The Hobbit: The Desolation of Smaug has received mixed reviews this holiday season, with some complaining that the movie does not stick to the book's plot line stringently enough.
Physicist Rhett Allain, however, has a quibble not with the movie's divergence from J.R.R. Tolkien's original story, but with the physics behind one of the more spectacular scenes in the movie. (Spoilers ahead.) The dwarves melt a giant gold statue in mere minutes, using that molten metal to drive the dragon Smaug out of the Lonely Mountain. Here's Allain writing for Wired:
How much gold was melted? I don’t exactly know. How long did it take to melt this gold? It was just a few minutes at most, but I don’t have an exact time. How much energy would this take and what about the power? This is exactly what I want to estimate.
First, Allain jotted down some basic information about gold (he assumed Middle-earth gold has the same properties as planet Earth gold), namely:
- Specific heat of gold = 0.126 J/(gm*K) *oh, this is for gold at 20°C since the specific heat isn’t actually constant.
- Melting temperature of gold = 1337.33 K (1064°C).
- Latent heat of fusion for gold = 63.5 J/g.
- Density of gold = 19.3 g/cm^3. (I’ll use the symbol ρ for density) Also, the other mass measurements are in grams but the volume is in meters cubed. I will actually the density gold as 1.93 x 10^7 g/m^3.
These figures allowed him to calculate the heat needed to melt the statue. He assumed the gold started at a temperature of 10°C and a final of 1064°C. Based on what was shown in the film, he also assumed the gold dwarf statue is about 15 meters tall and has a diameter of about 5 meters.
He calculated that 3.719 x 10^9 watts of power would be needed to melt the gold, which means about 7.44 x 10^9 watts would actually be needed to be produced to achieve that effect because energy transfer is an inefficient process. Producing this energy, he found, would require about 74 metric tons of coal.
Up until this point, Allain writes, this scenario is all perfectly reasonable. It's the short time in which the dwarves managed to melt the giant statue that represents the real departure from the laws of physics:
It just seems crazy to be able to burn 74 metric tons of coal in just 5 minutes. Of course there is something else they could use get the power they want. What about a nuclear power plant? A Nimitz class aircraft carrier has a power plant that can produce 194 Mega Watts (1.94 x 10^8). That’s not enough. Not nearly enough. The dwarves would need about 38 Nimitz class carriers in order to get that kind of power.
So unless those ancient dwarf furnaces were burning some sort of Middle-earth super fuel, in real life, Smaug probably would have just eaten the dwarves while they were standing around waiting for all of that gold to melt.
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