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indicate that two different types of ion pairs are present and undergoing propagation, the contact ion pairs and solvent-separated ion pairs. The observed ion-pair propagation constant kp is an apparent rate constant. kp is a composite of the rate constants for the contact ion pair (kc ) and solvent-separated ion pair (ks ) according to

(5.45)

5-101

Now comparing (5.27) and (5.28) with (5.22) and (5.23), we find that we can make the following: correspondences

5-102



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where x and (1 x) are the fractions of solvent-separated and contact ion pairs, respectively, and Kcs is the equilibrium constant for interconversion between the two types of ion pairs:

ck, m= 1,

P (C+)

(5.46) (5.47) (5.48) (5.49) (5.50)

5-103

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Qj{k) = - ck Vj,

The variation of kp with temperature depends on the interplay of the separate variations of kc , ks , and Kcs on temperature according to

Pj{k)

5-104 5-105 5-106

Fj{k) = 27r 2 ckp Ej{k),





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A consideration of Eqs 5-101, 5-102, and 5-104 to 5-106 indicates the reasons for the behavior observed in Fig 5-7 There are no solvent-separated ion pairs at the highest tem perature ($70 C in THF) Decreasing temperature decreases kp because kc is decreasing As the temperature continues to decrease, a temperature is reached at which solvent-separated ion pairs are formed Since ks > kc , kp goes through a minimum (at $30 C), and then increases (provided Es < Hcs ) However, a further decrease in temperature causes the rate to go through another in ection There is so much conversion of contact ion pairs to solvent-separated ion pairs that any additional increase in the fraction of the latter is insuf cient to counter the conventional effect of Es ks decreases with decreasing T, kp reaches a maximum (at about 75 C) and then decreases.

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(5.51) as these are continuous rather than discrete variables. So this interesting analogy tells us that each pair of field variables Vj{k) and Bj{k) corresponds to a Hamiltonian of the sort (5.38) by way of (5.46)-(5.50). Now, provided that we restrict k to a half space, pairs corresponding to different values of j and k are independent. Thus, the Hamiltonian for our single electric dipole interacting with the radiation field is just the sum (integral) of each of these individual Hamiltonians for each pair Vj{k) and Bj{k):

The overall effect of T on kp depends on the relative values of Ec , Es , Hcs , and Scs In a poor solvent such as dioxane, there is a neg ligible fraction of solvent-separated ion pairs at all temperatures; kc and kp decrease with decreasing temperature over the complete temperature range In a moderately good solvent (THF and 3-methyltetrahydrofuran), the behavior in Fig5-7 is observed For a suf ciently good solvent where the fraction of solvent-separated ion pairs is close to one, ks and kp would decrease continuously with decreasing temperature Data indicate that hexamethylphosphoramide is such a solvent for polystyryl sodium [Schmitt and Schulz, 1975] Experimental data of kp versus 1=T can be tted to the preceding equations to yield values of the various activation and thermodynamic parameters pertinent to ion-pair propagation (Table 5-12).

= .N2 2

Corresponding values of the parameters for free-ion propagation are included in Table 5-12 for comparison The solvent-separated ion pair is approximately half as reactive as the free ion, while the contact ion pair is more than 3 orders of magnitude.

1:3 105 L mol 1 s 1 4:0 10 8 mol L 1 (20 C) 79:1 10 8 mol L 1 ( 48 C) 16.6 kJ mol 1 1:0 108 L mol 1 s 1 36.0 kJ mol 1 6:3 107 L mol 1 s 1 24 L mol 1 s 1 19.7 kJ mol 1 2:08 108 L mol 1 s 1 5:5 104 L mol 1 s 1 19.7 kJ mol 1 142 J K 1 mol 1 2:57 10 3 (20 C) 2:57 10 2 ( 50 C)

2 + Vj{k) -

Data for Schmitt and Schulz [1975].

Most database tables place related data in columns but some place this data in rows. If you want to sort data across a row, click in the worksheet, click the Home tab, click the Sort & Filter button and choose Custom Sort to open the Sort dialog box.Then click the Options button. In the Sort Options dialog box that appears, click Sort Left to Right.

1 7r2P Ej{k) Vj{k)

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