THEORY OF RADIATION CHEMISTRY. VI. CYLINDRICAL DIFFUSION


THEORY OF RADIATION CHEMISTRY. VI. CYLINDRICAL DIFFUSION...

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NOTES

May, 1963

1155

tained for alcohol-carbon tetrachloride ~ y s t e m s ~ - ~tive heat-of-mixing results show that for these three systems the breriking of alcohol-alcohol association and for alcohol-ether systems.a bonds is the dominant factor in determining the net lobTABLE I served heat of mixing. The similarity of the three HEATO F M I X I N G FOR THE C.4RBON TETRACHLORIDE (l)-ISOcurves shows the interesting result that these three PROPYL ALCOHOL (2) SYSTEM AT 24.92 i- 0.01' proton acceptors are about equally capable of breaking 21 52 AHrn, cal./g. mole the alcohol-alcohol bond and that the amount of ~4010,057 0.943 11.5" vation is apparently a secondary effect. The data ,090 ,910 18.0" also show that monomeric isopropyl ether and the ,148 852 35.1" polymeric ether, poly-(propylene oxide), mix with ,317 ,683 84.1 isopropyl alcohol to produce similar heat-of-mix ing ,384 ,616 103 curves. It appears that polymer and monomeric ,463 ,537 133 ,664 336 164" ethers exhibit very similar energetic effects when ,773 ,227 158" mixed with isopropyl alcohol. ,877 ,123 130" Acknowledgment.-The authors are grateful for fin,870 ,121 132" ancial support given by the Kettering Foundation in ,945 ,055 84.1" the form of a grant and by the Woodrow Wilson Founa Calculated from heat of dilution. dation, in the form of a fellowship. TABLE I1 H E A T OF RfIXliYG FOR THE nIISOPROPYL

ETHER(1)-ISOPROPYL

ALCOHOL( 2 ) SYSTEM AT 24.92 =k 0.01' XI

a

AHm, cal./g mole

52

0.063 0 937 ,125 ,875 ,281 ,719 ,337 ,663 ,404 .596 .466 ,534 ,542 .458 .633 ,367 .795 ,205 Calculated from heats of dilution.

22.5" 46,9" 102 126 139 151 163" 160" 150"

BY ARYEHH. SAMUEL AND JAMES S. MILLS Stanford Ressarch Instatute, Menlo P a r k , California Recewed October 1 , 1962

TABLE I11

HEATOF MIXIXGFOR PROPYL

POLY-(PROPYLENE OXIDE) (1)-Iso-

ALCOHOL( 2 ) SYSTEM AT 24.92 i 0.01"

Xlb

a

THE

AHm, cal./g. mole (group basis)

XZB

0.075 0.925 .122 .878 .189 ,811 .282 ,718 .448 .552 .519 ,481 .586 ,414 .634 ,366 .039 .361 Calculated from heat of dilution. x2 = moles isopropyl alc.

moles isopropyl alc.

r

+ moles -

CH2-C--0

24.4" 40.0" 59.9" 88.5" 121 123 132 136 135

H l -

THEORY OF RADIATION CHEMISTRY. VI. CYLIKDRICBL DIFFUSION MODEL WITH A SCAVESGER'

groupti

I n these systems two types of hydrogen bonding are possible. The pure alcohol is highly associated and therefore as the pure alcohol is mixed with a second liquid, alcohol--alcohol hydrogen bonds are broken with an endothermic heat effect. The alcohol, however, is also capable of hydrogen bonding with the second liquid, causing an exothermic heat effecte6 The posi( 2 ) R. F. Blanks and J. h1. Prausnita, to be published. (3) H. Hirobe, a8 quoted by J. Timmermans, "Physico-Chemical

Con-

s t a n t s of Binary Systems in Concentrated Solutions," Intersoience, New York, N. Y . , 1959. (4) J.-E. A. Otterstedt a n d R. W. Missen, Trans. Faraday Soc., 58, 869 (1962). ( 5 ) G. Scatohard, S. E . Wood, and J. M. hIocht4, J . Am. Chem. Soc., 68, 1963 (1946); 74, 3724 (1952). ( 6 ) Alcohol-carbon tetrachloride solvation is discussed b y Otterstedt a n d Missen.4 Isopropyl alcohol-isopropyl e t h e r a n d isopropyl alcoholpoly-(propylene oxide) heats of solvation are repormd elsewhere.2

Previous papers of this series2have outlined a simplified mathematical model for the simultaneous diffusion and recombination of ions and radicals formed in Ihe track of ionizing particles. This model (the Mag;ee model) is useful to the degree that it is susceptible of analytic mathematical treatment while remaining reasonably similar to the actual events in irradiated materials. The main characteristic of the Magee model is a discontinuity of the concentration of active species (ions or radicals) at the boundary of the expanding track (cylindrical) or spur (spherical). The concentration is assumed to be uniform in space, though varying in time, both outside (background concentration, y) and inside the bouiidary. I n this paper, the cylindrical model of ref. 2a is modified by the introduction of a homogeneously distributed scavenger (solute) which ultimately reacts with all active species which do not undergo pairwise recombination. The presence of such a scavenger is a feature of most real systems. The cylindrical model is appropiiately used for irradiations with particles of high linear energy transfer (LET). The scavenger if; assumed to be present in c concentration c,. Depletion of the scavenger tracks is ignored; the inaccuracies introduced by this assumption have been discussed by Kuppermpnn arid Belforda for the case of irradiated water and are fourtd to be small. The bimolecular reaction rate of the scavenger with the active species is I%,.These two quantities always appear as the product The species formed in the scavenger reac sumed stable by comparison with the active species. It should be noted that this is a (1) Presented in p a r t a t the Second I n t e r n a t Research, Harrogate, Yorkshire, England, August 5-1 1, 1962. (2) (a) J. L. Magee, J . Am. Chem. Sac., 75, 3270 ( l Q 5 1 ) , (b) A. IT. Samuel, J . Phys. Chem., 66, 242 (1962). (3) A. Kuppermann andl G. G. Belford. J . Chem. Phys., 86, 1427 (1962).

NOTES

1156

T'ol. 67

and that it should not be used if the parameters are expected to be grossly different for the species formed. For example, it should not be used for ions unless it is believed that the electrons will attach to molecules to form negative ions. Solution of the Mathematical Model.-This model leads to the following differential equation for the number of active particles ( N ) per unit length inside the track dW/dt

- k N 2 / v - AN

+ y dvldt

(1) where k is the rate constant of recombination, and v is the track volunie per unit track length

v

= vo

2

+ Dt ( D

=

diffusion constant)

(2) At zero time (the moment of passage of the primary T = AT, = wo vOyo,where wois the particle), v = VO, h number of active species per unit track length formed by the primary ionizing particle (this is proportional to the LET). Outside the track boundary, the change in the background concentration, y, is described by

+

dy/dt

=

-/cy2 - Ay (y = yo when t = 0)

(3) a differential equation with separable variables having the solution y = yo exp(-At){l

+ ky0A-l[1 - exp(-AAt)]j-l

[Ei(--iqa) - Ei(-iq)] (9) This is a general solution giving the number of active species in the track at any moment of its life. Special Cases.-Nlagee2" has shown that when densely ionizing particles pass through liquids or solids, the background concentration, y, is very low for all normal dose rates. It is therefore useful to have a special form of (9) for the case of negligible y, Either by solving the Bernoulli equation produced by dropping the last term of (5), or by simple substitution of y = yo = 0, hence p = 0, in (9), one obtains

N

kN2

d N _ dv

Dv

AY D

+

+ k ~ / ~ A - ~-{ exp[l

- LI~)]

=

vy

=

1

AD-l(v - v0)1\

(5)

uyo exp[-AD-l(v - vO)] - exp[- AD-'(v -

+ kyoA-'{I

~n)])

(6) This is a solution of ( 5 ) . The general solution is found by substituting N = N' 1/z (7)

+

.A linear differential equation in 2 results, which is solved in the usual manner =

~D-I

1 + p[1 -

Ei(-q)]]-'

Another approximation is needed for sniall A, when p becomes large and q small. The tabulation of Ei. (- x) does not extend to very small arguments, but one may then use the approximation

Ei(-iqx) - Ei(-iq)

= 111 x

- iq(z - 1) (11)

If In x >> gi(x - I), the sumniatioii can be obtained in closed form as

2=1

This is a, Riccati equation and can be converted into a linear differential equation if a particular solution can be found. -4s the particular solution N' we take the case wa = 0. This corresponds to zero LET of the primary particle or an "empty track" so that (4) applies inside the track also lJTf

-

m

yo exp[-AD-l(v

1

+

(w0-' kD-1 e * [ [ E i ( - p )

(10)

(4)

On substituting (2) in (4) and the derivative of (2) in (I), the following differential equation is obtained

= e-q('-l)

e-q('--l)

11 x

Here p = kyo/A, q = Avo/D, x = u / u 0 , and Ei stands for the exponential integral. (It will be noted that p g = p of ref. 2a.) The constant of integration C is found by substituting the initial condition AV = S o , hence 2 = l/wo, when x = 1. This gives as the complete solution, after substitution in (7)

[l - p e * / ( p

+ l ) ] +ln x

(12)

This obviates the necessity of calculating many ternis of the summation when p >> 1. Use of the Formulas.-To use (9) for the study of real systems, it is necessary to relate it to an experimentally observable quantity. The most easily observed such quantity is the fraction of active species which react with the scavenger. We give here a fivestep procedure for the calculation of this fraction as a function of the track parameters. 1. The necessary parameters for a calculation are lc, k,c, = A , w0,D , uo, and yo. All except yo must be known or assumed in advance. Instead of yo, we shall use the dose rate as our sixth independent variable, as shown in steps 2 and 3. It is usually not too difficult to estimate w0,D , and e,. When not known experimentally, values of k and k , can be obtained from estimates of the activation energy and steric factor for the reconibiliation and the scavenger reaction, respectively. The value of YO in gases is available from cloud chamber experiments; in liquids there are conflicting views of its magnitude. 2. The next step is to fix the expansion ratio of the track. As in ref. 2a, we call the final track volume at the moment the track is traversed by another ionizing particle urn. Then x, = Y, 'vo. The value of xm is obtained from

zm(xn,- 1)

==

( W / ~ ) W ~ D / I V ~(13) ~

May, 1963 where W is the energy (in e.v.) necessary to form an ion or radical pair, and I is the dose rate (in e.v./cm.a/sec.). This formula is essentially the same as eq. 15 of ref. 2a. 3. The value of yo is then calculated by trial and , value of yo is assumed and N(xm) is calcuerror. 4 lated by (9). When the correct value is found, N(x,) = voyoxm. If the trial N is too high, raise yo, and vice versa. We have found that about six trials are necessary; but it is sometimes observed that the second term of (9) IS very insensitive to changes in yo, so that the labor is much redraced. 4. Using the value of yo thus obtained, calculate N and y for vo 5 v