In order to have the fission reaction to start a nuclear reactor, you need to be able to produce as many neutrons needed in order to maintain the reaction stable during the time. If you have too few fissions, then there will be a lack of neutrons and this will turn off the reaction chain. If you have too many fissions, then there will be an excess of neutrons, and you will end up with a Chernobyl-like issue. This is called the criticality of the reactor. Hence, you need to compute the exact amount of fuel to put inside your reactor, in order to maintain the reaction chain. Such a calculus of criticality is very complex, and out of reach for this class. It is, however, possible to estimate it, with the Oklo reactor. The Oklo reactor is located near Franceville, in Gabon. It is the only natural nuclear reactor known so far. At some point in history, what would become the mining pit of Oklo was a place containing a lot of uranium and water, which acted as a moderator, hence allowing the nuclear chain reaction to start and to maintain itself active for some time. At Oklo, the current abundance of uranium 235 is 0.7171%, while at start, it was exactly the same amount as in other places on Earth. 1) We will assume that when the Earth was formed, the abundance of uranium 235 was A. Up to a time to, both Oklo and other places worked the same way. What is the abundance at Oklo at time to? We will call this value Ao. 2) We will now express the abundance of uranium 235 as a function of the current abundance, and as the time elapsed since to. What is the equation you need to use? 3) For a place not at Oklo, you can see that you can express the value of Ao if t=to. Explain how to do so. 4) For Oklo, you can see that there will be an amount of uranium missing, which will grow as time goes back. Express this difference of amount, either as a difference of abundance, or a difference of mass, as a function of time elapsed. 5) Knowing that today, it is not possible to have another natural nuclear reactor starting a reaction chain, what is the minimal time elapsed since the reactor stopped. 6) It is possible to use the concentration of plutonium at Oklo to further constrain the time of operation of the natural reactor. Can you explain why?

In order to have the fission reaction to start a nuclear reactor, you need to be able to produce as many neutrons needed in order to maintain the reaction stable during the time. If you have too few fissions, then there will be a lack of neutrons and this will turn off the reaction chain. If you have too many fissions, then there will be an excess of neutrons, and you will end up with a Chernobyl-like issue. This is called the criticality of the reactor. Hence, you need to compute the exact amount of fuel to put inside your reactor, in order to maintain the reaction chain. Such a calculus of criticality is very complex, and out of reach for this class. It is, however, possible to estimate it, with the Oklo reactor. The Oklo reactor is located near Franceville, in Gabon. It is the only natural nuclear reactor known so far. At some point in history, what would become the mining pit of Oklo was a place containing a lot of uranium and water, which acted as a moderator, hence allowing the nuclear chain reaction to start and to maintain itself active for some time. At Oklo, the current abundance of uranium 235 is 0.7171%, while at start, it was exactly the same amount as in other places on Earth. 1) We will assume that when the Earth was formed, the abundance of uranium 235 was A. Up to a time to, both Oklo and other places worked the same way. What is the abundance at Oklo at time to? We will call this value Ao. 2) We will now express the abundance of uranium 235 as a function of the current abundance, and as the time elapsed since to. What is the equation you need to use? 3) For a place not at Oklo, you can see that you can express the value of Ao if t=to. Explain how to do so. 4) For Oklo, you can see that there will be an amount of uranium missing, which will grow as time goes back. Express this difference of amount, either as a difference of abundance, or a difference of mass, as a function of time elapsed. 5) Knowing that today, it is not possible to have another natural nuclear reactor starting a reaction chain, what is the minimal time elapsed since the reactor stopped. 6) It is possible to use the concentration of plutonium at Oklo to further constrain the time of operation of the natural reactor. Can you explain why?