Note: The text of this post is taken word for word from a Quora post that I wrote on the same topic. This does not count as plagiarism as that Quora post was written by me. It can be found here.
In episode 26 of Naruto Shippūden (Puppet Fight: 10 vs 100!), a truly phenomenal feat is witnessed. Around 29 seconds into the episode, the genius that is Sakura for some reason makes a 0 IQ move and throws a kunai at the S-rank rogue ninja Sasori. It is completely vaporized by Sasori’s flamethrowers in two seconds. Leave it to Sakura, the queen of intelligence, to make a move like that. I think all of you already get my point about Sakura, and this post is technically about Sasori.
Nearly 7 years ago, a Pokémon YouTuber with the username MandJTV (I promise I’m not getting sidetracked) uploaded a video in which he tried to determine how hot Charizard’s flames were, because its Pokémon Stadium Pokédex entry said that Charizard can quickly melt glaciers weighing ten thousand tons. This video was uploaded as the third installment of his Strange Pokémon Physics series. As the issue of how hot Sasori’s flamethrowers are is related to this, I came to the conclusion that the equations in Michael’s video would be relevant and decided to rewatch it. Told you that I wasn’t getting sidetracked.
To figure out how hot Sasori’s flamethrowers are, we first need to figure out how much heat energy, or thermal energy, is needed to vaporize Sakura’s kunai. As the Wikipedia article on sublimation (state of matter changing directly from solid to gas) was way less helpful than I expected, I will assume in my calculations that Sasori’s flamethrowers first melt the kunai and then vaporize the melted iron very quickly.
The SI base unit of energy (which I assume is also the unit that chakra is measured in, since chakra is twice in the first ten episodes explicitly called elemental life energy and is said to be composed of physical and spiritual energy, but this isn’t about chakra) is the joule, which is quite small (less than a quarter of one calorie) so the kilojoule (one thousand joules) is the one more commonly used, and the one used in both Michael’s and my calculations.
I’ll assume that the kunai is experiencing normal atmospheric pressure, giving it a melting point of 1538 degrees Centigrade and a boiling point of 2862 degrees Centigrade.
The formula for heat energy transfer to an object (the first equation in the video, and the one that appears in the thumbnail) is Q=mcΔT. Q is the amount of heat energy in joules, m is the mass of the object being heated in kilograms, c is the specific heat capacity (resistance to change in temperature) of the object’s substance in kilojoules per kilogram kelvin, and ΔT is the change in temperature in degrees Centigrade or Kelvin (doesn’t matter which one you use, as the intervals are the same size and both will yield the same result for this equation. So this can be read as “amount of heat energy equals mass times specific heat capacity times change in temperature).
I’m guessing that the mass of Sakura’s kunai is 300 grams, or three-tenths of a kilogram. Kunai are made of iron, and one source I found places the specific heat capacity of iron at 0.462 kilojoules per kilogram kelvin. Assuming that the kunai started at 21 degrees Centigrade and was raised to its melting point of 1538 degrees Centigrade, the ΔT is 1517 degrees. This leaves us with Q equaling 210.256 kilojoules, but that’s not all.
When a substance reaches its melting point, it requires more heat energy to be inputted in order to actually melt. This energy goes to breaking the molecular bonds and changing the state of matter, during which the temperature remains constant. The amount of energy required to actually change a melting point solid into a liquid is called the latent heat of fusion, which for iron is 107.321 kilojoules per kilogram, and is 6.01 kilojoules per mole for all substances in general (including iron). The amount of heat energy to actually turn a 300-gram melting point iron kunai to 300 grams of liquid iron is 32.196 kilojoules. So far we have the amount of energy needed to get Sakura’s kunai to its melting point and actually melt it at 242.452 kilojoules. We still aren’t done.
After melting the kunai, Sasori’s flamethrowers also caused it to reach its boiling point, which is 2862 degrees Centigrade. I’ll bring back the Q=mcΔT equation, with the only changes variable being the change in temperature. The kunai went from a liquid at its melting point to its boiling point. 2862–1538=1324, so the change in temperature is 1324 degrees. The amount of energy required to raise a now-liquid 300-gram kunai from its melting point to its boiling point is 183.506 kilojoules, but that’s not all.
When a substance reaches its boiling point, it requires more heat energy to be inputted in order to actually boil. This energy goes to breaking the molecular bonds and changing the state of matter, during which the temperature remains constant. The amount of energy required to actually change a boiling point liquid into a gas is called the latent heat of vaporization, which for iron is 725.887 kilojoules per kilogram, and is 40.65 kilojoules per mole for all substances in general (including iron). The amount of heat energy to actually turn a 300-gram boiling point liquid iron kunai to 300 grams of iron gas is 217.766 kilojoules. So far we have the amount of energy needed to get Sakura’s kunai to its boiling point and actually boil it at 401.272 kilojoules.
So the amount of heat energy required to vaporize Sakura’s kunai is 632.722 kilojoules.
In his video, Michael assumes that Charizard’s fire transfers hear via convection (good enough reason for me to do the same with Sasori’s arm flamethrowers), the equation for this being q=hAΔT, with q being the rate of heat transfer in what I’m guessing to be kilojoules per second (the SI base unit of time is the second) and h being the convective heat transfer coefficient. This time, ΔT is the difference in temperature between the fluid (fire in this case) and the surface being heated. A is the surface area (ow what I’m guessing would be the kunai in this case). Michael gives h a value of 0.05 kilowatts per square meter kelvin, which I’ll also use. For A, I’ll use a surface area equal to twice the surface are of the human hand, since that seems to match what this picture of a kunai from the Narutopedia article on kunai indicates.
I like how Apple lets me just hold any picture I find on the internet and then gives me the option to add that picture to my photo library.
One source I found places the surface are of the human hand at 448 square centimeters, setting the kunai at 896 square centimeters, which is roughly 0.09 square meters.
The equation is q=hAΔT. Since I want to determine how hot Sasori’s flamethrowers are, I need to rearrange this equation so that ΔT is by itself on one side of the equals sign. I can do this by dividing both sides by hA, leaving me with ΔT=q/(hA).
With h being 0.05 and A being 0.09, the denominator is 0.0045. I’m getting a feeling that the quotient will be very big, because that’s usually what happens when the divisor is this much less than 1. The smaller the divisor, the bigger the quotient. If the divisor is 1, the quotient will be equal to the dividend. If the divisor is a positive number less than 1, the quotient will be larger than the dividend.
The dividend or numerator is q, the heat energy per time. The heat energy is 632.722 kilojoules and the time is two seconds, so q is 316.361 kilojoules (over 75 thousand calories) per second.
We now know that ΔT (assuming all my math and assumptions are right, though I believe my math to be correct and my assumptions to be reasonable) is equal to 316.361/0.0045. I’m multiplying both the numerator and denominator by ten thousand in order to make the denominator a whole number. This leaves us with 3163610/45, for which the quotient is 70,302.444. This means that ΔT is 70,302.444 degrees Centigrade, meaning that Sasori’s flamethrowers are 70,302.444 degrees Centigrade hotter than the kunai that they incinerated. Since I previously estimated that the kunai started at 21 degrees, this makes Sasori’s arm flamethrowers 70,323 degrees Centigrade, provided that all my math and assumptions were right.
In conclusion, Sasori’s arm flamethrowers are 70,323 degrees Centigrade. By comparison, lightning bolts are only 27000 degrees. Page 200 of the second Databook (Tō no Sho) says that Amaterasu is as hot as the sun. I interpret this to mean the surface of the sun rather than the core, an assumption that I believe to be fairly reasonable. The surface of the sun is 5505 degrees Centigrade, making Amaterasu about the same temperature. This means that being hit by Amaterasu is like taking a bath in liquid helium (only a few degrees warmer than absolute zero, also over 130 degrees Fahrenheit colder than liquid nitrogen) when compared to Sasori’s flamethrowers.
