Dude, what?
A first-hand look at exponentially decaying thought processes
The most common isotope of uranium, uranium 238, has a half-life of 1.41x10 17 second, or 4.468 billion years. This means that half of the atoms in a sample will decay in that amount of time.
1 : Write the equation for the decay of U238.
y = a(.5) x
Where X is 4.468 billion years.
2: An unrealistic experiment involves 500 grams of newly-formed uranium 238. The researchers would like to know when the 500g sample will decay to 99.9999% of its value. The experiment starts in January 2010.
500g x .999999 = 499.9995
Inputting that value as y and intersecting it with the original equation gives us 1.4427e-6, or .0000014427.
.0000014427 * 4,468,000,000 = 6,445.9836.
6,445.9836 + 2010 = 8455.9836.
By the end of the year 8445, the sample would have decayed to 99.9999% of its original value.
3: The researchers would also like to know at what year it will be reduced to 400 grams.
y = 400 intersects with 500(.5) x at x=.32192809.
.32192809 * 4,468,000,000 = 1,438,374,706
1,438,374,706 + 2010 = 1,438,376,716 years.
4: The scientists soon realize that, despite predicted advancements in cloning technology, they will be unable able to see the experiment through. This is because, they calculate, in roughly 1 billion years the sun will have grown in intensity that water will evaporate from the earth and soon after that, the atmosphere will burn out. At that point, 1 billion years from now, what will be remaining of the 500g u238 sample?
1 billion years is 22.38137869% of 4.468 billion years. Inputting the decimal version of that percentage, .2238137869, into our graph as the x value gives us 428.1494. By the time the world ends (at least by the time it theoretically ends because of the sun becoming too powerful) the sample will be 428.1494 grams, or 85.62988% of its original value.
5: While out drinking, the scientist suggest that in a billion years or so, time travel would have had to be invented, because, that seems like quite enough time to figure that out. Then, logic would dictate that their clones would be able to utilize time travel to reverse-decay the U238 into its previous isotope, protactinium 238. This would require the U238 to go back to 2010 when it was formed, and then travel even further than that to the point where it was a newly-formed p238 isotope sample. p238 has a half-life of 2.27 minutes. p238 has an isotope mass of 238.054506. u238 has a mass of 238.050783.
How many years would the clones and the uranium sample have to travel in order to reach that point? How much p238 would you have once you have reached the point when p238 was newly-formed?
Okay, let me just figure out this timeline... time travel always complicates things.
2010: experiment starts with newly formed isotope.
~1,000,002,010: u238 sample enters time travel device accompanied by the scientist clones who I am imagining look like Muppets or something. They travel back in time one billion years to when the u238 was first formed.
They then continue on a bit more and realize that they don’t really understand how these isotopes are formed. If these things have half-lives, it’s not like they simply turn into a different isotope after a period of time, right? Okay, well I am going to assume that as an isotope decays, the mass it loses gets converted into the next isotope in the decay daisy-chain. This is probably not the case, but whatever, it is 6:26am and I am very tired.
So... they travel 1 billion years and 4.54 minutes and they reach their final destination: the u238 has reverted back to new p238. I am not sure how much p238 they have, but I am pretty sure it should be around 500g, as that seems pretty reasonable.
6: The clones then travel back in the time machine to the year 2010 in order to alert their former selves what a pointless experiment it all is, and how they should probably stop as it is a huge waste of their time and energy. One of the scientists asks his clone for the lottery numbers, the clone reads off some kind of time travel protocol about how that would altar the fabric time and space, and how the universe could possibly implode on itself. The scientist then says “Whatever, just give me the numbers.”
And that’s how the universe ended.
uh... 42?
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