Kurzgesagt – In a Nutshell
Sources – Gold Apocalypse
Prof. Matthew Caplan
Special thanks to Matt Caplan, Professor of Physics at Illinois State University, for bringing his expertise to our wacky ideas.
– Here in the Kurzgesagt labs we only work on the most important scientific problems like “What if we nuke stuff” or “how about we make this elephant explode” or who could forget “look at this thing, it is really big!”.
Here are the links to the videos we mention:
#What If We Detonated All Nuclear Bombs at Once?, 2019
#How to Make an Elephant Explode – The Size of Life 2, 2017
#The Largest Star in the Universe – Size Comparison, 2020
– Continuing this proud tradition, let us explore the scientific mystery of what would happen to you, if Earth suddenly turned into gold! The “Midaspocalypse”, based on the ancient tale of King Midas who was cursed so everything he touched turned into gold.
#Midas, Ancient History Encyclopedia, retrieved 2020
Quote. “Midas was a mythical king of Phrygia who was famous for his ability to change anything that he touched into solid gold.”
– An atom of gold has 79 protons and 118 neutrons in its nucleus The electric force of the protons on the electrons around them shapes the atom and gives gold its chemical properties, like that it doesn’t rust, and that it’s kind of shiny and bendy. So, to make not-gold into gold, we have to change atoms.
Here are all the chemical properties of gold, as well as information on its history, sources and uses:
#Gold, Royal Society of Chemistry, retrieved 2020
https://www.rsc.org/periodic-table/element/79/gold
– Let’s say Midas touches a duck. All the light elements like hydrogen, carbon, and oxygen gain electrons and protons and neutrons to become gold. Not only is the duck suddenly 33 times more massive it is also much too dense.
The following are calculations performed by Professor Matt Caplan:
We know that ducks and humans are made of the same elements in the same portions. Oxygen, carbon, hydrogen, calcium and more. An average 70kg human has about 7*10^27 atoms, so an average 1kg duck would have 10^26 atoms.
The mass of one gold atom is 3.27*10^-25 kg. So, one duck’s-worth of gold atoms would add up to 32.7 kg!
Now, these atoms would be packed together the same way that the original oxygen, carbon, hydrogen and other atoms were. Gold atoms, however, are much larger than any of those other atoms. In the reference below, we find that gold atoms are 135 picometres (a trillionth of a metre) in radius.
#Gold: radii of atoms and ions, WebElements, retrieved 2020
https://www.webelements.com/gold/atom_sizes.html
For comparison, carbon, oxygen and hydrogen atoms are 70, 60 and 25 picometres in radius respectively.
#WebElements, retrieved 2020
https://www.webelements.com/carbon/atom_sizes.html
https://www.webelements.com/oxygen/atom_sizes.html
https://www.webelements.com/hydrogen/atom_sizes.html
To put it into numbers, a normal duck has a density of about 1000 kg/m^3. That’s why they float on water! Their mass fits inside a 0.001m^3 volume.
When we transform a duck into gold, those new 33kg of mass are squeezed into the same 0.001m^3 volume. This gives it a density of 33,000 kg/m^3. However, solid gold should have a density of 19,300 kg/m^3 (see reference below). It is too dense, and it won’t stay that way for long.
#Gold, Royal Society of Chemistry, retrieved 2020
https://www.rsc.org/periodic-table/element/79/gold
– The gold atoms are far closer together than they like and repel each other violently, causing the golden duck to explode with the energy of half a ton of TNT, leaving only gold dust and a very dead Midas. This is clearly not a good way for Midas’s power to work.
As mentioned above, all those gold atoms don’t want to be packed so densely together. Professor Matt Caplan uses the free electron model for metals to calculate the energy released when those gold atoms repel each other into a more natural configuration.
The equation is Total Electron Energy = 3/5 * (Number of Electrons) * (Fermi Energy)
The number of atoms in a gold duck, from the calculations above, is 10^26. Each gold atom has 79 electrons, so the total number of electrons is 7.9*10^27. In a simple block of gold, the Fermi Energy of the electrons around each atom is 5.5 electronVolts, which is about 8.8*10^-19 Joules. We can calculate that the Total Electron Energy is 4.17 GigaJoules.
However, this is not a simple block of gold. It is too many atoms packed in too small of a space. More specifically, the density is 33,000/19,300=1.7 times higher than it should be. Fermi Energy in this case scales with density by the power 2/3, so the Fermi Energy rises by a factor 1.7^(2/3)=1.4 to 1.23*10^-18 Joules per electron.
This means the duck’s Fermi Energy actually adds up to 5.84 GigaJoules.
The difference between these two energies, 1.67 GigaJoules, gets released explosively. It is the equivalent to 399kg of TNT.
– Let’s freeze time for just a moment, and rearrange all the matter in the Earth. Just like the duck, the Earth is now solid gold, but with many tiny atomic-scale gaps.
While these gaps weren’t a huge deal for the duck, they’re a big problem for the Earth. A spongy planet is not a thing that can exist, as gravity compresses Earth, squeezing it together to close up the gaps. As a result, the Earth contracts, shrinking to 2/3rds its radius.
Another calculation tells us by how much the Earth shrinks.
The Earth has a mass of 5.97*10^24 kg. If it was made of gold, it would have a density of 19,300 kg/m^3. If we divide mass by density, we get volume: 3.09*10^20 m^3. This fits inside a sphere with a radius of 4,195,245 metres or 4195km.
This is 65.8% of the Earth’s current radius of 6371 km.
#Planetary Fact Sheet, NASA, retrieved 2020
https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
– But the ride doesn’t last forever, it only takes 10 minutes for everything to crash down, and a weird 10 minutes indeed. And then, as abruptly as it started, it stops. The collapsing earth has reached its desired size and gravity is suddenly turned back on for you. Hopefully you enjoyed your minutes of floating because the ground and you with it crash into the planet at 30,000 kilometers/hour, making your body splatter like a water balloon as it hits the ground. In one instant basically all of humanity gets smashed to red puddles.
Professor Matt Caplan goes through multiple steps to calculate how long it would take to fall from Earth’s current surface (6371km from the planet’s centre) to its new gold-shrunk surface (4195km from the planet’s centre). It is not a straightforward calculation because gravity increases as we get closer to the centre.
The first step is to get the characteristic freefall timescale for our current Earth. The equation to use is Characteristic Freefall Time = ((3 * pi)/(32 * G * rho))^0.5
pi is 3.142
G is the gravitational constant, equal to 6.67*10^-11 m^3/kg/s^2
Rho is the average density, in this case 5500 kg/m^3
We calculate a value of 896 seconds, or roughly 14.9 minutes.
#Timescales in Stellar Interiors, MIT, 2019
http://www.mit.edu/~iancross/8901_2019A/lec011.pdf
Quote:
#The mean density of the Earth, David W. Hughes, 2006
https://www.researchgate.net/publication/234529392_The_mean_density_of_the_Earth
Quote: “The Earth’s mean density of around 5520kg m^3 hovers between the density of laboratory iron (7000kg m−3) and the density of crustal surface rock5 (2700kg m^3)”
The next step is to integrate the fall time between the current surface (6371km) and the new, lower surface (4196km). It can be done using Wolfram Alpha, to obtain a value of 0.699*896=626 seconds or 10.4 minutes.
Integral for full freefall (1.5708):
Integral for freefall to 2/3rds radius (1.098):
1.098/1.5708=0.699
A further calculation tells us how fast we are going when we hit the new surface.
The initial gravity is 9.81 m/s^2. We know that gravity scales with the square of radius. A 1.5 times smaller Earth would have 1.5^2=2.25 times higher surface gravity, or roughly 22m/s^2. Anything falling between these two surfaces would experience varying gravity between them. One way to calculate the final velocity of a falling object is through a step calculation, where we see what happens every 10 km throughout the fall. This table displays the results: https://drive.google.com/file/d/1zuuWPyvo6MX8yTZ_Uy9bOH0hJa0oUTcX/view?usp=sharing
We obtain a final velocity of over 8,200m/s, which is a whopping 29,500km/h.
#Gravity, Hyperphysics, 2020
http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grvcon
– This is only the start of our problems though, since earth imploded supersonically. The kinetic energy of the implosion is basically equivalent to detonating a planet made of TNT. Crushed together with incredible forces, earth’s core reaches a million degrees Celsius - a temperature closer to the core of a star than anything we’re used to finding on earth.
The very practical kinetic energy equation tells us how much energy is released by an object striking at a certain speed.
Each kilogram that falls from Earth’s current radius to its new, smaller radius reaches 9978m/s and gains a kinetic energy of 49.78 MegaJoules. It is all released upon impact.
An average human would explode like a bomb with 3.48 GigaJoules of energy, equivalent to about 832kg of TNT. But there’s more than just people falling. All the oceans, the mountains...the whole crust of the Earth falls. That’s 2.77*10^22 kg of mass releasing 1.37*10^30 Joules of energy upon impact.
#Kinetic Energy, The Physics Hypertextbook, retrieved 2020
https://physics.info/energy-kinetic/
Quote: “If kinetic energy is the energy of motion then, naturally, the kinetic energy of an object at rest should be zero. Therefore, we don't need the second term and an object's kinetic energy is just… K = 0.5 * m* V^2“
– As the earth ‘crashes into itself’ it generates an enormously powerful shockwave which plows upward, catapulting the atmosphere up and off. The earth’s surface temperature reaches hundreds of thousands of degrees, and everything on it is instantly vaporized to a fluffy plasma cloud that starts to expand, but not by much.
From our calculations above, we learn that each kg of matter on the surface gains 49.76 MegaJoules of energy. Immediately upon impact, that energy becomes heat. If we divide that energy by the average heat capacity of air (about 1300 J/kg/K at high pressure), we find a temperature of nearly 40,000 Kelvin.
Even matter with a high heat capacity, like water (2000J/kg/K when steam) or people (3500J/kg/K) would be pushed into temperatures of tens of thousands of Kelvin. That’s more than vaporization; it’s enough to break everything down into ionized plasma.
At these temperatures, it doesn’t matter whether we measure it in degrees Celsius or Kelvins as the difference between the two is much smaller than the range of uncertainty in our estimates. We used Kelvin in our calculations because that is the common standard.
#Air - Specific Heat at Constant Pressure and Varying Temperature, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/air-specific-heat-capacity-d_705.html
#Human Body and Specific Heat, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/human-body-specific-heat-d_393.html
#Water Vapor - Specific Heat, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/water-vapor-d_979.html
– Many of the atoms that may have been you get mixed into this cloud, while others boil off, escaping from the atmosphere. The golden plasma outshines the sun while the enormous radiation lifts tonnes of material off into space. Over the coming days the plasma cloud cools, and eventually freezes into a shiny little golden ball.
#Sun, Encyclopædia Britannica, retrieved 2020
https://www.britannica.com/place/Sun
Quote: “The Sun is classified as a G2 V star, with G2 standing for the second hottest stars of the yellow G class—of surface temperature about 5,800 kelvins (K)—”
That’s about 5530°C.
– Even though our new pure gold earth is not expanding or contracting, it’s suddenly much more massive. The density of gold is three and half times greater than the earth, meaning the earth is going to get three and half times more massive! For starters, everyone now has to contend with surface gravity that is more than three times stronger, so if you’re not a champion weightlifter who is used to carrying around a few times your bodyweight on your shoulders, you’re probably going to be slammed to the ground by your own weight.
The density of gold is 19,300 kg/m^3. The density of Earth is 5520 kg/m^3. The new gold Earth would have a mass that is 19300/5520: 3.49 times higher than it is currently. As a result, gravity would also be 3.49 times higher.
#Technical data for Gold, PeriodicTable, retrieved 2020
https://periodictable.com/Elements/079/data.html
Quote:
#The mean density of the Earth, David W. Hughes, 2006
https://www.researchgate.net/publication/234529392_The_mean_density_of_the_Earth
Quote: “The Earth’s mean density of around 5520kg m^3 hovers between the density of laboratory iron (7000kg m−3) and the density of crustal surface rock5 (2700kg m^3)”
– Depending where you were when earth turned into gold, this alone could seriously hurt or even kill you. Trees and artificial structures collapse under stress they were never meant to sustain while birds and planes and all things that were able to fly or float splash to the ground all around you.
Trees have to support their own weight under our gravity. Under a 3.49 times higher gravity, they would have to carry the weight of two other trees and them some. It would reduce the maximum height of a tree to 28% of what it is today. Not really a problem for young trees that are still growing, but older trees would quickly collapse.
In most modern countries, buildings have an extra margin of strength to withstand their own weight plus other loads, like strong winds or minor earthquakes. This can amount to 15% greater than their maximum expected load, up to 200% more in certain cases. It is not enough to withstand our new gravity. The smallest disturbance would send buildings tumbling to the ground.
#Design of Masonry Structures, A.W.Hendry et al., 2004
Quote:
Birds and planes travel at a speed such that the air passing over their wings generates a lift equal to their weight. In higher gravity, they must generate more lift. Since they cannot make their wings bigger, they must travel faster - about 87% faster. Birds would quickly get exhausted and fall. Planes can sustain higher speeds but would run out of fuel quickly and have to land without delay.
#The Lift Equation, NASA, retrieved 2020
https://www.grc.nasa.gov/www/k-12/airplane/lifteq.html
Quote: “The lift equation states that lift L is equal to the lift coefficient Cl times the density r times half of the velocity V squared times the wing area A.
L = Cl * A * .5 * r * V^2”
– And you’re not the only thing weighed down by the greater gravity. The weight of the atmosphere, and also atmospheric pressure, nearly quadruples, which is a bad thing if you like living. On its own, this won’t kill you - scuba divers can comfortably breathe air at these pressures for a while, but unfortunately squeezing the atmosphere this much raises its temperature to 150 degrees Celsius, which is like the insides of an oven. The entire earth’s surface bakes, roasting anything and everything. There is no escape.
We can calculate the conditions in an atmosphere under higher gravity.
The mass of the atmosphere is 5.148*10^18 kg. The pressure it exerts on the surface is simply its weight divided by the planet’s surface area. If gravity increases 3.49 times, the atmosphere’s weight becomes 3.49 times higher and so does the atmospheric pressure.
Currently, the atmospheric pressure is 100 kiloPascals. On gold-Earth, it would be 349 kiloPascals.
#How Much Does Earth’s Atmosphere Weigh?, Encyclopaedia Britannica, retrieved 2020
https://www.britannica.com/story/how-much-does-earths-atmosphere-weigh
Quote: “The total mass of Earth’s atmosphere is about 5.5 quadrillion tons, or roughly one millionth of Earth’s mass.”
#Atmospheric Pressure vs. Elevation above Sea Level, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/air-altitude-pressure-d_462.html
Quote:
Higher pressure in a gas means higher temperature. The gas laws for adiabatic (no energy loss) compression state that:
New Pressure/Old Pressure = (New Temperature/Old Temperature)^(y/(1 - y))
The ratio between new pressure and old pressure in our case is 3.49.
The New Temperature to Old Temperature ratio is what we are trying to find.
y is the adiabatic constant. For our atmosphere, composed nearly entirely of two-atom molecules like nitrogen and oxygen, this value is 1.4
The equation can be rearranged so that
New Temperature = Old Temperature * (New Pressure/Old Pressure)^((y - 1)/y)
Inputting the values we have, and using a standard 300 Kelvin as the Old Temperature, we get:
New Temperature = 300 * (3.49)^((1.4 - 1)/1.4) = 300 * 1.43 = 429 Kelvin
The atmosphere’s temperature rises to 429 Kelvin or 156°C.
#Adiabatic Expansion – Adiabatic Compression, Nuclear-Power, retrieved 2020
Quote:
#Specific Heat Ratio of Air, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/specific-heat-ratio-d_602.html
Quote:
– Gold may be a metal but it is about three times weaker than steel, and is also very malleable, which makes it very bad to make up mountains – the tallest mountains that can be supported are now only about two kilometers high. So whole ranges compress, as their own weight basically crushes their base. It’s hard to say what happens here. We are probably in for giant earthquakes and landslides as the planet is squeezed into a new form.
#Yield and Ultimate Properties, Structx, retrieved 2020
https://structx.com/Material_Properties_003a.html
We can use the methods devised by Victor F. Weisskopf, a theoretical physicist famous for his work on the Manhattan Project, to estimate the maximum height of mountains. The equation we can use is:
Maximum height = Specific heat of fusion/Gravity
Specific heat of fusion is the energy needed to melt a material. As the theory goes, a mountain cannot be so tall that the pressure at its base is enough to melt the materials it is made of. If it is taller than its maximum height, it will ‘flow’ into a smaller shape.
In reality, rocks break and metals bend under lower pressures than is needed to melt them. That is why no natural mountain actually reaches its maximum height.
Gold is a weak metal with a low specific heat of fusion. Specifically, it only needs 63.5 kJ/kg to melt it. For comparison, it takes 1787 kJ/kg to melt rock. If we input the values for gold, as well as the increased gravity of 3.49*9.81: 34.23m/s^2, we obtain a maximum mountain height of 1855m.
#How high could a mountain be?, Victor Weisskopf, 1975
http://www.hk-phy.org/articles/mount_high/mount_high_e.html
#Metals - Latent Heat of Fusion, Engineering ToolBox, retrieved 2020
https://www.engineeringtoolbox.com/fusion-heat-metals-d_1266.html
Quote:
– And it’s not just mountain ranges- the differences between the continents and the ocean floor level out, causing the ocean basins to overflow, sending massive tidal waves over the earth’s surface. What remains is a planet made from gold, entirely covered by an ocean 3 kilometers deep, a super hot atmosphere and a lot of dead people.
On the new gold-Earth, the ocean floor moves up while the continental shelves are flattened. The oceans will overflow. Their volume is 1.33 billion cubic kilometres. Our planet has a surface area of 510 million square kilometres. If we divide that volume by the surface area, we get an average depth of 2.6 kilometres.
#The Volume of Earth’s Ocean, Matthew A. Charette and Walter H .F. Smith, retrieved 2020
https://www.whoi.edu/science/MCG/groundwater/pubs/PDF/Charette_TOS.pdf
Quote:
#By The Numbers, NASA, retrieved 2020
https://solarsystem.nasa.gov/planets/earth/by-the-numbers/
Quote: “Surface Area: 1.97 x10^8 square miles (5.1 x 10^8 km^2)”