Balancing equations nuclear reactions

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

To summarize, the equation for a nuclear reaction is balanced when the total charge and total mass number of the products equals the total charge and total mass number of the reactants. This conservation requirement is one reason why the symbol for any nuclide includes its charge number (Z) as a subscript and its mass number as a superscript. These features provide a convenient way to keep track of charge and mass balances. Notice that in the equation for neutron decay, the sum of the subscripts for reactants equals the sum of the subscripts for products. Likewise, the sum of the superscripts for reactants equals the sum of the superscripts for products. We demonstrate how to balance equations for other reactions as they are introduced. 

A chemical equation describes a chemical reaction in many ways as an empirical formula describes a chemical compound. The equation describes not only which substances react, but the relative number of moles of each undergoing reaction and the relative number of moles of each product formed. Note especially that it is the mole ratios in which the substances react, not how much is present, that the equation describes. In order to show the quantitative relationships, the equation must be balanced. That is, it must have the same number of atoms of each element used up and produced (except for special equations that describe nuclear reactions). The law of conservation of mass is thus obeyed, and also the "law of conservation of atoms. Coefficients are used before the formulas for elements and compounds to tell how many formula units of that substance are involved in the reaction. A coefficient does not imply any chemical bonding between units of the substance it is placed before. The number of atoms involved in each formula unit is multiplied by the coefficient to get the total number of atoms of each element involved. Later, when equations with individual ions are written (Chap. 9), the net charge on each side of the equation, as well as the numbers of atoms of each element, must be the same to have a balanced equation. The absence of a coefficient in a balanced equation implies a coefficient of 1. 

Most nuclear reactions involve the breaking apart of the nucleus into two or more different elements or subatomic particles. If we know all but one of the particles, then the unknown particle can be determined by balancing the nuclear equation. When chemical equations are balanced, we add coefficients to ensure that there are the same number of each type of atom on both the left and right of the reaction arrow. However, in order to balance nuclear equations we ensure that there is the same sum of both mass numbers and atomic numbers on the left and right of the reaction arrow. Recall that we can represent a specific isotope of an element by the following symbolization ... 

Writing the emitted j8 particle as e in the nuclear equation makes clear the charge balance of the nuclear reaction The subscript in the nucleus on the left (53) is balanced by the sum of the two subscripts on the right (54 - 1 = 53). 

Just as an ordinary chemical equation is a shortened version of the complete thermochemical equation which expresses both energy and mass balance, each nuclear equation has a term (written or implied) expressing energy balance. The symbol Q is usually used to designate the net energy released when all reactant and product particles of matter are at zero velocity. Q is the energy equivalent of the mass decrease (discussed above) accompanying the reaction. Q is usually expressed in MeV. 

Nuclear transmutations are represented by nuclear equations. Nuclear equations show the change in the nucleus as well as the particle emitted during the decay process. Just like chemical equations, these equations must follow the Law of Conservation of Mass and the Law of Conservation of Charge. That is, they are balanced by equating the sum of mass numbers on both sides of a reaction equation and the sum of atomic numbers on both sides of a reaction equation. 

Back on Earth, however, chemical reactions are everywhere in our daily lives. We rely on chemical reactions for everything from powering a car to making toast. In this chapter, you will learn how to write balanced chemical equations for these reactions. You will look for patterns and similarities between the chemical equations, and you will classify the reactions they represent. As well, you will learn how to balance and classify equations for nuclear reactions. 

The principles of balancing nuclear equations apply to all nuclear reactions. Nuclear fission occurs when a highly unstable isotope splits into smaller particles. Nuclear fission usually has to be induced in a particle accelerator. Here, an atom can absorb a stream of high-energy particles such as neutrons, Jn. This will cause the atom to split into smaller fragments. 

Complete and balance each nuclear equation. Then classify the reaction. 

The superscripts represent the mass numbers, and the (optional) subscripts represent the atomic numbers or charges. Knowing that the superscripts and subscripts must balance allows us to deduce one species involved in a reaction if all the others are given. The process is often called balancing a nuclear equation. ... 

All balances on atomic species (C, H, O. etc.) take the form input = output, since atomic species can neither be generated nor consumed in chemical reactions (as opposed to nuclear reactions). The number of degrees of freedom is determined directly by subtracting equations from labeled unknowns no additional degrees of freedom are contributed by the reactions. 

Write the balanced equations for the following nuclear reactions. 

Write a balanced equation for the nuclear reaction that produces... 

The radioactive decay processes you have just read about are all examples of nuclear reactions. As you probably noticed, nuclear reactions are expressed by balanced nuclear equations just as chemical reactions are expressed by balanced chemical equations. However, in balanced chemical equations, numbers and kinds of atoms are conserved in balanced nuclear equations, mass numbers and atomic numbers are conserved. 

You are given all of the particles involved in an induced transmutation reaction, from which you must write the balanced nuclear equation. Because the alpha particle bombards the aluminum atom, they are reactants and must appear on the reactant side of the reaction arrow. Obtain the atomic number of aluminum and phosphorus from the periodic table. Write out the nuclear reaction, being sure to include the alpha particle (reactant) and the neutron (product). 

The equations for nuclear reactions are balanced using the same methods developed for chemical reactions. Charge, mass, and atomic number are conserved. The electron is assigned an effective atomic number Z = — 1 for this purpose. 

Complete and balance the following equations for nuclear reactions that are thought to take place in stars ... 

Energy released during fission is calculated by using Einstein s equation. -> For a balanced nuclear reaction,... 

In chemical reactions, atoms in molecules and ions are rearranged, but matter is neither created nor destroyed, and atoms are not changed into other atoms. In earlier chapters, we learned to write balanced chemical equations to represent chemical reactions. Such equations must show the same total number of atoms of each kind on both sides of the equation and the same total charge on both sides of the equation. In a nuclear reaction, a different kind of transformation occurs, one in which a proton can change into a neutron, or a neutron can change into a proton, but the total number of nucleons remains the same. This leads to two requirements for the equation for a nuclear reaction ... 

To show the quantitative relationships, the equation must be balanced. That is, it must have the same number of atoms of each element used up and produced (except for special equations that describe nuclear reactions). The law of conservation of mass is obeyed... 

A properly written chemical equation must contain properly written formulas and must be balanced. If the reaction represents a closed system, as is usually the case, then the law of mass conservation tells us that no mass can be gained or loss during the reaction. Furthermore, if no nuclear decay is occurring, the number of atoms of each element must remain constant. That is, there will be same number of each type of atom on the left and right hand side of the equation. 

Potassium-40 can decay to form calcium-40. Write a balanced equation for this nuclear reaction and determine the type of decay. 

Balance the following equations, which represent nuclear reactions in the uranium-238 decay series. 

In order to discuss nuclear reactions in any depth, we need to understand how to write and balance the equations. Writing a nuclear equation differs somewhat from writing equations for chemical reactions. In addition to writing the symbols for various chemical elements, we must also explicitly indicate protons, neutrons, and electrons. In fact, we must show the numbers of protons and neutrons present in every species in such an equation. 

Write the balanced equation for the nuclear reaction 26Fe(d,a)2sMn, where d represents the deuterium nucleus (that is, iH). 

Radioactive nuclei emit a particles, (3 particles, positrons, or y rays. The equation for a nuclear reaction includes the particles emitted, and both the mass numbers and the atomic numbers must balance. 

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