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I had an interesting question today and would like to see the approach some of you take in solving it, here it is:

Everyone has to refill their gas tank at some point and most do so before the tank is empty so as to keep some gas in the tank before the fuel gauge hits E. When we reach this point we refill, thus diluting the gas currently in the tank by increasing the volume of gas in the tank. The gas then burns down once more after usage of the vehicle then turning into the few gallons in your tank before you hit E, this of course prompts you to refuel again.

So my questing is simple, given a fixed first fill up, about how long (in any time unit) would it take for every molecule of the original tank to be burnt off in the engine? (or the best possible answer)

Of course many factors vary but lets make this the ideal scenario for calculations. The individual always refills their tank with no molecule more than 1 gallons left, the individual uses their vehicle for exactly 4 hours each day, their car uses exactly 25 mpg, the tanks maximum gasoline capacity is 10 gallons, the car never needs to be repaired, always works at maximum efficiency, weather or heat from the machine do not effect the car...ect, eradicate the complex and tricky factors as necessary.

Things you might want to know:

There are 4.5x10^26 atoms in every gallon of gasoline

Hope you're interested and excited to see your thought :P

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All you care about is the fraction of the tank that remains when you fill. In your model, we go down to $0.1$ tank and fill to a full tank, so the chance any molecule remains after one cycle is $0.1$. It therefore takes about $27$ tankfulls until you lose the last molecule.

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  • $\begingroup$ If the tank is sometimes burned a little emptier (the problem only says "no molecule more than 1 gallon"), the expected number of fillings could be even fewer. $\endgroup$
    – David K
    Commented Feb 11, 2015 at 4:01
  • $\begingroup$ Perhaps I'm over looking something obvious but I'm not exactly following how you got 27 tankfuls. Also, how long would this take? $\endgroup$ Commented Feb 11, 2015 at 4:13
  • $\begingroup$ The number of original molecules remaining is reduced by a factor of $10$ each fill. We need to reduce $4.5\cdot 10^{27}$, the number of molecules in the original tank, to less than $1$ by dividing by $10$ some number of times. We can argue about one more or less. It is easy to compute the number of miles this takes from your data, but you did not specify the speed, which gives the number of miles per day. How many miles per tankful? $\endgroup$ Commented Feb 11, 2015 at 4:19
  • $\begingroup$ I see, that makes perfect sense haha, and dont worry I just calculated the time at an average of 55 mph. I think I was really over thinking this so thanks for simplifying it a bit for me :P $\endgroup$ Commented Feb 11, 2015 at 4:28

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