells online for a price of $10.97, so the price for one AA would be $0.685. Oh, and yes - I know you could get a better deal on batteries at Amazon, but I am trying to compare going to the store to buy batteries. This would give an money-energy density of:

And now for the Duracell batteries. Here is a 20 pack for $12.97. This would put the price of one AA at $0.649. The money-energy density would be:

Interesting. Very interesting. So it seems that all three of these batteries have about the same money-energy density. How does this answer the question? Well, there is a bit more to batteries than just the energy stored in it. It depends on what you are using it for. Suppose that I was using these batteries for a flashlight. In this case, it wouldn't matter too much which battery I used. If I used the cheaper DG, I would just have to replace the batteries more often. However, suppose that I am using the batteries for my Wii remote or my awesome Syma indoor RC helicopter (these really do fly nice). For these electronic devices, if the voltage drops too low they might not work properly. Yes, the battery will still have energy in it, but if it won't run the device correctly who cares?

Let include a quick note while I am thinking of it. If you look at the above voltage plots, you can see that the voltage for all batteries drops quite quickly below the 1.2 volt mark. This does not mean they are "dead". This is voltage of the battery while it is being used. If you unplugged it, you would get a much higher voltage. Yes, I will write a later post about this voltage drop.

I hate it when I don't give a definite answer to the question "which battery should I buy?" Maybe this advice will work: if you have Amazon Prime, buy batteries from Amazon. If you wait until you need batteries right NOW (like I do) for your kids toys, just go to the Dollar Store.

Some Other Stuff —————-

I can't help myself. I was curious. If the better batteries have more energy, do they also have a greater mass? Let me find out. It is probably a silly idea to make a plot of just three data points, but I did it anyway.

Maybe I should just compare the mass-energy density for these three batteries:

DG = 1.98 x 10 5 Joules/kg

Joules/kg Energizer Duracell = 3.84 x 10 5 Joules/kg

Joules/kg Duracell Energizer = 4.53 x 105 Joules/kg

*Note: *I incorrectly labeled the energy densities for the Duracell and Energizer. Oops. Thanks to Craig for pointing out my error.

**Ok. What does this mean? I guess it means if you are worried about the total mass of your portable electronics AND you are worried about battery life - go with the Duracell or Energizer. Oh, and these energy densities seem to be in agreement with the data posted on Wikipedia. I love that Wikipedia page on energy density. Not only is it useful, it shows you just how terrible alkaline batteries are. Alkalines have an energy density around 590 kiloJoules per kg. Burning wood has about 16 MEGA Joules per kg (yes, difficult to use all that energy unless you are trying to heat up a room or something). Oh, and look at gasoline, 47 MegaJoules per kg. This is why we don't make battery powered cars that run on AA batteries.

Buying a Pack of Batteries ————————–

Help me. I can't stop. Looking at the prices of Duracell and Energizer AA's on several online stores, I can make this plot. (Walmart, Walgreens, KMart)

The Durcell data is in blue and the Energizer is in red. I think some of the data would fit better, but there are a couple of cases where 10 batteries cost more than 14 (or something like that). If I assume a linear model fits this data (and it isn't a terrible fit), then the function would tell you the price for one battery. Like this:

Just for fun, this says that if I purchased a zero-pack of batteries (you know, just the packaging or something) it would cost $2 for the Duracell and $5 for the Energizer. The slope tells you how much one battery would cost (without the packaging - I guess you would have to bring your own box to the store). For Duracell, this is $0.378 per battery and for Energizer it is $0.755.

I admit, I need more data. Although I am sort of surprised the linear function works as well as it does. It seems like you would get a discount for buying a whole bunch of batteries. Well, in a sense you do - you only pay for the packaging (the y-intercept) once.

One last warning. There will be more battery related posts in the future. I can't help myself.

UPDATE:

It should be clear that I usually fly by the seat of my pants. I pick up some cheap batteries at a store and I just assume they are alkaline. Apparently not. The dollar-store batteries I used say "heavy duty". The interwebs tell me these batteries are zinc chloride and not alkaline. Oops. Anyway, even though they are different types of batteries, I can still compare the cost per energy. So.....