question: How does splitting an atom create so much energy using the law of conservation of energy?
comment: Normal energy in chemical reactions comes from the bonds between electrons and the positively charged nucleus. Energy in nuclear reactions come from the bonds between nucleons (protons and neutrons). It's not that splitting an atom inherently liberates energy, but that some atomic nuclei are in high-energy states, and letting them fall into a lower energy state liberates energy. This is akin to the fact that splitting molecules in general doesn't liberate energy, but splitting hydrocarbons does, because they have so much energy stored in their bonds. You can't burn ash, and similarly you can't get more energy from the byproducts of nuclear decay (this isn't entirely true - usually the results are sometimes still unstable, but less so, and they can fall apart a handful more times before getting to fully stable products).
passages: Section::::Fission and fusion.
  
  Nuclear energy is released by the splitting (fission) or merging (fusion) of the nuclei of atom(s). The conversion of nuclear mass-energy to a form of energy, which can remove some mass when the energy is removed, is consistent with the mass-energy equivalence formula:
  
  Δ"E" = Δ"m" "c",
  
  in which,
  
  Δ"E" = energy release,
  
  Δ"m" = mass defect,
  
  and "c" = the speed of light in a vacuum (a physical constant 299,792,458 m/s by definition).
   Energy transformations in the universe over time are characterized by various kinds of potential energy that has been available since the Big Bang later being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available. Familiar examples of such processes include nuclear decay, in which energy is released that was originally "stored" in heavy isotopes (such as uranium and thorium), by nucleosynthesis, a process ultimately using the gravitational potential energy released from the gravitational collapse of supernovae, to store energy in the creation of these heavy elements before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs or in civil nuclear power generation. Similarly, in the case of a chemical explosion, chemical potential energy is transformed to kinetic energy and thermal energy in a very short time. Yet another example is that of a pendulum. At its highest points the kinetic energy is zero and the gravitational potential energy is at maximum. At its lowest point the kinetic energy is at maximum and is equal to the decrease of potential energy. If one (unrealistically) assumes that there is no friction or other losses, the conversion of energy between these processes would be perfect, and the pendulum would continue swinging forever.
   Section::::Applications.
   Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that eventually drew the wider recognition.
   If the mass of the nucleus following a fusion reaction is less than the sum of the masses of the separate particles, then the difference between these two values can be emitted as a type of usable energy (such as a gamma ray, or the kinetic energy of a beta particle), as described by Albert Einstein's mass–energy equivalence formula, formula_1, where formula_2 is the mass loss and formula_3 is the speed of light. This deficit is part of the binding energy of the new nucleus, and it is the non-recoverable loss of the energy that causes the fused particles to remain together in a state that requires this energy to separate.
   The basic idea of a balance is that nothing can get lost or annihilated - this fits to the first law of thermodynamics, which assigns energy this property. But energy splits up during usage and its output does not have the same potential for the physical performance as before.
   Section::::Creating reactions utilizing atom economy.
   BULLET::::- Assume the possible excitation energy of the molecule to be , which enables the reaction to occur.
  
  BULLET::::- The rate of intra-molecular energy distribution is much faster than that of reaction itself.
  
  Section::::Derivation.
  
  Assume that is an excited molecule:
  
  where stands for product, and for the critical atomic configuration with the minimum energy along the reaction coordinate.
  
  Unimolecular rate constant formula_2 is obtained as follows.
  
    formula_3
   Section::::Compatibility (and applicability) of DEB theory/models with other approaches.
   The energy released in a nuclear reaction can appear mainly in one of three ways:
  
  BULLET::::- kinetic energy of the product particles (fraction of the kinetic energy of the charged nuclear reaction products can be directly converted into electrostatic energy);
  
  BULLET::::- emission of very high energy photons, called gamma rays;
  
  BULLET::::- some energy may remain in the nucleus, as a metastable energy level.
  
  When the product nucleus is metastable, this is indicated by placing an asterisk ("*") next to its atomic number. This energy is eventually released through nuclear decay.
   Section::::Quantum theory.
   The energy of the nucleus is negative with regard to the energy of the particles pulled apart to infinite distance (just like the gravitational energy of planets of the solar system), because energy must be utilized to split a nucleus into its individual protons and neutrons. Mass spectrometers have measured the masses of nuclei, which are always less than the sum of the masses of protons and neutrons that form them, and the difference—by the formula "E" = "m" "c"—gives the binding energy of the nucleus.
  
  Section::::Introduction.:Nuclear binding energy.:Nuclear fusion.
   Available energy (particle collision)
  
  In particle physics, the available energy is the energy in a particle collision available to produce new matter from the kinetic energy of the colliding particles. Since the conservation of momentum must be held, a system of two particles with a net momentum may not convert all their kinetic energy into mass - and thus the available energy is always less than or equal to the kinetic energy of the colliding particles. The available energy for a system of one stationary particle and one moving particle is defined as:
  
  where
  
  Section::::Derivation.
   Einstein began by postulating simple proportionality relations for the different reaction rates involved. In his model, the rate formula_18 for a system to "absorb" a photon of frequency formula_17 and transition from a lower energy formula_20 to a higher energy formula_21 is proportional to the number formula_22 of atoms with energy formula_20 and to the energy density formula_16 of ambient photons of that frequency,
   But in cases where the choice of state basis is "not" determined by a coordinate system, and the perturbed Hamiltonian is "not" diagonal, a level splitting may appear counter-intuitive, as in examples from chemistry below.
  
  Section::::Examples.
  
  In atomic physics:
  
  BULLET::::- The Zeeman effect – the splitting of electronic levels in an atom because of an external magnetic field.
  
  BULLET::::- The Stark effect – splitting because of an external electric field.
  
  In physical chemistry:
  
  BULLET::::- The Jahn–Teller effect – splitting of electronic levels in a molecule because breaking the symmetry lowers the energy when the degenerate orbitals are partially filled.
   The charge of a uranium nucleus, we found, was indeed large enough to overcome the effect of the surface tension almost completely; so the uranium nucleus might indeed resemble a very wobbly unstable drop, ready to divide itself at the slightest provocation, such as the impact of a single neutron. But there was another problem. After separation, the two drops would be driven apart by their mutual electric repulsion and would acquire high speed and hence a very large energy, about 200 MeV in all; where could that energy come from? ...Lise Meitner... worked out that the two nuclei formed by the division of a uranium nucleus together would be lighter than the original uranium nucleus by about one-fifth the mass of a proton. Now whenever mass disappears energy is created, according to Einstein's formula "E"= "mc", and one-fifth of a proton mass was just equivalent to 200 MeV. So here was the source for that energy; it all fitted!
   Section::::Methods of production.:Electron impact excitation.
  
  Much early experimental work on Rydberg atoms relied on the use of collimated beams of fast electrons incident on ground-state atoms. Inelastic scattering processes can use the electron kinetic energy to increase the atoms' internal energy exciting to a broad range of different states including many high-lying Rydberg states,
  
  Because the electron can retain any arbitrary amount of its initial kinetic energy, this process always results in a population with a broad spread of different energies.
  
  Section::::Methods of production.:Charge exchange excitation.
   This term is the most significant, but it gives trivial behavior. This form of the free energy is ultralocal, meaning that it is a sum of an independent contribution from each point. This is like the spin-flips in the one-dimensional Ising model. Every value of "H" at any point fluctuates completely independently of the value at any other point.
   Section::::Principle.
   Thus, conservation of energy ("total", including material or "rest" energy), and conservation of mass ("total", not just "rest"), each still holds as an (equivalent) law. In the 18th century these had appeared as two seemingly-distinct laws.
  
  Section::::History.:Conservation of energy in beta decay.
   Part of the rest energy (equivalent to rest mass) of matter may be converted to other forms of energy (still exhibiting mass), but neither energy nor mass can be destroyed; rather, both remain constant during any process. However, since formula_13 is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass (for example, 1 kg) from rest energy to other forms of energy (such as kinetic energy, thermal energy, or the radiant energy carried by light and other radiation) can liberate tremendous amounts of energy (~formula_14 joules = 21 megatons of TNT), as can be seen in nuclear reactors and nuclear weapons. Conversely, the mass equivalent of an everyday amount energy is minuscule, which is why a loss of energy (loss of mass) from most systems is difficult to measure on a weighing scale, unless the energy loss is very large. Examples of large transformations between rest energy (of matter) and other forms of energy (e.g., kinetic energy into particles with rest mass) are found in nuclear physics and particle physics.
   The transfer of energy from an uncharged primary particle to charged secondary particles can also be described by using the "mass energy-transfer coefficient".
  
  Section::::Biological effects.
  
  Many studies have attempted to relate linear energy transfer to the relative biological effectiveness (RBE) of radiation, with inconsistent results. The relationship varies widely depending on the nature of the biological material, and the choice of endpoint to define effectiveness. Even when these are held constant, different radiation spectra that shared the same LET have significantly different RBE.
   Stimulated emission occurs when an excited electron can drop by the same amount of energy as a passing photon. This causes a second photon to be emitted, closely matching the original's energy, momentum, and phase. Now there are two photons, doubling the chance that they will cause the same reaction in other atoms. As long as there is a large population of atoms with electrons in the matching energy state, the result is a chain reaction that releases a burst of single-frequency, highly collimated light.
   Section::::Theory.
   The energy can be added to the nucleus by an incident high-energy gamma ray. If the energy of the incident photon exceeds the separation energy, a photodisintegration might occur. Energy in excess of the threshold value becomes kinetic energy of the ejected particle.
  
  By contrast, nuclear binding energy is the energy needed to completely disassemble a nucleus, or the energy released when a nucleus is assembled from nucleons. It is the sum of multiple separation energies, which should add to the same total regardless of the order of assembly or disassembly.
  
  Section::::Physics and chemistry.
  