Conservation of mass number

Note that the daughter nucleus has two fewer protons and two fewer neutrons than the parent, resulting in a different element. In this particular case, uranium has decayed to thorium. In radioactive decay equations, the total mass number A on the left side must equal the total mass number on the right. In the example above, Afefttotal = 235, while Anghttotal = 231 + 4 = 235. In addition, the total proton number on the left side of the equation must be equal to total proton number on right side (92 = 90 4- 2). So, in any nuclear reaction, there is conservation of mass number and charge. 

The total number of protons plus neutrons in the products and in the reactants must be the same (conservation of mass number). 

In these cases, the heavier product nuclide can be inferred from the conservation of mass numbers and atomic numbers. For example, when sB reacts with a neutron and emits an alpha particle, the heavier reaction product must be 3 Li ... 

The reaction must proceed with conservation of mass number and of charge. The mass numbers are denoted by the superscripts, and the charges by the subscripts (i.e. the number of protons). 

Solution. According to the Nuclear Wallet Cards, the only stable isotope of iodine is 1. Therefore, l lies to the higher side of the band of stability and will need to increase Z in order to become a stable isotope. The only form of radioactive decay that increases Z is the emission of a beta particle. Using the principles of conservation of mass number and conservation of atomic number during a nuclear reaction, the nuclear equation for beta decay is... 

Note that the sum of the superscripts on the left-hand side (14+1 = 15) is equal to that on the right-hand side and also that the sum of the subscripts on the left-hand side (7+0=7) is equal to that on the right-hand side (6+1=7). This conservation of mass number and electric charge always holds true for nuclear reactions. Check this with (19.1)-(19.3) above as weU]. By the way, the resultant nucleus C is radioactive, though. jN is not, and the radioactivity of fl is used to determine the age of the archaeological artifacts. 

Think About It The rules of summation that we apply to balance nuclear equations can be thought of as the conservation of mass number and the conservation of atomic number. 

Count the number of species whose concentrations appear in the equilibrium constant expressions these are your unknowns. If the number of unknowns equals the number of equilibrium constant expressions, then you have enough information to solve the problem. If not, additional equations based on the conservation of mass and charge must be written. Continue to add equations until you have the same number of equations as you have unknowns. 

In this chapter we will apply the conservation of mass principle to a number of different kinds of systems. While the systems are different, by the process of analysis they will each be reduced to their most common features and we will find that they are more the same than they are different. When we have completed this chapter, you will understand the concept of a control volume and the conservation of mass, and you will be able to write and solve total material balances for single-component systems. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

One molecule (or mole) of propane reacts with five molecules (or moles) of oxygen to produce three molecules (or moles) or carbon dioxide and four molecules (or moles) of water. These numbers are called stoichiometric coefficients (v.) of the reaction and are shown below each reactant and product in the equation. In a stoichiometrically balanced equation, the total number of atoms of each constituent element in the reactants must be the same as that in the products. Thus, there are three atoms of C, eight atoms of H, and ten atoms of O on either side of the equation. This indicates that the compositions expressed in gram-atoms of elements remain unaltered during a chemical reaction. This is a consequence of the principle of conservation of mass applied to an isolated reactive system. It is also true that the combined mass of reactants is always equal to the combined mass of products in a chemical reaction, but the same is not generally valid for the total number of moles. To achieve equality on a molar basis, the sum of the stoichiometric coefficients for the reactants must equal the sum of v. for the products. Definitions of certain terms bearing relevance to reactive systems will follow next. 

Use the law of conservation of mass to determine which numbered box(es) represent(s) the product mixture after the substances in the box at the top of the next column undergo a reaction. 

Now there are four H atoms, two Na atoms, and two O atoms on each side, and the equation conforms to the law of conservation of mass. The number multiplying an entire chemical formula in a chemical equation (for example, the 2 multiplying H20) is called the stoichiometric coefficient of the substance. A coefficient of 1 (as for H2) is not written explicitly. 

The average number of particles produced on a single breakup event is then Jo b r)dr, and Jo rb(r)dr = 1, because of conservation of mass. 

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. 

The chemical equation then represents a conservation of atoms, which ensures conservation of mass and an alternative view of the species as molecules or moles. The stoichiometric coefficients correspond to the number of molecules or moles of each species. 

The Law of Conservation of Mass states that the total mass remains unchanged. This means that the total mass of the atoms of each element represented in the reactants must appear as products. In order to indicate this, we must balance the reaction. When balancing chemical equations, it is important to realize that you cannot change the formulas of the reactants and products the only things you may change are the coefficients in front of the reactants and products. The coefficients indicate how many of each chemical species react or form. A balanced equation has the same number of each type of atom present on both sides of the equation and the coefficients are present in the lowest whole number ratio. For example, iron metal reacts with oxygen gas to form rust, iron(III) oxide. We may represent this reaction by the following balanced equation ... 

The fundamental basis for virtually all a prion mathematical models of air pollution is the statement of conservation of mass for each pollutant species. The formulation of a mathematical model of air poUution involves a number of basic steps, the first of which is a detaUed examination of the basis of the description of the diffusion of material released into the atmosphere. The second step requires that the form of interaction among the various physical and chemical processes be specified and tested against independent experiments. Once the appropriate mathematical descriptions have been formulated, it is necessary to implement suitable solution procedures. The final step is to assess the ability of the model to predict actual ambient concentration distributions. 

The temporal derivatives on the right-hand sides represent the changes in concentration of molecules in state A or B, which are the rates or the velocities of the reaction(s). The second equation follows immediately from Equation (1.2a) and the independent requirement that A(t) + B(t) = const (i.e., the conservation of mass). The rate is negative when molecules leave a state and positive when they populate a state. In addition, the rate is considered proportional to the concentration of molecules in state A. Hence, the larger the number of molecules in state A, the larger the velocity of the reaction. Integration of Equation (1.2a) shows that the molecules in A vanish according to simple exponential law. 

The process by which we make a chemical equation conform to conservation of mass is called balancing the equation. A chemical equation is balanced by placing whole number coefficients in front of reactants and/or products to make the number of atoms of each element equal on both sides of the equation. In our example, this is done by placing a 2 in front of both hydrogen and water... 

Conservation of mass and charge are used when writing nuclear reactions. For example, let s consider what happens when uranium-238 undergoes alpha decay. Uranium-238 has 92 protons and 146 neutrons and is symbolized as After it emits an alpha particle, the nucleus now has a mass number of 234 and an atomic number of 90. 

When you balance a chemical reaction equation, the primary concern is to obey the principle of conservation of mass The total mass of the reactants must equal the total mass of the products. (See Chapter 8 if you need to review this process.) In redox reactions, you must obey a second principle as well the conservation of charge. The total number of electrons lost must equal the total number of electrons gained. In other words, you can t just leave electrons lying around. The universe is finicky about that type of thing. 

The Navier-Stokes equations express the conservation of momentum. Together with the continuity equation, which expresses conservation of mass, these equations are the fundamental underpinning of fluid mechanics. They are nonlinear partial differential equations that in general cannot be solved by analytical means. Nevertheless, there are a number of geometries and flow situations that permit considerable simplification and solution. We will explore many of these and their solution, usually by computational techniques. While... 

The pressure rise across a detonation wave may be computed from the experimental Mach numbers of detonation by using only the conservation of mass and momentum and a state equation, the use of the conservation of energy not being required. For the Chapman-Jouguet case, this pressure rise is... 

BALANCE. (1) Exact equality of the number of atoms of various elements entering into a chemical reaction and the number of atoms of those elements in tlie reaction products. For example, in the reaction NaOH + HC1 -j- NaCl + H2O. the atoms in tlie input side are H[2], Na[l], 0(1], and 0(1]. Each of these is also present in the products, though in different combination. The atoms of catalysts (when present) do not enter into reactions and therefore are not involved. The balance of chemical reactions follows the law of conservation of mass. 

Since the principle of conservation of masses applies to chemical reactions, coefficients must often be used in writing chemical reactions so that the number of atoms of products is equal to the number of atoms of reactants, An example is the reaction of two moles of hydrogen with one mole of oxygen to form ru n moles of water... 

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