200
n. R.
DAVIDSON AND D. L. FULLER
A SIMPLE ANALOG COMPUTER FOR THERMODYNAMIC CALCULATIONS H. R. DAVIDSON AND D. L. FULLER Central Research Laboratory, General Aniline and Film Corporation, Easton, Pennsylvania Received January 1 1 , 1960
For many chemical reactions one can estimate, with accuracy sufficient for most purposes, the standard free-energy change (AFO) as a linear function of the absolute temperature. For an ideal gas reaction, for example, mM
+
e n N
+ ...
(1)
we can estimate AF" = u
+ b(t + 273')
(2) where a and b are constants and 1 is the temperature in degrees Centigrade. The function AFO is of great importance, because from it we can calculate the extent to which the reaction can go to the right or left (see, for example, reference 1). This follows because the equilibrium constant K for the reaction is
where the P's are the partial pressures at chemical equilibrium and R is the gas constant. When K has been determined from equation 3, one may calculate the extent of the reaction algebraically, but a simpler technique is to set up tables or curves as illustrated below, for various types of reactions. As an example, consider a reaction of the type
A
e B + 3 C
O where the moles a t the start are 1 X and the moles a t equilibrium are 1 - X For this reaction one may write the equation
n n
+ 3X
J = -K= -
where X is a measure of the extent of reaction possible at equilibrium and P = the total pressure. When K has been calculated from equation 3, J may readily be calculated from equation 4, especially for selected pressures such as p = 1, loll3, 10, etc. From table 1 we may then find X , which depends upon the original dilution factor n. Heretofore, we have carried out a large number of laborious calculations of this type. In this paper we describe a device,for carrying out the first part of the calculations, the solution of equation 3, almost instantaneously with as high an
0.W8 11. I W 1 0.1'10
j
0.m
! 5.01 57. I
0.519 0.i79
0.i4i I1.21 2.16
I17 I.% S.17 7.14 7X.8
~
~
,
~
!
1.5 2.35 1. !I. 99.
FIG.1. The computer
given in figure 2 . The analog method of calculation is u e d , and the baaic arialog unit is a simple Ohm's law circuit. In this type of circuit E = I R , where E is the voltage 1tcrhs8 the resistance R, and I is the current flowing through that resistance. By properly ehoming values of E, I, or R, this relationship may 'm used &s an analog of the mathematical proeevs of multiplication or division. Since voltages acmes resiston may he combined so a8 to add or subtract, depending upon the polarity of the voltages, an analog for the mathematical process of addition and subtraction is provided. Thus the electrical relationships between
202
H. R. DAVIDSON AND D. L. FULLER
resistances, currents, and voltages in simple circuits may be used to represent the simple mathematical operations. In this computer, a voltage proportional to the exponent of equation 3 is obtained by the general method described above and then placed across the voltmeter, which is calibrated to read K directly. The details of the circuit are given in figure 2.
2
sw. c I
FIQ.2. Circuit diagram of the computer Potentiometer R1is set to represent the value a by making the resistance between the center tap and the slider proportional to a. Settings on one side of the center tap represent positive values, and settings on the other side represent negative values of a. In like manner, the value b is set on potentiometer Rn.The value of t is set on RIso that the sum of the resistance R1,the &ohm resistor, the @ohm potentiometer,l and the value of Rt introduced into the circuit is proportional to t 273. Thus the current which flows through R1 is inversely 273) and the voltage which appears between the slider and proportional to ( t center tap of RI is proportional to a divided by (t 273). The voltage appearing between the slider and center tap of Ra is proportional to b. By adding these voltages as shown in the circuit diagram, a total voltage proportional to the exponent of equation 3 is obtained. The constants of proportionality are adjusted to be the same in both circuits. Any one of four resistors may be inserted in series with the meter, which is then calibrated to read the negative exponential
+ +
+
1 The 6-ohm potentiometer is used for adjustment of the voltage across RI. Changes in but since the total resisthis adjustment will introduce errors in the value of t set on Rz, tance adjustment is small in comparison with R1 and Ra, the error is small.
ASSOCIATION IA’ CELLULOSE ACETATE SOLUTIONS
203
of the voltage on four different scales. The various switches shown in the diagram are used for adjustments and choice of meter scales. Batteries were chosen for the voltage supplies in order that the computer might be readily moved from one desk to another without the need for electrical outlets. Transformers might, of course, be used in place of the batteries, so that thecomputer could be operated from a 110-v. power outlet. In this case, however, either a meter rectifier or an A.C. meter would be required. While the accuracy of this computer is sufficient for most problems, it could be improved by the use of more precise electrical components and somewhat different values of resistors and supply voltages. The authors wish to acknowledge the contributions of Mr. Carl Helstrom in the construction and calibration of the computer. REFERENCE
(1) MACDOUGALL, F. H . : Thermodynamica and Chemistry, 3rd edition, p. 218 IT. John Wiley and Sons, Inc., New York (1939).
SEDIMENTATION EQUILIBRIA OF POLYDISPERSE NON-IDEAL SOLUTES. IV ASSOCIATION IS DILUTECELLULOSE ACKTATE SOLUTIONS’ hZ. WALES
AND
D. L. SWANSON
Department of Chemistry, University of Wisconsin, Madison, Wisconsin Received January 19, 1960 INTRODUCTIOS
It has been known for a long time (8, 19) that the presence of metallic ions can cause association of cellulose derivatives in solution. This effect is believed to be produced by ion association between the metal ions and sulfuric acid half-ester, or carboxyl, groups present on the polymer molecules. It has been suggested (20) that some abnormalities observed in plots of reduced osmotic pressure vs. concentration in dilute solutions may be caused by this sort of association, dissociation occurring on dilution. It has been found by several workers (3, 16, 21), including the authors, that the first fraction collected during a fractionation of a cellulose derivative either will not redisperse or gives a very turbid suspension. This fraction also contains a disproportionately large amount of inorganic matter, relative to the whole polymer (16). In addition, difficulty has been encountered in removing traces of so-called This research waa supported by the Office of Naval Research, Contract X3-onr-76300, and the Wisconsin Alumni Research Foundation.
