Over the past year or so, I’ve been dabbling in bread baking. One of the first steps to every batch is to get about 750 mL of water to a specific temperature, ranging from 85 °F to 105 °F. The standard drill was to just mix hot and cold water from the tap until I got within a degree or so of the target temperature. Then I’d take that and repeatedly pour out a little bit, weighing it each time, until I got to the exact amount I needed. This method worked fine enough, but the tediousness of the whole thing gave me an itch that I needed to scratch.
#science
Two properties that make water a remarkable substance are its high specific heat capacity and high heat of vaporization. In case your chem101 knowledge is as hazy as mine, specific heat capacity refers to the amount of energy it takes to increase the temperature of a substance per unit of mass. In the case of water, it’s 4.186 J/g/°C, which means it takes 4.186 Joules of energy to raise the temperature of 1 gram of water by 1 degree Celsius. Since water is apparently an important substance on this planet, this quantity of energy has been given its own special name: the Calorie, abbreviated with a (lowercase) c. Yes, this is the same calorie that’s shaming you as you order that Big Mac with fries at Big Papa McD’s – well almost the same. In the context of nutrition facts, the word calorie technically refers to the kilo-calorie, which is 1000 calories. (Makes total sense, I know.) So that 540 Cal Big Mac actually contains 540,000 calories, or 2,260,440 Joules of energy.
Here’s some fun food math: since a Watt is 1 Joule/second, if you scarf that thing down in under 5minutes, the power transfer is over 7.5 kW (2300 MJ / 300 s) – which is over twice the amount of power an average electric oven draws, or equivalent to charging 1500 phones at once! Does this make an sense in reality? No clue.
Back to the chemistry. The heat of vaporization refers to the amount of energy it takes to change a substance from liquid to vapor per unit mass. For water, this is equal to 540 calories per gram, or 2260 Joules per gram. Note that this doesn’t include the energy needed to bring the water to it’s boiling point, just the energy needed to change 1 gram of just-under-boiling water into steam. Keeping the whole calorie versus kilo-calorie nonsense in mind, this means that the 540 kcal of energy that your body absorbs from a Big Mac could vaporize 1 kilogram of (almost boiling) water. If this doesn’t sound impressive, think about the fact that it takes 540 times more energy to vaporize a given amount of water than it does to raise its temperature by a single degree Celsius. Put another way, it takes over five times more energy to turn a given amount of just-below-boiling water into steam than it does to heat the same amount of liquid water from right above freezing (~0 °C), to just below boiling (~100 °C)!
Getting to the (boiling) point
So how does all this Big Mac science help me get my bread dough water to that Goldilocks temperature faster? Armed with the basic chemistry knowledge, I set out to measure the output power of my microwave by nuking about a liter of water for specific time intervals and recording its temperature and mass along the way. Using this data and the physical constants introduced above, I could calculate the amount of energy that the water would have to absorb in order to cause the changes in temperature and mass I observed. A change (increase) in temperature would correspond to the water absorbing an amount of energy that is proportional to the mass of water and the specific heat capacity. A change (loss) in mass would correspond to the amount of energy that is proportional to the heat of vaporization. Combing the two would give me the total amount of energy absorbed by the water. Divide that by the time spent applying the energy (nuking it) and you have the total absorbed power in Watts (Joules per second)!
Ain’t it fun living in the real world
If you squint closely at my chicken scratch, you’ll see that I started with about 750 grams of room temperature water and microwaved it a bunch of times for 30 seconds, and then did a couple longer runs, once it started boiling.
Like most real-world experiments, the results weren’t as tidy as the theory. The chart below shows the Total Power Absorbed for each heating cycle and an average of all the heating cycles, as well as the Heating Power, and Vaporization Power for each heating cycle (all in Watts), versus the starting temperature. If the microwave were truly outputting 900 W and all of it were being absorbed by the water, then I’d expect the Total Power Absorbed to be a horizontal line at 900 W. Since I don’t live in an ideal world, I was satisfied to see that the number was at least somewhat in the ballpark, with an average of 571 W, minimum of 452 W, and maximum of 773 W, which all seemed reasonable enough for a half-baked kitchen experiment. And it was neat to see the transition from heating power to vaporization power as the temperature increased.
Cosmic Microwave Background?
What’s with all the variation between samples and why is it so far off from the microwave nameplate power rating of 900 W?
I don’t have a way of directly measuring the radiated power, but if I assume for a second that the 900 W figure is legit, there’s still a couple sources of error that come to mind. First off, despite all this fancy talk of heat of vaporization, the idea to track the mass didn’t actually occur to me until about halfway through the experiment, at which point 5 grams of water had already evaporated.
To get a feel for the how much error is due to the resolution of my scale and temperature probe, I took a SWAG at the resolution for each (1 g and 1 °C, respectively) and calculated the impact an error of that magnitude would have on the power calculation. A 1 °C error in temperature would correspond to anywhere from a 9 W to 105 W of error in my heating power calculation, depending on the duration of the heating cycle (less time, more error) and mass of water being heated (more mass, more error). Similarly, an error of 1 gram would correspond to a error in the vaporization power calculation ranging from 9 W to 75 W, again depending on the duration of the heating cycle (less time, more error). These errors are additive, so you start to see how things could get out of whack pretty quick.
On top of that, I needed to remove the water from the microwave to weigh it and measure the temperature, so it inevitably cooled a bit between heating cycles. I’m guessing some of the bumps in the lines are a result of this cooling. Since I could pull the plastic container of boiling water out of the microwave without getting third-degree burns, I assume it has some insulative properties. But beyond that, I have no way to quantify how much heat was transferred from the water to the environment while it was in the microwave. The downward trend of the calculated heating power versus temperature is consistent with more energy being lost to the environment as the temperature increased – which conveniently follows Newton’s Law Of Cooling. On the other hand, it could also be due to vaporization that’s too small for me to measure. Seems like a stretch, but maybe with a little deeper analysis, I could coax some kind of quantitative measure of the thermal properties of the container, or system, out of the data.
On top of the measurement error and thermal leakage, the specific heat capacity varies a bit with temperature and pressure. I tried to see what the value would be over the temperature range using this online calculator, but it’s not clear to me if my situation is isochoric or isobaric. In any case, it didn’t look like it would account for all of the variation I see in my data and skimming through it was making my head hurt.
Final thoughts
About three cycles into the experiment, I was getting pretty consistent results showing that a 30-second nuking would raise the temperature by about 10°F. This was really all I needed to know to easy my bread making struggles, but at that point I just wanted to see if the experiment would work. In the end, I’d say my data made sense qualitatively, but there was too much error to accurately determine the microwave’s output power for anything useful. Repeating it a couple times over the entire temperature range would probably be a good way to average out some of the error, but my attention span has its limits. Maybe someday when I get my hands on one of those super fast Thermapens, I’ll make some tweaks and run the experiment again. Or better yet would be some kind of setup to record the temperature and mass continuously without removing the water from the microwave. Not being able to put any electronics inside the microwave presents a fun challenge. Maybe an infrared temperature probe on the outside and a plastic beam balance for the mass could do the trick?
