MATHEMATICAL PROBLEM PAGE
T
HIS page is offered a t the suggestion of the Committee on Mathematics of the Division of Chemical Education' in the hope that it may arouse interest in the solving of mathematical problems and that it may help chemists and teachers to keep pace with the increasing use of mathematics in the chemical literature. It is to be tried as an experiment for three months, after which time i t will be continued if there is evident demand. The solutions to each set of problems will be published the following month. It is planned to give a miscellaneous selection of problems calling for different degrees of mathematical preparation. For the present i t is planned to emphasize mathematical aspects of physical chemistry taken largely from current journals. Suggestions for problems and for the conduct of the page will be appreciated by the editor.
5.
(a) Determine, approximately, the slope of line B in the accompanying figure from the Journal of the American Chemical Society [55, 218 (1933)l.
PROBLEMS
During a certain storm the rainfall was 1 in. (2.64 cm.) ,. (a) Assuming that each drop was 0.05 cc. in volume, how many raindrops fell on a square kilometer (0.386 square mile)? (b) Assuming for the sake of simplifying the calculation that l cc. of water weighs l g., how many tons? (1 kilogram = 2.20 pounds.) (c) Employing t h e preceding assumption, how many molecules? The valence factor, w, for calaum chloride according to the Debye-Hiickel theory, is given by the expression: \ - ~ - -
~
What is the numerical value of w? A solution has a pH value of 4.7. What is the effective concentration, x, of hydrogen ions? = x; x = ? I n other words, if Light having a wave number of 42,940 [J. Am. Chem. Soc., 55, 55 (1933)l corresponds to what value in: (a) wave-length in an., (b) wave-length in h g s t r o m units, (c) wave-length in mp. (d) What is the frequency of this light? (e) What is the value in ergs of one quantum having a wave number of 42,940 cm-'?
1/T X lo6. Courlssy J . Am. Ckrm. Soc.
(b) Calculate the energy of activation, E, in calories per mol by multiplying this slope by 2.303 (the conversion factor to change loglo to log.) and by the gas constant, 1.987. 6. Taking the value of E in problem 5 as 41,500, k as 0.00220, T as 573'K., and R as 1.987, calculate the value of s in the formula
7. Using the constants and formula of problem 6, calculate the value of k a t 290°C. (563'K.) and check the calculation with the data of Table I1 [ J .Am. Chem. SOG.,55,219 (1933.1 8. Write the exponential equation of problem 6 as a logarithmic equation. 9. Show how to evaluate the constants n and K in the formula P = KO, by plotting the log V against log 0 [Ind. Eng. Chem., 25, 157 (1933)l.
A , B, V are constants [J. Am. Chem. SOL, 54,
