Problems — Hints to Solutions

In keeping with the didactic approach of this book, and the view that any science is best learned by solving problems, I have provided solutions to 50 exercises in the text and posed 112 exercises for the student. Answers to all the quantitative problems, and hints and solutions to selected problems, are given in Appendix VII. 

Problems (with hints on solution) appear at the end of each chapter and form a valuable supplement to the text. 

Homework problems are also handled differently. At the end of each chapter are short, mostly derivation type, problems which we call Review Problems. Hints or solutions are provided for these exercises. To enhance the skill of problem solving, we take the extreme approach, more so than... 

Solving problems is essential for a sound understanding and for relating principles to real engineering situations. Numerical answers and hints to the solution of problems are given in the final appendix. 

Revise your solution to Problem 8.52 to apply the pib basis functions to the onedimensional quartic oscillator with V = cx. See Problem 8.55 for hints. Take the box to extend from x, = —3.5 to 3.5, where x, is as found in Problem 4.32. Increase the number of pib basis functions until the lowest three energy values remain stable to three decimal places Compare the lowest three energies with those found by the Numerov method in Problem 452. Check the appearance of the lowest three variational functions. Now repeat for the box going from X, = -4.5 to 4.5. For which box length do we get faster convergence to the true energies ... 

Basic Physical Chemistry for the Atmospheric Sciences covers the fundamental concepts of chemical equilibria, chemical thermodynamics, chemical kinetics, solution chemistry, acid and base chemistry, oxidation-reduction reactions, and photochemistry. Over 160 exercises are contained within the text, including 50 numerical problems solved in the text and 112 exercises for the reader to work on with hints and solutions provided in an appendix. 

Except for the techniques of second quantization, the presentation in this book makes no use of techniques that should not be familiar to any student of quantum chemistry after one or two years of undergraduate study. At the end of each chapter, problems and exercises are provided, complete with hints and solutions. The reader is strongly ur ed to work through these exercises as a means of deepening and consolidating the understanding of the material presented in each chapter. 

One synthesis of phenobarbital begins with ethyl phenylacetate and diethyl car bonate Using these starting materials and any necessary organic or inorganic reagents devise a synthesis of phenobarbital (Hint See the sample solution to Problem 21 3a )... 

Use the algebraic procedure to synthesize an MEN which has the minimum number of units satisfying the MOC solution for Problem 3.4. Hint Due to the absence of a pinch, there is no need to examine the matching feasibility criteria given by Eqs. (5.8) and (5.12). All you need is to check the practical feasibility Eq. (3.5) at the two ends of each exchanger. 

Hints First convince yourself that there is an optimal solution by considering the limiting cases of ij near zero, where a large hole can almost double the catalyst activity, and of ij near 1, where any hole hurts because it removes catalyst mass. Then convert Equation (10.33) to the form appropriate to an infinitely long cylinder. Brush up on your Bessel functions or trust your S5anbolic manipulator if you go for an anal5dical solution. Figuring out how to best display the results is part of the problem. 

We can find the solution to Eq. (4-1), which is simply a set of first order differential equations. As analogous to how Eq. (2-3) on page 2-2 was obtained, we now use the matrix exponential function as the integration factor, and the result is (hints in the Review Problems)... 

In this textbook many exciting topics in astrophysics and cosmology are covered, from abundance measurements in astronomical sources, to light element production by cosmic rays and the effects of galactic processes on the evolution of the elements. Simple derivations for key results are provided, together with problems and helpful solution hints, enabling the student to develop an understanding of results from numerical models and real observations. 

Solutions to most problems are contained in the literature cited. Therefore only hints are given in a few cases the complete solution or numerical answer is provided. 

One could perhaps think of an approach consisting of straightforward calculations of the NMR shifts of the heavy atoms, a task that can be performed with accuracy thanks to methods developed by a number of authors [165,169,250,298]. Evidently, this would stiU leave us with the problem of assigning the charges of the hydrogen atoms attached to the various atoms, paying particular attention to those attached to atoms other than carbon but that problem could surely be overcome. Alternative examples exist for sigma systems [34-37] they could perhaps offer useful hints, but we prefer to keep an open mind on that matter and try harder for simpler solutions. 

Obtain a general formula for the most probable three-dimensional translational quantum number j = jmax for a gas (assume a Boltzmann distribution). Evaluate this expression for NO2 at 1000 K (assume a cubic container 0.1 m on each side). Determine the translational energy that this corresonds to (J/mole). Find the fraction of molecules having a translational energy level greater than jmax. Hint Solution to this problem will involve the error function, erf(x). 

In contrast to the generalization made in Problem 9.11 there is reason to beEeve the solution data for CHjOtCH NH, may be more indicative of inherent basicity than the gas-phase work. Can you suggest a reason Hint Consider (he possibilities for hydrogen bonding.) ... 

Calculate the pH and the concentrations of all species present in 0.25 M solutions of each of the salts in Problem 15.122. (Hint The principal reaction is proton transfer from the cation to the anion.)... 

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