Premise[]
Recently, Elizhaa asked me about the energy to vaporize a planet. So... I figured I'd try to find out.
Not sure if we have a calc for this and this will be relatively rough, but let's try. I should note that this calc does not account for other factors of dispersing the Earth into vapor- the pressure, GBE, etc. We are purely looking at the energy needed to reduce the base elements to gaseous form.
Calc[]
Based on this we're looking at the most prevalent elements in the Earth. All of this comprises the mass of the Earth (5.98e24 kg).
Iron: 32.1% (1.91958e24 kg), Heat Capacity of 460
Oxygen: 30.1% (1.79998e24 kg), Heat Capacity of 919
Silicon: 15.1% (9.0298e23 kg), Heat Capacity of 710
Magnesium: 13.9% (8.3122e23 kg), Heat Capacity of 1050
Sulfur: 2.9% (1.7342e23 kg), Heat Capacity of 700
Nickel: 1.8% (1.0764e23 kg), Heat Capacity of 440
Calcium: 1.5% (8.97e22 kg), Heat Capacity 630
Aluminium: 1.4% (8.372e22 kg), Heat Capacity of 870
The last 1.2% is a mixture of tons of lesser elements. For the sake of this calc, we'll be ignoring it.
All layers of the Earth are solid save for one, which is a layer of molten iron as hot as the surface of the sun. Considering how miniscule the oceans are in relation to the rest of Earth we'll be ignoring these as the only other quote unquote major source of liquid.
Let's get on with the liquid bit first. The inner core is about 1% of the Earth's total volume at 1.0837e21 m^3 total. 1% of that is 1.0837e19 m^3 for the Inner Core.
Density of liquid iron is 6980 kg/m^3. Mass of the Inner Core is 7.564226e22 kg. This means the iron content of Earth is now divided into the categories "Liquid" and "Solid".
Solid Iron: 1.84393774e24 kg
Liquid Iron: 7.564226e22 kg
Now we can actually calc the energy needed to vaporize. For the purpose of this calc we will assume the Earth is heated uniformly to the same heat. Let's look at the heat each element needs to vaporize (AKA Boiling Point).
Iron = 2862 C
Oxygen = -183 C (so this is pointless)
Silicon = 3265 C
Magnesium = 1091 C
Sulfur = 444.6 C
Nickel = 2913 C
Calcium = 1484 C
Aluminium = 2470 C
So Silicon's heat will be our assumed heat. As a high-end we'll use the boiling point of Tungsten in order to account for truly all elements on Earth- 3414 C is our high-end.
Let's handle heat change first. The core is assumed to maintain heat similar to the surface of our sun all throughout as a starting point- 5778 C, so not relevant. We'll assume the other stuff is the average of their ambient temperatures.
Mantle: 4000 C - 200 C (2100 C average) and is 84% of Earth's Volume
Crust: 400 C - 300 C (350 C average) and is 1% of Earth's Volume
For the purposes of this calc we will assume all elements are roughly evenly distributed through the sections of Earth aside from the Inner Core- we even know the Outer Core isn't entirely iron.
Outer Core represents 15% of total Earth volume and is comprised of iron and nickel for the most part. The assumptions have to be hefty in order to make up for this- we'll assume half of the world's nickel is present here (0.9%) and the rest is Iron (14.1%, or about 45.338% of remaining iron). Adjusted values below.
Iron in Inner Core: 7.564226e22 kg
Iron in Outer Core: 8.36004493e23 kg
Iron in Mantle/Crust: 1.00793325e24 kg
Nickel in Core: 5.382e22 kg
Nickel in Mantle/Crust: 5.382e22 kg
Anything in the Core isn't relevant for heat change since everything there would vaporize from heat anyways if the pressure wasn't so high. So we're ignoring them aside from just shifting states of matter.
We'll put all the things the Crust is made of here. Everything else is assumed to be in the Mantle. We're looking at Oxygen, Silicon, Aluminium, Iron, Calcium, and Magnesium. Divide this 1% volume between them for the following masses:
Total Volume = 1.0837e19 m^3
Volume Each = 1.80616667e18 m^3
Oxygen = 2.06083617e15 kg
Silicon = 4.20475601e21 kg
Aliminium = 4.89471168e21 kg
Iron = 1.42217564e22 kg
Calcium = 3.61233334e21 kg
Magnesium = 3.14273001e21 kg
Subtract this from values established at the beginning to get the following table of values.
Mass of Elements[]
Mass of Earth = 5.98e24 kg
Mass of Crustal Iron = 1.42217564e22 kg
Mass of Crustal Oxygen = 2.06083617e15 kg
Mass of Crustal Silicon = 4.20475601e21 kg
Mass of Crustal Calcium = 3.61233334e21 kg
Mass of Crustal Magnesium = 3.14273001e21 kg
Mass of Mantle-Based Iron = 9.93711494e23 kg
Mass of Mantle-Based Nickel = 5.382e22 kg
Mass of Mantle-Based Oxygen = 1.79998e24 kg
Mass of Mantle-Based Silicon = 8.98775244e23 kg
Mass of Mantle-Based Sulfur = 1.7342e23 kg
Mass of Mantle-Based Magnesium = 8.2807727e23 kg
Mass of Mantle-Based Aluminium = 7.88252883e22 kg
Mass of Mantle-Based Calcium = 8.60876667e22 kg
Mass of Outer-Core Iron = 8.36004493e23 kg
Mass of Outer-Core Nickel = 5.382e22 kg
Mass of Inner-Core Iron = 7.564226e22 kg
Good shit. Now we need Specific Heat energy since we've spent all that time setting this shit up. We won't do anything for the Core elements since their heat doesn't need to change at all for this event to happen.
Specific Heat Energy[]
As said earlier, as a low-end we assume 3265 C end temperature, as a high-end we assume 3414 C end temperature based on Tungsten. EDIT: Coming back to it, just using the high-end. Results don't change much and it's the only thing that makes logical sense.
Calculation for this is just mass times specific heat times temperature change (which varies). I'm beaten by this so far so I'm using this.
Low-End Temp Change for Crust Elements = 2915 C
High-End Temp Change for Crust Elements = 3064 C
Low-End Temp Change for Mantle Elements = 1165 C
High-End Temp Change for Mantle Elements = 1314 C
Let's get to it.
Crustal Iron = 2.005e28 Joules
Mantle Iron = 6.006e29 Joules
Crustal Oxygen = 5.803e21 Joules
Mantle Oxygen = 2.174e30 Joules
Crustal Silicon = 9.147e27 Joules
Mantle Silicon = 8.385e29 Joules
Crustal Magnesium = 1.011e28 Joules
Mantle Magnesium = 1.143e30 Joules
Mantle Sulfur = 1.595e29 Joules
Mantle Nickel = 3.115e28 Joules
Mantle Aluminium = 9.011e28 Joules
Mantle Calcium = 7.127e28 Joules
Total Energy of Heat Change = 5.14743701e30 Joules, Small Planet level. But we're far from done.
Shifts in Matter[]
Now we get into truly changing the matter from solid/liquid to gas. For this we classify everything by solid or liquid. Everything in the inner core is liquid (a small amount of iron)- everything else is held to be solid. This includes the outer core which shifts between liquid and solid.
For solids, they must undergo a state of fusion and vaporization, so we need to multiply them by their values for that in J/kg. For the liquid, it must only undergo the value for vaporization. Oxygen is already gaseous so it need'nt be accounted for.
Let's start.
Solid Iron =1.84393774e24 kg
Liquid Iron = 7.564226e22 kg
Solid Silicon = 9.0298e23 kg
Solid Magnesium = 8.3122e23 kg
Solid Sulfur = 1.7342e23 kg
Solid Nickel = 1.0764e23 kg
Solid Aluminium = 8.372e22 kg
Solid Calcium = 8.97e22 kg
Let's start with the irons.
Liquid Iron Vaporization = 4.700e29 Joules
Solid Iron Fusion & Vaporization = 4.557e29 Joules & 1.146e31 Joules
Solid Silicon Fusion & Vaporization = 1.614e30 Joules & 1.154e31 Joules
Solid Magnesium Fusion & Vaporization = 3.062e29 Joules & 4.357e30 Joules
Solid Sulfur Fusion & Vaporization = 9.288e27 Joules & 5.299e28 Joules
Solid Nickel Fusion & Vaporization = 3.204e28 Joules & 6.793e29 Joules
Solid Calcium Fusion & Vaporization = 1.911e28 Joules & 3.469e29 Joules
Solid Aluminium Fusion & Vaporization = 3.320e28 Joules & 9.091e29 Joules
Total Fusion + Vaporization Energy = 3.2284828e31 Joules, Small Planet level
Keep in mind I had to calculate the latent heat of fusion/vaporization for Magnesium, Sulfur, and Nickel since they aren't present on our Calculations page.
Magnesium has Fusion of 8954 J/mol and 127400 J/mol for Vaporization. Magnesium weighs 24.305 g/mol so energy is...
- Fusion: 368401.563 J/kg
- Vaporization: 5241719.81 J/kg
Sulfur has Fusion of 1717.5 J/mol and 9800 J/mol for Vaporization. Sulfur weighs 32.07 g/mol so energy is...
- Fusion: 53554.724 J/kg
- Vaporization: 305581.54 J/kg
Nickel has Fusion of 17470 J/mol and 370400 J/mol for Vaporization. Nickel weighs 58.69 g/mol so energy is...
- Fusion: 297665.501 J/kg
- Vaporization: 6311126.26 J/kg
I'd like these to be added to the calculations page for future reference if that's okay.
Total Energy[]
We're adding together all values denoted in this blog as well as the GBE of Earth.
GBE of Earth = 2.24e32 Joules
Matter Shifting of Earth = 3.2284828e31 Joules
Heat Change of Earth = 5.14743701e30 Joules
Total Energy = 2.614e32 Joules, Planet level.
Underwhelming but that was a helluva long path to get here. Oof.
Tallies[]
The World Gets Vaporized: 62.48 Zettatons of TNT, Planet level