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The work expended in recharging ion k is ^ ^
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The overall electrical work connected with the described process for ion k is then -zle2K and, for all the ions present in volume of the solution Vy
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Work Wel is then identified with the correction for non-ideal behaviour AG E and the activity coefficient is obtained from the equation (1.3.2!)
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Differentiation assuming that V is independent of n, (which is fulfilled for point charges) yields
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34 It is convenient to introduce In 10 x 4jzV2( kT)3/2 (1.3.23)
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yielding the usual form of the equation for the activity coefficient (cf. p. 11): log y* =-,4z 2 VZ (1.3.24)
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termed the Debye-Hilckel limiting law. Coefficient A has the numerical value A = 5.77057 x 10 4 (Dr)" 3/2 = 1.82481 x 10\DT)-3/2 and, for water at 25 C, A = 1.61039 x 10"2 m3/2 mol" m = 0.50925 dm 3/2 -mor 1/2 In these expressions, the first value is valid for basic SI units and the second for / in moles per cubic decimetre substituted into Eq. (1.3.24). Equation (1.3.24) is a very rough approximation and does not involve the individual characteristics of ion k. It is valid for a uni-univalent electrolyte only up to an ionic strength of 10~3 mol dm" 3 (see also Fig. 1.8). In view of the definition of the mean activity coefficient and of the electroneutrality condition, v+z+= v_z_, the limiting law also has the form logy =^z + z_V7 (1.3.25) m3/2 moP 1/2 dm3/2 mol"1/2
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1.3.2 More rigorous Debye-Hiickel treatment of the activity coefficient A more rigorous approach requires integration of Eq. (1.3.13) not over the whole volume of the solution but over the effective volume after the volume of the central ion has been excluded, as this region is not accessible for the ionic atmosphere. Thus, integration is carried out from r = ay i.e. from the distance of closest approach, equal to the effective ion diameter, which is the smallest mean distance to which the centres of other ions can approach the central ion. This value varies for various electrolytes (Table 1.3). In this refined procedure the integration constant appearing in Eq. (1.3.13) attains the value kx = [zkel4ne{\ + KO)\ exp {KO) and the potential \pk = zke/4jt [r(l + Ka)]'1 exp [ic(a r)]. This expression can again be separated into the contribution of the isolated central ion ip k and the contribution of the ionic atmosphere tya. As a result of the principle of linear field superposition, these two quantities can be added algebraically. The contribution of the
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Fig. 1.8 Dependence of the mean activity coefficient y >c of NaCl on the square root of molar concentration c at 25 C. Circles are experimental points. Curve 1 was calculated according to the Debye-Hiickel limiting law (1.3.25), curve 2 according to the approximation aB = 1 (Eq. 1.3.32); curve 3 according to the Debye-Hiickel equation (1.3.31), fl = 325nm; curve 4 according to the BatesGuggenheim approximation (1.3.33); curve 5 according to the Bates-Guggenheim approximation + linear term 0.1 C; curve 6 according to Eq. (1.3.38) for a=0.4nm, C = 0.055 dm 5 -mor 1 ionic atmosphere is then given by the expressions -1 (1.3.26)
Alter \l + Ka
(1.3.27)
because at a distance of r < a, no ion of the ionic atmosphere can be present
36 Table 1.3 Effective ion diameters (According to B. E. Conway) Ion Rb , Cs , NH4 , Tl , Ag
a (nm)
+ + + +
K , cr, Br, r, o r , NO 2 -, NO 3 OH~, F " , CNS", NCO~, H S " , C1O 3 ", C1O 4 ", BrO 3 ~, IO 4 ", MnO 4 ", HCOCT, citrate", CH 3 NH 3 + Hg 2 2+ , SO 4 2 -, S 2 O 3 2 ", S 2 O 6 2 ", S 2 O 8 2 ", Se 4 2 ", CrO 4 2 ~, HPO 4 2 ~, PO 4 3 ~, Fe(CN) 6 3 ~, Cr(NH 3 ) 6 3+ , Co(NH 3 ) 6 3+ , Co(NH 3 ) 5 H 2 O 3+ , NH 3 + CH 2 COOH, C 2 H 5 NH 3 + N a \ CdCl + , C1O2", IO 3 ", HCO 3 ", H 2 PO 4 ~, H S ( V , H 2 AsO 4 -, Pb 2 + , CO 3 2 ", SO 3 2 ", MoO 4 2 ", Co(NH 3 ) 5 Cl 2+ , Fe(CN) 6 NO 2 ", CH 3 COO", CH 2 C1COCT, (CH 3 ) 4 N + , (C 2 H 5 ) 2 NH 2 + , NH 2 CH 2 COO" Sr 2+ , Ba 2 + , Ra 2 + , Cd 2+ , Hg 2 + , S2~, S 2 O 4 2 ", WO 4 2 ", Fe(CN) 6 4 -, CHC1 2 COO~, CC1 3 COO + , (C 2 H 5 ) 3 NH + , C 3 H 7 NH 3 + Li + , Ca 2+ , Cu 2 + , Zn 2 + , Sn 2+ , Mn 2 + , Fe 2 + , Ni 2 + , Co 2 + , Co(ethylendiamine) 3 3+ , C 6 H 5 COO', C 6 H 4 OHCOCT, (C 2 H 5 ) 4 N + , (C 3 H 7 ) 2 NH 2 + Mg 2+ , Be 2+ H + , Al 3 + , Fe 3 + , Cr 3+ , Sc 3+ , La 3 + , In 3 + , Ce 3 + , Pr 3+ , Nd 3 + , Sm 3+ Th 4 + , Zr 4 + , Ce 4 + , Sn 4+
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