Problems Involving Volume

State Charles s law, and use it to solve problems involving volume and temperature. 

If you master the following strategy, you will need to remember only one gas law—the ideal gas law. Consider the example of a fixed amount of gas held at constant pressure. You need Charles s law to solve problems involving volume and temperature. 

The last definition has widespread use in the volumetric analysis of solutions. If a fixed amount of reagent is present in a solution, it can be diluted to any desired normality by application of the general dilution formula V,N, = V N. Here, subscripts 1 and 2 refer to the initial solution and the final (diluted) solution, respectively V denotes the solution volume (in milliliters) and N the solution normality. The product VjN, expresses the amount of the reagent in gram-milliequivalents present in a volume V, ml of a solution of normality N,. Numerically, it represents the volume of a one normal (IN) solution chemically equivalent to the original solution of volume V, and of normality N,. The same equation V N, = V N is also applicable in a different context, in problems involving acid-base neutralization, oxidation-reduction, precipitation, or other types of titration reactions. The justification for this formula relies on the fact that substances always react in titrations, in chemically equivalent amounts. 

The second method, based on measurements of IR reflection spectra, is simpler and enables working with larger volumes of molten salt. No special problems involving temperature and atmosphere control exist. The method was used successfully by Fordyce and Baum [336-338] in the investigation of fluoride melts containing tantalum and niobium. 

In Illustrations 8.3 and 8.6 we considered the reactor size requirements for the Diels-Alder reaction between 1,4-butadiene and methyl acrylate. For the conditions cited the reaction may be considered as a pseudo first-order reaction with 8a = 0. At a fraction conversion of 0.40 the required PFR volume was 33.5 m1 2 3, while the required CSTR volume was 43.7 m3. The ratio of these volumes is 1.30. From Figure 8.8 the ratio is seen to be identical with this value. Thus this figure or equation 8.3.14 can be used in solving a number of problems involving the... 

The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. As discussed in Chapter 1, you must be able to translate a verbal statement or concept of the desired objective into mathematical terms. In the chemical industries, the objective function often is expressed in units of currency (e.g., U.S. dollars) because the goal of the enterprise is to minimize costs or maximize profits subject to a variety of constraints. In other cases the problem to be solved is the maximization of the yield of a component in a reactor, or minimization of the use of utilities in a heat exchanger network, or minimization of the volume of a packed column, or minimizing the differences between a model and some data, and so on. Keep in mind that when formulating the mathematical statement of the objective, functions that are more complex or more nonlinear are more difficult to solve in optimization. Fortunately, modem optimization software has improved to the point that problems involving many highly nonlinear functions can be solved. 

The first part of this problem appears in numerous problems involving solutions. Moles are critical to all stoichiometry problems, so you will see this step over and over again. This is so common, that anytime you see a volume and a concentration of a solution, you should prepare to do this step. 

Be sure, especially in stoichiometry problems involving gases, that you are calculating the values such as volume and pressure of the correct gas. You can avoid this mistake by clearly labeling your quantities that means, mol of 02 instead of just mol. 

Another type of gas law problem involves stoichiometry. Gas stoichiometry problems are just like all other stoichiometry problems—you must use moles. In addition, one or more gas laws are necessary. Let s look at a gas stoichiometry problem. What volume, in liters of oxygen gas, collected over water, forms when 12.2 g ofKCl03 decompose according to the following equation ... 

Stoichiometry problems (including limiting-reactant problems) involving solutions can be worked in the same fashion as before, except that the volume and molarity of the solution must first be converted to moles. 

The problems involved in the drying or dehumidification of gases are referred to in Volume 1, Chapter 13, and the most important methods available are now summarised. 

The domain in which GEORGE operates is a small but important one for introductory chemistry. He works with problems involving the fundamental quantities mass, volume, and number of moles. He can also work with derived quantities such as density, molar mass, molar... 

J Jt any mathematical problems involve units of length, weight, volume, F lor money. You incorporate the units into your computations and then report them in the answers so the solution makes sense and is useful. Sometimes you re confronted with problems that have two or more different units — such as feet and inches or pounds and ounces — and you have to make a decision as to which unit to use. 

Mixture problems involving actual substances occur when you take two or more different solutions or granular compounds or anything that will combine or mix and create a new combination that s no longer purely one or the other. When you pour chocolate syrup into milk, you add volume to the liquid in the glass, and the color of the milk mixture isn t as dark as the chocolate or as white as the milk. The more chocolate, the darker the mixture. Yum ... 

In the calculus problem involving the open box, you get to determine the size of squares in the corners that gives you the largest possible volume. If the squares are small, then the box isn t very deep, but the area of the base is big. If the squares are large, then the box is deep, but the area of the base is small. Calculus allows you to balance the depth and the base area to find the best dimensions. In the next problem, you get to do somewhat the same process with a table of the possible values. 

No discussion of natural gas liquids recovery would be complete without some reference to the conservation problems involved. Formerly considerable volumes of oil well gas were flared without processing. The liquid content of this rich gas was lost entirely. 

We have discussed at length the usefulness of powers of 10 as part of scientific notation, but many practical problems involve the powers of other numbers. For example, the area of a circle involves the square of the radius, and the volume of a sphere involves the cube of the radius. Nearly every calculator yields the square of a number when you simply enter the number through the keyboard and then press the ke ihe S( uafe appears in the lighted display. 

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