--------- #1278 - Hot Lava from the Center of Earth?
- Volcanoes spew hot lava from the interior of the Earth. The ocean floor spreads apart and has the power to move continents. The iron core that produces the Earth’s magnetic field remains liquid and rotates like a dynamo. Where does this huge amount of energy come from that can melt the interior of the Earth? This review looks at radioactivity as the source.
- Attachments - none
- Geologist have sunk over 20,000 bore holes deep into the Earth’s crust to try to estimate how much heat energy is being generated inside the Earth’s interior. The answer they get is that the Earth is producing about 44,000,000,000,000 watts, (44 trillion watts) , of heat energy and dissipating it into space. Where does all this energy come from?
- The answer is that about half comes from radioactive decay of Uranium, Thorium and Potassium in the Earth’s crust and the Earth’s mantel. The rest of the heat may be primordial heat left over from the original formation of the planet. Or, it may be fission reactions going on deep inside the Earth. Or, it may be something else we have not yet figured out.
- The conclusion about the radioactive decay comes from detecting neutrinos coming from inside the Earth and passing through detectors on the surface. Physics understands radioactive decay very well and the finger print of neutrino and anti-neutrinos in our detectors is hard evidence as to what is going on deep inside the Earth.
------------------------------ 8 terawatts from Uranium 238
------------------------------ 8 terawatts from Thorium 232
------------------------------ 4 terawatts from Potassium 40
---------------------------- 20 terawatts total from radioactive decay.
- Neutrinos are neutral particles and are very small, near zero mass. They react only with the Weak Nuclear Force and are almost not affected by the force of Gravity. Neutrinos pass through the Earth as if it were transparent. Obviously these little guys are going to be very hard to detect. The KamLAND detector in Japan is a giant sphere filled with 1,000 tons of scintillating mineral oil. The sphere is surrounded by over 1,800 photomultipliers. When a neutrino collides with a proton in this tank a flash of light is recorded.
- When an anti-neutrino collides with a proton to create a neutron and an anti-electron the anti-electron quickly annihilates with a normal electron in the hydrogen rich mineral oil. The annihilation creates a flash of electromagnetic radiation, a flash of light. A couple hundred millionths of a second later the neutron is captured by a proton in this hydrogen rich environment. This capture results in a flash of Gamma Rays. The total light curve is a tell tale signature of an anti-neutrino arriving from the Earth‘s interior.
- After years of study the data concluded that 20 terawatts of radioactive decay is occurring inside the Earth. Another 3 terawatts comes from isotope decay, when an isotope looses a neutron that converts back to a proton. The rest of the heat energy is most likely left over from the Earth’s creation as a planet. It is still cooling off.
- Radioactive decay is a type of nuclear fission. However, we usually refer to “nuclear fission” when a nuclei splits into more or less equal pieces. There could be natural nuclear fission occurring deep inside the Earth. These fission reactors have actually been discovered in deep mines in Africa. It is inconclusive as to how common true nuclear fission could be occurring. The more common type of fission is “radioactive decay” when a nucleus emit’s particles from a parent particle and is converted into a daughter particle.
- There are 3 different types of radioactive decay depending on the particle that is emitted from the nucleus:
----------------- Alpha Particle -- which is 2 protons and 2 neutrons bound together as the nucleus of helium (4).
----------------- Beta Particle -- electrons or anti-electrons emitted plus a neutrino. This decay is due to the Weak Nuclear Force interaction. The mass remains the same but the atomic number is reduced.
----------------- Gamma Particle - is actually not a particle but a photon, a Gamma Ray of electromagnetic energy is emitted.
- Different laws of physics govern each type of decay and there are different probabilities of each event of decay occurring. Each particular element will have a constant decay rate. This is normally referred to an element’s “half-life”. A half-life is half of the amount of time it takes for 100% of the atoms to decay. Take a sample Uranium 238 for example. 238 refers to the atomic number, or the number of protons in the nucleus. When 69.3% of it decays to Uranium 235 it takes 704,000,000 years for that to happen.
------------- The half-life of Uranium 238 is 704,000,000 years. The total lifetime is It takes that long for ½ of the sample of U238 to decay to U235.
------------ The half-life is actually half the life time it takes for 100% of the U238 to decay to U235.
Because the decay is an exponential function, it does not decay at a linear rate. When you reach the half life, 704,000,000 years, actually 69.3% of the Uranium has decayed.
- The half-life for U 238: t½ = 7.04*10^8 years. The total lifetime is 1.016*10^9 years
------------------ t½ = 0.693T
--------------- where T is the total lifetime of the element before it is 100% decayed.
- The math for change is calculus. The decay rate is a constant rate of change for the creation of the number of atoms of U235. The constant decay rate, “X”, for U238 to U235 is 930,000 decays per second. ( X = 9.3*10^5 decays / sec). When the sample of U238 was pulled up from the borehole 0.72% of it had decayed to U235. How much U 238 was in the ore that was sampled? And, therefore how much energy is created from nuclear fission to heat up the Earth?
----------------- T = t½ / 0.693
----------------- T = 1,016,000,000 years which is the total lifetime for the U238.
------------ The decay rate equation = dN / dt = - X No e^-xt = - X* N
-------------by differentiating this exponential equation we get the
------------ decay rate = - X*N
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Some Calculus: If you plot “N”, the number of U238 atoms on the vertical axis and time ,”t”, on the horizontal axis the decay equation is an exponentially curved line falling down and to the right. The slope of his curved line is the rate of change and it is the exponent in the equation, “-x*t” . The slope falls faster in the beginning because you have more U238 atoms. There are chain reactions that exponentially accelerate the decay process. As there are fewer and fewer atoms the decay process slows down. When you get to the half way point on the time line, “t½” you find that not 50% but 69.3% of the atoms have decayed.
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- where: X is the constant decay rate = 9.3^10^5 decays / second
- -------- N is the number of U235 atoms created over the time (dt)
---------- No is the number of atoms at time zero
--------- dt is the duration of time.
---------- dN / dt is the rate of change in the number of U238 decays over the time “dt”.
---------- e^-xt is the exponential decay function of the number of atoms decaying over time. The negative sigh means a decaying rather than growing number.
------------- The constant decay rate ,“X”, is the reciprocal of the lifetime ,”T“.:
--------------------------- X = 1 / T
---------------------------- decays / second = 1 / seconds / decay
---------- Solving the decay rate equation for the number of U235 atoms, or in other words, the number of U238 atoms that have decayed.
--------------- N = - 1/X * dN / dt = - T * dN / dt
-------------- Substituting t½ /0.693 for T
----------- The rate of decay in years = 930,000 decays / sec * 31,600,000 sec / year
----------- The rate of decay in years = 29.39 * 10^12 decays / year
-------------- The number of atoms = N = -T*dN/dt = 10.16*10^8 years * 29.39 * 10^12 decays / year
-------------- The number of atoms = N = 3 *10^22 atoms of U235 created.
------------ Avogadro’s number is the number of atoms in a mole of any element. A mole is the atomic number of element in grams Therefore, a mole of U235 is 235 grams.
------------- Avogadro’s Number for any element = 6.02^10^23 atoms per mole.
--------- 3 * 10^22 atoms * 1 mole / 6.02*10^23 atoms * 235 grams / mole = 12 grams.
-------------- Percentage of U235 in the U238 sampled = 0.72%
------------ 12 grams / 0.0072 = 1.6 kilograms of U 238 and 12 grams had decayed to U235.
------------ These decay formulas are very accurate in predicting how a large sample of Uranium will behave. Each decay results in the release of neutrinos. Detecting these neutrinos tells geologists how much Uranium is decaying deep inside the Earth. The decay process releases radiation and energy that heats up the Earth.
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(1) We should mention that the detectors in Japan also detect all the nuclear fission occurring in all the power generation plants located in Japan. These power planet neutrinos have different finger prints and have to be removed from the data. There are also neutrinos arriving from the nuclear fusion going on in the Sun. These too have to be analyzed and removed from the data.
(2) To learn more about the calculus see reviews:
------------ #745 The calculus of a circle
------------ #737 Everyday Calculus
----------- #734 Spacetime, energy, and calculus.
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707-536-3272, Saturday, July 23, 2011
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