After watching The Hobbit: The Desolation of Smaug I quickly realized my initial estimation for the amount of gold under the mountain was WAY too low. There are some other things I noticed, but perhaps I should give a SPOILER ALERT.
I don't consider these to be REAL spoilers since the story essentially follows the book which published in 1937. But it's better to be safe and I don't want any ranting emails from angry parents saying I ruined their kids movie experience. Here is a random picture of trees in order to give you an opportunity to bolt out of here before seeing the spoilers.
Ok, now it's just us. We are free to say whatever we like about The Hobbit: The Desolation of Smaug. First, let me make a few comments about the movie.
- Of course there is a lot of material added to the movie that wasn't in the book. How else could you make a 300 page book into three movies that are each about three hours long?
- In this movie, we finally get to see Gandalf use magic. Sure, he used magic in The Lord of the Rings movies, but that was just simple stuff. He made his staff glow and he blinded some orcs. Oh, that's right; he also made an impressive and threatening voice when talking to Bilbo. Of course he still isn't quite at the same magic level as Albus Dumbledore or even Harry Potter, but he is a wizard. Personally, I kind of like that Gandalf doesn't use a ton of magic.
- The barrels down the river wasn't so much a stealthy escape as it was an epic battle. Oh well, I have already looked at dwarves in floating barrels.
- The dwarves fight Smaug. Yes, in the book the dwarves just kind of avoid the dragon. Well, this fight is what I want to talk about in this post.
You could consider this the spoiler part of the post. In an attempt to defeat Smaug, the dwarves try to drown him in liquid gold. Yup, that's right. Apparently, there was a whole bunch of gold just waiting to be melted down and then poured into a GIANT cast of a dwarf statue.
The Physics of Melting Gold
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.
To start, I need to know some things about gold. If I want to take room temperature gold and make melted gold I have to do two things. First, I have to increase the temperature. Second, I have to make the gold phase change from a solid to a liquid. The specific heat (we use the symbol c) tells me how much energy it takes to raise a mass of gold a certain temperature. The latent heat of fusion (we use the symbol lf) tell me how much energy per mass I need to make the gold go from solid to liquid. Actually, it seems that the common name for this is the specific heat of fusion.
- 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/cm3. (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 107 g/m3.
Those are not really estimates. How much gold do the dwarves need to melt? I don't know. Let's just say that there is enough gold to make a HUGE dwarf statue. I am going to leave the actual estimations for later. For now, I will say that the statue is like a cylinder with a radius of r and a height of h. From this I can get the volume and mass of the gold.
Now to melt the gold. I will assume a starting temperature of Ti and a final temperature of Tf. Using these values and the mass, I can calculate the energy needed to melt the gold.
Now, what about power? If the gold is melted in some time interval (Δt) then the power would be:
That's the physics - now for some values.
Wild Estimations
Let's start with the values that are the easiest. The initial and final temperatures of the gold should be easy to guess. I will use a starting temperature of 10°C and a final of 1064°C. The starting temperature could probably even be colder since it is inside a mountain - but I will use that value anyway.
The other value that's easy to guess is the time. Actually, if I had the clip from the movie I could measure the time directly. Instead, I will just make a wild guess based on my memory. How about 5 minutes to melt the gold.
The size of the statue is difficult to estimate. Again, with the actual video from the movie this would be fairly straight forward. Instead, I will just have to guess. I am going to say the golden dwarf statue is like a cylinder with a height of 15 meters and a radius of 2.5 meters. Actually, to get this I estimated the height of the statue at 15 meters. Then I looked at a dwarf and estimated a height to radius ratio of about 6. Who cares? Well, if I want I can just change the height and then generate new values for the radius.
The other values in the calculation (such as the properties of gold) are known. Well, that assumes Middle Earth gold is the same as Earth gold. Putting in these values and estimates, I get a required power of 3.719 x 109 Watts. That's not the power required from the source, this is the power TO the gold. Typically, a heater that you use will just give some of the energy to the gold and some to the surroundings. If you assume a heating efficiency of 0.5, the power from the source would have to be twice as high at 7.44 x 109Watts.
Of course this isn't good enough. What if my estimates are way off? How about a plot of power vs. height of statue from 10 to 20 meters (with a 0.5 efficiency)?
It's not a linear function since as the height of the statue increases so does the radius.
Heating Sources
I have to make another comment about the movie. Actually, it's a complaint. After getting to the giant furnace to melt the gold, one dwarf comments with something like "Oh no! The furnaces are cold!" WHAT? Is this any surprise? The mountain has been abandoned for 150 years. Are you really surprised that the furnace isn't still going? Ok. That is all I wanted to add.
How do you heat up gold? I suspect the dwarves used coal or something similar. According to Wikipedia, coal has an energy density of 24 megajoules per kilogram and a coal power plant is about 40% efficient. How much coal would the dwarves need to burn to melt all this gold? In this case, I just need the energy to melt this gold - the time doesn't matter. The gold would need 7.14 x 1011 Joules to heat and melt. I will use e to represent the furnace efficiency and σ for the energy density of coal (in Joules per kilogram). I can now solve for the mass of coal needed.
74 metric tons of coal (81.99 short tons). Coal has a density of 640 - 929 kg/m3. If I go with 800 kg/m3, this coal would fill a space of 93 m3. That's a lot of coal, but not a crazy amount.
The problem with coal isn't the amount of coal you would need. Instead, the problem is with the time. 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 108). That's not enough. Not nearly enough. The dwarves would need about 38 Nimitz class carriers in order to get that kind of power. Well, they could use the power plant in the S.H.I.E.L.D. Helicarrier (which I previously estimated). That would do the trick.
One last thing. Could you float in liquid gold if you had a wheel barrel? Yes. I'm pretty sure a wheel barrel would even float in water. However, the wheel barrel would get pretty darn how fairly quickly. I don't think you could ride in a river of molten gold for very long without bad things happening.
