Process of Balancing Equations

Many equations can be balanced by inspection. This is a trial-and-error method by which we try various coefficients on both sides, going back and forth, from the left side, to the right side, to the left side again, and so on until the equation is balanced. This process will be described here. 

Equations are balanced by placing coefficients in front of the individual symbols and formulas. The equation representing the water-formation reaction 

While the oxygens are now balanced, the hydrogens have become unbalanced. However, we see that placing a 2 in front of the H2 would balance the hydrogens and not affect the oxygens. 

As a general rule, the coefficients used should be whole numbers and not fractions. For example, this equation is balanced if the fraction V2 were as the coefficient for O2  

However, fractions in general should be avoided. Also, the coefficients used should be the smallest possible numbers. For instance, this same equation is balanced as follows  

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The basic process of balancing equations is, by nature, a trial and error affair, but you can streamline the process by using a few different strategies. 

I will go through the solution slowly, and if I lose you, go back to the beginning and try to follow me again. It is very important that you learn the process of balancing equations correctly. 

To illustrate the process of balancing equations, consider the reaction that occurs when methane (C1T4), the principal component of natural gas, bums in air to produce carbon dioxide gas (CO2) and water vapor (H2O). Both of these products contain oxygen atoms that come from O2 in the air. We say that combustion in air is "supported by oxygen," meaning that oxygen is a reactant. The unbalanced equation is... 

There are a number of past and present commercial routes to phenol using benzene as a feed stock. Outline two such processes, writing balanced equations for the reactions involved. Compare the two routes in terms of atom economy. 

In the previous section we showed that process variables could be divided into vectors x and u, corresponding to measured and unmeasured variables, respectively. Accordingly, linear systems of balance equations can be represented in terms of compatible... 

The half-reaction method of balancing equations can be more complicated for reactions that take place under acidic or basic conditions. The overall approach, however, is the same. You need to balance the two half-reactions, find the LCM of the numbers of electrons, and then multiply by coefficients to equate the number of electrons lost and gained. Finally, add the halfreactions and simplify to give a balanced net ionic equation for the reaction. The ten steps listed above show this process in more detail. 

It is reasonable to assume that in the vast majority of cases encountered in reactive processing Re < < 1 and (H/L)Re < < 1. Thus, we can consider the flow to be quasi-stationary and that temperature changes occur quickly after alterations in the temperature and degree of conversion distributions.202 Now we can rewrite the system of balance equations in the following dimensionless form ... 

The equation representing a chemical reaction can be written correctly by carrying out a process of balancing the equation. This is done by introducing numerical coefficients before the formulas of the reactants and products until there are exactly the same number of atoms of each element on the left side of the equation as on the right side of the equation. 

The process of balancing a more complicated equation is il crroecl by the examples given belowc... 

This result appears to be counterintuitive, especially since we normally allow the energy to depend on mole numbers, as specified by the relation E = E S, V, N( ). However, this problem is apparent rather than real from the viewpoint of chemistry the fundamental species in any chemical reaction are the participating atoms whose numbers are strictly conserved—witness the process of balancing any chemical equation. Thus, while the arrangement or configuration of the atoms changes in a chemical process their numbers are not altered in this process. Under conditions of strict isolation the system behaves as a black box no indication of the internal processes is communicated to the outside. One should not attempt to describe processes to which one has no direct access. However, under conditions illustrated in Remark 1.21.2, even an isochoric reaction carried out very slowly in strict isolation, produces an entropy change dS = dO = 1 Hi dNi > 0. See also Eq. (2.9.3) which proves Eq. (1.21.3) under equilibrium conditions. 

This description makes the process of balancing an equation seem very long-winded, but as you get used to doing more of them, it gets faster. Now try the supplementary questions on this topic for extra practice. You ll find them after the answers to the diagnostic test. Some of them may not need changing in order to be balanced - look at each example logically and add up the number of atoms on each side first. Don t just jump to conclusions ... 

How does the process of balancing an equation illustrate the law of conservation of mass ... 

The equation satisfies the law of conservation of mass however, we have altered one of the reacting species. Hydrogen chloride is HCl, not H2CI2. We must remember that we cannot alter any chemical substance in the process of balancing the equation. We can only introduce coefficients into the equation. Changing subscripts changes the identity of the chemicals involved, and that is not permitted. The equation must represent the reaction accurately. The correct equation is... 

In this chapter we continue the quantitative development of thermodynamics by deriving the energy balance, the second of the three balance equations that will be used in the thermodynamic description of physical, chemical, and (later) biochemical processes. The mass and energy balance equations (and the third balance equation, to be developed in the following chapter), together with experimental data and information about the process, will then be used to relate the change in a system s properties to a change in its thermodynamic state. In this context, physics, fluid mechanics, thermodynamics, and other physical sciences are all similar, in that the tools of each are the same a set of balance equations, a collection of experimental observ ations (equation-of-state data in thermodynamics, viscosity data in fluid mechanics, etc.), and the initial and boundary conditions for each problem. The real distinction between these different subject areas is the class of problems, and in some cases the portion of a particular problem, that each deals with. 

Since the left-hand side of the equation has the units of reaction rate (mol L s ), these units must also apply to the right-hand side of the equation. The process of balancing dimensions in equations is called dimensional analysis, it is a most useful tool when checking kinetic (or any other physical) equations. Hence, the units of / [A] must be mol L s, implying that k has imits of s . Simple dimensional analysis leads directly to the general expression for the units of a particular rate constant in a particular reaction scheme. 

The application of the formal macroscopic theory to transformation processes in open systems is based on the formulation of balance equations for a number of conserved quantities and an additional thermodynamic constraint allowing the formulation of a useful efficiency measure. 

Most processes consist of many interconnected units. In analysing such processes material balance equations may be written for each unit, for groups of units, or for the whole plant. To obtain a unique solution the number of equations describing a process must be equal to the number of unknown variables. If the analysis leads to fewer equations it is necessary to specify extra design variables. In the case of an actual plant where values of process variables are obtained by direct measurements the number of equations may exceed the number of unknowns. In such circumstances calculations should be based on the most reliable measurements. 

Example 7.3 illustrated the technique of working forwards through a process, solving balance equations unit by unit. This is usually possible for processes without recycle streams provided that the feed is specified. If no individual balance yields enough equations it is necessary to solve simultaneously equations arising from balances on two or more units. Most multiple unit processes involve recycle streams but the treatment of these will be postponed until section 7.2.S.3. 

To illustrate the process of balancing an equation, consider the reaction that occurs when methane (CH4), the principal component of natural gas, burns in air to produce... 

The process of balancing chemical equations is relatively straightforward for simple equations. It is discussed in Chapter 4. 

Streams are usually classified as process streams (streams 1 to 8) and utilities, such as electricity or cooling water (streams 9 to 11). This classification is useful, because process streams usually don t mix with utilities. Systems of balancing equations of process streams and of individual utilities are often independent and can be solved separately. The classification of the stream (12)... 

Among the mathematical models of industrial chemical processes, the system of balance equations is basic. If some measured data do not satisfy the balance equations, this fact is attributed to measurement errors, and not perhaps to an inadequate description of the process by the model. In practice, measurement errors are always present. Hence before using the measured data, they are adjusted to obey the balance constraints. The adjustment by methods using statistical theory of errors is called reconciliation. 

The classification was developed mainly for use in balancing complex plants in process industries. The methods developed are based on the solvability analysis of sets of balance equations (Vaclavek et al. 1972, Crowe 1989) and on the analysis of stracture of balanced system (Vaclavek and Loucka 1976, Vdclavek and VosolsobS 1981). Systematic research by Mah and his coworkers in this area, poblished in a number of papers, is summarized by Mah in his monograph (Mah 1990), where also a comprehensive survey of the state of the art of classification can be found. 

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