VOL. 19, 1933 CHEMISTRY: SKA U A ND LA NGDON 943 a marked influence on the dielectric behavior of an electrolyte at higher concentrations. The conductance and the freezing point curves of the same electrolytes exhibit similar differences. I Vingee, Thesis, Brown University, 1931. 2 Batson, Unpublished observations in This Laboratory. I Fuoss and Kraus, J. Am. Chem. Soc., 55, 21, 1019, 2387, 3614 (1933). 4 Williams and Allgeier, Ibid., 49, 2416 (1927). Wyman, Phys. Rev., 35, 623 (1930). 6 Hedestrand, Z. phys. Chem., B 22, 1 (1933). THE PURIFICATION AND PHYSICAL PROPERTIES OF CHEMI- CAL COMPOUNDS. IV. A DEVELOPMENT OF A THEO- RETICAL BASIS FOR THE BEHA VIOR OF CONTROLLED TIME- TEMPERA TURE CUR VES By EVALD L. SKAU AND WENDELL H. LANGDON TRINITY COLLEGE, HARTFORD, CONN. Communicated October 7, 1933 In order to make a study of the agreement attainable between the theoretical time-temperature curve and the experimental data as determined by means of an apparatus already described in the literature,1 it was necessary to develop the proofs of the two propositions given below. Let it be assumed (1) that a one-gram sample of a chemical compound hermetically sealed in a container of neg11*ible heat capacity be suspended in a vacuum within a copper shield of high heat capacity; (2) that the sample is at constant pressure and that it is at all times homogeneous and at a uniform temperature throughout; and (3) that the sample is thermally insulated so that all heat transfei to or from the surroundings is through radiation and that the rate of such heat transfer can be expressed by Newton's Law of Cool"? dh =K(o-O) (1) where H is the Vat content per gram of the sample at time t, As and 0 are the tempgritures of the shield and of the sample, respectively, and K is a constant of the apparatus. dh Since the specific heat at constant pressure is given by Cp = -, equation (1) may be written in the form do -P = K(Os - 0). (2)
944 CHEMISTRY: SKA U AND LANGDON PROc. N. A. S. We shall further assume that the temperature of the shield is being d6a raised or lowered at a constant rate, i.e., that - is a positive or a negative constant. PROPOSITION I. If Cp is constant uith respect to the temperature, As- 0 rapidly approaches the constant value Kp s PROOF: Let 0-0 = u and = a Then a -d = due -, or - = a - du do Replacing s- 0 and in equation (2) by their values, and rearranging the terms, we obtain du K du + u a(3) = which is a linear differential equation of the first order whose solution is Kx K di u = e JP ef[jpj a + C] where C is the constant of integration. But since K, Cp and a are constants, we have u = e cp af +G C = e C[Pk K: e P+ C] acp + K K + Ce~~~~~ Therefore, Os - 0 - dds- + Ce c, (4) K ~~~~~~CO, In the usual set-up C is such that 0.693-, the half value period of the cp ~K Ki exponential e CV, is about two minutes. Thus, Os- 0 rapidly approaches the constant value P t. That is, the rate of temperature rise of the K sample approaches a steady state where d- =.
VOL. 19, 1933.VCHEMISTR Y: SKA U A ND LA NGDON 945 Figure 1 shows this behavior graphically for a time-temperature curve where d-s is positive, i.e., for a heating curve. The upper broken line is the time-temperature curve of the shield. The full curves are the timetemperature curves of the sample, the lower one being for a case where the value of C in equation (4) is positive, the upper one where it is negative. The lower broken line shows the condition approached by these two curves, 6 ~~~~~~~~Cp s a condition in which Os - 0 is constant and equal to CK - It will be noted that the numerical value of C is the difference between the 0 intercepts of this limiting line and of the curve for the sample; for c = Os C0ds) - =0] and CP dos is the vertical distance between the broken lines. K 6,- -- 848 FIGURE 1 FIGURE 2 FIGURE 3 PROPOSITION II. For the case where d2 is a positive constant (a heating curve) if Cp is constant for a sufficiently long period to attain the steady state mentioned in Proposition I, and if at some later time, ti, Cp starts to increase with the temperature as shown in figure 2, the cunre becoming concave upward at O1 (corresponding to t1), it follows that the temperature curve for the sample is concave downward for all values of t greater than ti, as shown in figure 3. d2cp d That is, if the following conditions hold: 2 = 0 when t < t1; =, Cp Cs du d2c u =.tland = O when t 2 ti; but d2 > O when t 5 ti; then for all K 2 values of t > ti, 2 > 0. PROOF: Differentiating equation (3), which applies here also, we have
946A CHEMISTR Y: SKA U AND LA NGDON. PROC. N. A. S. so that when t ti, - = 0 since 2 = K / dc- du\ 2 CP2 CP, dcp - dcp * d - o = 0.~ ~ ~ ~ ~~0 Further d3u K { d2cp _ C( + 2 dcp) But dcp - dcp dk Whence, d2cp d12 dcp dj+ (d20 2 dd9 2 \di) d2cp 2 ldo\ 2d2CP dcp sic d2os = \d0/2 d0 2 2 d3u K 5 I(\2d2 c Consequently, 3 = C 2 dcp + Thus, when t =tl, d3u = Ku ( d2u udc 2 d2cp > 0. + 2CpdCp\) K )' Therefore, when t increases from ti, 2 becomes positive, also - must remain > 0, for if is or becomes less than the positive function, C (/2 / d2co 2 XdCp +2Cp dcp ' d3> and is increasing. Thus, the curve being continuous, d2 is 13 ~ 2 greater than zero for all values of t greater than t1. It can easily be shown from the foregoing equations that if in Proposition II an insufficient time is allowed for the time-temperature curve of the sample to attain the steady state mentioned in Proposition I before Cp becomes variable: (A) if C > 0 the time-temperature curve being already concave downward becomes even more so at t = ti, and (B) if d2u
VOL. 19, 1933 GENETICS: W. E. CASTLE 947 C < 0 the curve being already concave upward, will begin to flatten out more rapidly at t = ti, and will soon become concave downward. The experimental significance of these propositions will be discussed further in a forthcoming paper. Skau, Proc. Amer. A cad. Arts and Sci., 67, 551 (1933). THE LINKAGE RELATIONS OF YELLOW FAT IN RABBITS BUSSEY INSTITUTION, By W. E. CASTLE HARVARD UNIVERSITY Communicated September 20, 1933 The fat of ordinary domestic rabbits is white, like the fat of sheep and pigs, but in certain breeds of rabbits an occasional individual is found, when slaughtered, to have yellow fat. This is considered an undesirable characteristic, as it renders the appearance of the carcass unusual and so open to objection on the part of the purchaser. Beef with yellow fat is not unusual and so not objected to by the public. The color of yellow fat in rabbits is due to an element in their food (carotene, found in yellow carrots, or xanthophyl, occurring in the chlorophyl of green plants) which, being fat soluble, is deposited in the fat of the rabbit. Ordinary white-fat rabbits, according to the biochemists of Cambridge University, England, have in their livers an enzyme which reduces the assimilated carotene to a colorless state. This reducing mechanism is lacking in yellow-fat rabbits, and so the carotene passes unreduced and yellow into the fat storage tissue. The inheritance of the yellow fat character was first studied and described by Pease (1927),1 who found it occurring sporadically in a race of Flemish Giant rabbits in Cambridge, England. He discovered the character to be a simple recessive in inheritance but genetically linked with albinism-that is, borne in the same chromosome as the color gene, C, and its allelomorphs. Some years previously I had shown that a loose linkage exists between the color gene (C) and brown hair and skin pigmentration (b), with 35.8% crossingover in a total of 1006 back-cross young.2 It was to be expected, therefore, that as yellow fat (y) is linked with C, it should also be linked with b. Evidence that this is so will be presented in this paper. In some preliminary experiments3 on the inheritance of yellow fat, which I found occurring in three different stocks of our rabbits, I was able to verify Pease's discovery that this character is a good recessive and linked with albinism, and to show that it is also linked with the albino allelomorph chinchilla, with a crossover percentage of 7.8 4.7, an esti-