| Mean distance of closest approach of ions in NaCl (aq.) at 25° C calculated from degrees of association using Bjerrum’s theory |
R. Heyrovská
Academy of Sciences of the Czech Republic, J. Heyrovský Institute of Physical Chemistry, Dolejškova 3, 182 23 Prague 8, Czech Republic
This paper demonstrates, for the first time, the applicability to 1:1 strong electrolytes of Bjerrum’s pioneering theory, which relates the degree of ionic association with the mean distance a of closest approach of ions. It can be noted that only recently, strong electrolytes like NaCl were shown to be incompletely dissociated in aqueous solutions. The present data on the degrees of dissociation of NaCl (aq.) at 25° C have been used to calculate the values of a for concentrations from zero to saturation, using Bjerrum’s theory. It is found that both Q(b), the Bjerrum’s integral, and a are simple functions of the molal volume and concentration and that at saturation, a is less than q, the critical distance.
I
T was generally believed1–3 that 1:1 strong electrolytes like NaCl dissociate completely into ions in aqueous solutions. However, recent X-ray diffraction and molecular dynamic simulation studies of saturated solutions4,5 and investigation of thermodynamic data at all concentrations6,7 have shown that even 1:1 strong electrolytes are incompletely dissociated in aqueous solutions as originally supposed8,9.Taking the dissociation of NaCl (aq.) as a typical example,
NaCl Û Na+ + Cl– (1)
(1 – a )m a m + a m,
the degree of dissociation, a , evaluated7a from osmotic coefficients (j ) decreases with molality m from unity at infinite dilution to a minimum at about 1.5 m (corresponding to mean ionic activity, a± = 1) and then increases with concentration to a value < 1 at saturation.
The values of a are given in Table 1. With these values of a , it has been possible to give a quantitative and unified explanation for other solution properties like the EMF of concentration cells (which is directly proportional to ln(a±)) and the densities/molal volumes of solutions, using simple mathematical equations valid for the whole concentration range from very dilute to saturated solutions7a–h.
Therefore, a fresh investigation of the Bjerrum’s theory of ionic
association was undertaken7i. This theory predicts the degree of association
for various distances of closest approach of oppositely charged ions, and it predicts
negligible ionic association for 1:1 electrolytes like NaCl (aq.)1,2. Here,
with the present knowledge of
Table 1. Degree of dissociation (a ), molal volume (V) and molar concentration (c)
at molality m and the corresponding values of Q(b),
b and a of Bjerrum’s eqs (2) and (3), for NaCl (aq.) at 25oC
the degrees of association7a–c, it has been demonstrated that Bjerrum’s theory can be used to calculate the mean distance of closest approach of ions, using NaCl (aq.) as an example.
Bjerrum’s integral Q(b) and ionic association
Bjerrum derived the basic relation connecting the degree of
association, (1–a ), with the distance, a of
closest approach of oppositely-charged ions at any concentration, c (refs 1, 2).
This relation was obtained by integrating the number of oppositely-charged ions in all the
spherical shells around a central ion, from a distance r = a, up
to the critical distance r = q(> a). For a 1:1
electrolyte, (1–a )/c is proportional to the
integral Q(b) as given
by1 (the proportionality constant = 2.755 cm3 mol–1
at 25° C)
(1–a )/c = (4p N/1000)(e2/e kT)3Q(b), (2)
where e is the electronic charge, e is the dielectric constant of the solvent and N is the Avogadro number. The integral Q(b) depends1 on r,
Q(b) =
(3)
where x (= e2/e kTr) is the ratio of the mutual electrical potential energy (e2/e r) to the thermal energy (kT), and has the values x = 2 and x = b for r = q and r = a respectively. The critical distance q = e2/2e kT for x = 2 is the distance at which the probability of finding oppositely-charged ions is minimum. Therefore, eq. (2) gives finite degrees of association for all values of a £ q. At 25° C, a = 7.14/b Å and q = 3.57 Å for a 1:1 electrolyte in aqueous solution. This critical distance was considered too large to form ion pairs1,2.
Linear dependence of cQ(b) on the volume of NaCl (aq.)
For the values of a in Table 1, the corresponding values of Q(b) were calculated as Q(b) = (1–a )/2.755c (see eq. (2)). For calculating the molar concentrations c (= 1000m/V) the molal volumes V were obtained using eqs (4) and (5b) below. For NaCl (aq.) at 25oC, it was demonstrated7a,c,d that the molal volume10 is the sum of the volumes of solvent (A) and solute (B), as given by the pair of equations,
V = VA + m[(1–a )Vcr + a (V+ + V– + d Vel)],
(m < 1.5), (4)
Figure 1. The linear dependence of b (= 7.14/a) on 1/m for NaCl (aq.) at 25oC from 0.1 m to saturation (m = 6.14), where a (= 7.14/b) is the mean distance of closest approach of oppositely-charged ions (see eqs (8) and (9)).
V = V'A + m{(1–a )Vcr + a (V+ + V– + d Vel)
+[a d Vd–(1–a )Vcr]} (5a)
= V'A + a m(Vcr + d Vel), (m > 1.5 up to satn.), (5b)
where Vcr = 26.8 cm3 mol–1
is the volume of un-
dissociated NaCl (the same as in the crystal), VA =
1002.86 cm3 and V'A = 1002.38 cm3
are the volumes of 1 kg of water at 25oC in the solution, (V+ + V–)
= 18.31 cm3 mol–1 is the volume of ions in the
solution, d Vel = –2.06 cm3
mol–1 is the change in volume
due to electrostriction and d Vd = Vcr – (V+ + V–) =
8.49 cm3 mol–1 is the volume change accompanying
association/dissociation. The last term in eq. (5a) is the volume (due to interionic
interaction) shared by the ions and ion pairs.
From eqs (4) and (5b), one finds that (1–a ) (and hence cQ(b), see eq. (2)) is directly proportional to the volume of the electrolyte in the solution,
(1–a ) = 1 – Vcr/(d Vd–d Vel) + (V–VA)/m(d Vd–d Vel) (6a)
= – 0.540 + 0.0948 VB (m < 1.5). (6b)
(1–a ) = 1 – (V–V'A)/m(Vcr + d Vel) (7a)
= 1 – 0.0404 V'B (m > 1.5 up to satn.), (7b)
where VB = (V–VA)/m and V'B = (V–V'A)/m are the volumes per mole of the electrolyte at molality m.
Dependence of b (= 7.14/a) on the molality, m
Table 1 gives the values of b corresponding to the values of Q(b). Q(b) vs b data obtained by numerical integration of eq. (3) was furnished11. The last column gives the values of a (= 7.14/b) (estimated in this way for the first time). Note that Bjerrum’s theory gives a definte value for a at any given concentration. Therefore, in the graph of the degree of association (1–a ) vs a, only one value of a (= 2.44 Å) is possible at the given concentration (c = 0.1). It can be seen from Table 1 that a increases with m until at saturation (6.144 m) it reaches the value 3.53 Å (corresponding to b = 2.02 and Q(b) = 0.009). Note that at saturation, a is less than the critical distance q (= 3.57 Å)!
The relations below show the linear increase of b (= 7.14/a), the ratio of the electrical to thermal energy at the distance a of closest approach, with the amount of solvent per mole of solute, 1/m (see Figure 1). For the concentration ranges on the two sides of the a vs m curve7a, linear regressions give the pair of equations,
b = (7.14/a) = 2.00 + 0.185/m, (m < 1.5), (8)
b = (7.14/a) = 1.98 + 0.207/m, (m > 1.5). (9)
From eqs (8) and (9), it can be seen that the reciprocal of the distance a varies linearly with 1/m.
On combining the pairs of eqs (6) and (8) and (7) and (9), one gets,
(1–a ) = –0.540 + 0.512(V–VA)(b–2.00), (m < 1.5),
(10)
(1–a ) = 1 – 0.195(V–V'A)(b–1.98), (m > 1.5) (11)
which shows that (1–a ) and
hence the Bjerrum’s factor cQ(b) (see eq. (2)) are proportional to the
product,
V(b–2.00) and V(b–1.98) for m < 1.5
and m > 1.5 respectively.
Thus, it is shown here for the first time that Bjerrum’s theory can be used for evaluating the mean distance of closest approach of ions from the available data on the degrees of association.
6, Praha 6, private communication.
AKNOWLEDGEMENTS. I thank Dr. L. Novotný for moral support and Emil and Albert for encouragement and computer assistance. This work was supported by the grant, GA AV CR 440401 from the Academy of Sciences of the Czech Republic.
Received 18 May 1998; revised accepted 16 November 1998