Equation 1
kpE = kDE + kIE.
This relationship is obtained when the reaction volume is optically thin so that a negligible fraction of the incident light is absorbed and is sufficiently dilute in test chemical; thus the direct and indirect photoreaction processes become first-order.
Equation 2
(kp)SHW = (1/t)Pn(Co/Ct)SHW (in d-1)
Equation 3
(kp)W = (1/t)Pn(Co/Ct)W (in d-1),
where the subscript identifies a reaction in SHW or PW; t is the photolysis time in calendar days; Co is the initial molar concentration; and Ct is the molar concentration in the irradiated tube at t. In this case t = 1 day.
Equation 4
R = (kp)SHW/(kp)W.
The coefficient R, defined by Equation 4, is equal to [(kI + kD)/kD]. If R is in the range 0 to 1, the photoreaction is inhibited by the synthetic humic water and Phase 3 does not apply. If R is in the range 1 to 2, the test chemical is marginally susceptable to indirect photolysis. In this case, Phase 3 studies are optional. If R is greater than 2, Phase 3 measurements are necessary to measure kpE and to evaluate kIE.
Equation 5
kpE = 0.45(kp)SHW
Equation 6
kDE = 0.45(kp)W.
The factor 0.45 is an approximate geometric correction for scattered light in tubes versus horizontal surfaces. A rough value of kIE, the rate constant for indirect photolysis in natural waters or SHW, can be estimated from the difference between kpE and kDE using Equation 7:
Equation 7
kIE = kpE-kDE.
(kp)SHW = (1/1d)Pn(1.53 * 10-5/1.13 * 10-5) (kp)SHW = 0.30 d-1.
kpE = 0.45(0.30 d-1) = 0.14d-1.
Equation 8
-(d[C]/dt) = kI[C] + kD[C].
Equation 9
kI = kIo exp(-kt),
in which kIo is the initial indirect photoreaction rate constant and k is the SHW photobleaching rate constant. After substituting equation 9 for kI in Equation 8 under paragraph (d)(1)(v) of this section, and rearranging, one obtains
-(d[C]/[C] = kIo[exp(-kt)]dt + kD dt.
This expression is integrated to give Equation 10:
Equation 10
Pn(Co/C)SHW = (kIo/k)[1-exp(-kt)] + kD t.
The term (kIo/k) can now be evaluated. Since in pure water, Pn(Co/C)W = kD t, then subtracting this equation from Equation 10 gives
Equation 11
Pn(Co/C)SHW-Pn(co/C)W = (kIo/k)[1-exp(-kt)].
The photobleaching fraction, [1-exp(-kt)], is equivalent to the expression [1-(A370/A°370)], where A°370 and A370 are the absorbances at 370 nm, and are proportional to humic sensitizer content at times zero and t. Therefore, (kIo/k) is derived from the slope of a linear regression using [Pn(Co/C)SHW-Pn(Co/C)W] as the dependent variable and [1-(A370/A°370)SHW] as the independent variable.
Equation 12
Pn(A°370/A370) = (k/kA)Pn(Co/C)PNAP,
where the slope is (k/kA) and the value of kA is calculated from the concentration of pyridine and the absorption of light by PNAP: kA = 2.2(0.0169)[PYR]ka. Values of ka are listed in the following Table 1.
Table 1-Day Averaged Rate Constant (ka)1 for Sunlight Absorption by PNAP as a Function of Season and Decadic Latitude2
Latitude | Season | |||
Spring | Summer | Fall | Winter | |
20° N | 515 | 551 | 409 | 327 |
30° N | 483 | 551 | 333 | 232 |
40° N | 431 | 532 | 245 | 139 |
50° N | 362 | 496 | 154 | 6 |
1 ka = @ ega Lg in the units of day-1, (Mill et al. (1982) under paragraph (f)(10) of this section).
2 For use in Equation 15 under paragraph (d)(2)(i) of this section.
The value of kIo is then given by Equation 13:
Equation 13
kIo = (kIo/k)(k/kA)kA.
Equation 13a
Pn(Co/C)W = (kD/kA)Pn(Co/C)PNAP.
The slope is (kD/kA), and kD is obtained by multiplication of this slope with the known value of kA: i.e., kD = (kD/kA)kA.
Equation 14
(kp)SHW = kIo + kD.
Equation 5a
kpE = 0.455(kp)SHW.
Equation 15
[PYR]/M = 26.9[(kp)SHW/ka].
This pyridine concentration makes the actinometer rate constant match the test chemical rate constant.
Equation 16
V/mL=[PYR]/0.0124.
The PNAP/PYR solutions should be wrapped with aluminum foil and kept out of bright light after preparation.
Table 2-Category and Sampling Procedure for Test and Actinometry Solutions
Category | kp (d-1)SHW | Sampling procedure |
A | 5.5 J Kp J 0.69 | Sample at 0, 1, 2, 4, and 8h. |
B | 0.69[GREATER THAN]kp J 0.017 | Sample at 0, 1, 2, 4, and 8d. |
C | 0.17[GREATER THAN]kp J 0.043 | Sample at 0, 4, 8, 16, and 32d. |
Equation 17
Pn(Co/C)SHW-Pn(Co/C)W = (kIo/k)[1-(A370/Ao370)],
plot the quantities [Pn(Co/Ct)SHW-Pn(Co/C t)W] versus the independent variable [1-(A370/Ao370)]. Obtain the slope (S1) by least square linear regression. Under the assumptions of the protocol, S1 = (kIo/k).
Equation 18
kA = 0.0372[PYR]ka,
calculate kA using ka values found in Table 1 under paragraph (d)(1)(vii) of this section. The value of ka chosen must correspond to the date closest to the mid-experiment date and latitude closest to that of the experimental site.
Equation 19
kIo = (S1)(kA)(S2),
by incorporating the quantities kA, S1, and S2 determined as described in paragraphs (d)(2) (ix), (vi), and (vii) of this section, respectively.
Equation 20
kD = (S3)(kA),
using the quantities S3 and kA determined as described above.
Equation 14
(kp)SHW = kIo + kD.
Equation 5a
kpE = 0.455 (kp)SHW
As determined, kpE is the net environmental photoreaction rate constant. It applies to clear sky conditions and is valid for predicting surface photoreaction rates in an average humic containing freshwater body. It is strictly valid only for the experimental latitude and season.
Table 3-Chemical Analytical Results for Illustrative Example, Phase 3
Day | 105[C]SHW, M | 105[C]W, M | ASHW 370 | 105 [PNAP], M |
0 | 1.53 | 1.53 | 0.0500 | 1.00 |
1 | 1.03 | 1.40 | 0.0470 | 0.810 |
2 | 0.760 | 1.30 | 0.0440 | 0.690 |
4 | 0.300 | 1.01 | 0.0370 | 0.380 |
8 | 0.130 | 0.800 | 0.0320 | 0.220 |
Data for solutions A through D are given in column 2 through 5, respectively. No significant chemical loss was found in the dark controls.
Table 4-Photoreaction Function for Illustrative Examples, Phase 3, Derived From Table 3
Day | Pn(Co/C)SHW | Pn(Co/C)W | 1-(A370 /Ao370) | Pn(Ao370 /A370) | Pn(Co /C)PNAP |
0 | 0 | 0 | 0 | 0 | 0 |
1 | 0.396 | 0.0888 | 0.0600 | 0.0618 | 0.211 |
2 | 0.700 | 0.163 | 0.120 | 0.128 | 0.371 |
4 | 1.629 | 0.415 | 0.260 | 0.301 | 0.968 |
8 | 2.465 | 0.648 | 0.360 | 0.446 | 1.514 |
FIGURE 1-GRAPHIC DETERMINATION OF S1 = (K Io/k) based on Equation 17 under paragraph (d)(2)(vi) of this section.
FIGURE 2-GRAPHIC DETERMINATION OF S2 = (K/K A) based on Equation 12 under paragraph (d)(1)(vii) of this section.
Equation 19
kIo = (4.96)(0.300)(0.295) = 0.439d-1.
Equation 20
kD = (0.428)(0.300) = 0.128d-1.
Equation 21
kpE = (0.455)(0.439 + 0.128)d-1 = 0.258 d-1.
Equation 22
t1/2E = 0.693/kpE.
Substituting the value of kpE from Equation 21 under paragraph (d)(6)(iii)(H) of this section in Equation 22 yielded
Equation 23
t1/2E = 0.693/0.258d-1 = 2.7 d.
Equation 24
kp = (kp)obs-(kp)loss.
40 C.F.R. §795.70