Ion-Ion Interactions
Describe and give examples of ionophore, ionogen, strong and waek electrolyte. Why is the charge density around a reference ion not zero when the whole solution is electroneutral? Explain succinctly the Debye-Hückel approach to their ionic-cloud theory. M
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ION·ION INTERACTIONS EXERCISES
Review of Sections 3.1 to 3.3 of the Textbook. Describe and give examples of ionophore, ionogen, strong and waek electrolyte. Why is the charge density around a reference ion not zero when the whole solution is electroneutral? Explain succinctly the Debye-HOckel approach to their ionic-cloud theory. Mention the main steps involved in this theory as well as the assumptions involved in its development. Write the linearized Poisson-Boltzmann equation and the time-average spatial distribution of the excess charge density around a reference ion. What is the total ionic-cloud charge around a reference ion? Draw the variation of the electrostatic potential and the excess-charge density as a function of the distance from the central ion. What does 1(-1 represent? Write expressions for the Debye-Huckel length, the total electrostatic potential, the potential due to the isolated central ion, and the potential due to the ionic cloud. What is the physical meaning of effective thickness of the ion atmosphere?
3.1 (a) Write down the full Taylor expansion of t!' and e-x • (b) Find the percentage difference between t!' and l+x for x = 0.1, 0.5 and 0.9. (Cf. Exercise 3.32 in the textbook) (Bockris-GamboaAldeco) Answer: 89
M. E. Gamboa-Adelco et al., A Guide to Problems in Modern Electrochemistry © Springer Science+Business Media New York 2001
CHAPTER 3
90 (a) The expansions are: 2 3 eX =l+x+~+~+ ...
e
-x
x
(3.1)
3!
21
2
x
3
=1-x+---+ ... 21
31
(3.2)
(b)
x
eX
l+x
e
x
-(I+x)xlOO x
0.1 0.5 0.9
1.1052 1.6487 2.4596
1.1 1.5 1.9
e 0.47% 9.02% 22.57 %
3.2 Evaluate the electrostatic potential f// and the work of charging Cu +2 ions from a state of zero charge to a charge of Zil!o in water at 25°C. (Contractor) Data: z;=2 &=
78.3
T=298 K r Cu +2 = 72 pm
Answer: The work of charging an electrical conductor is (cf. Eq. 3.3 in the textbook), z ·eo W=-'-f// 2
(3.3)
and, the electrostatic potential If/, is given by the equation (cf. Eq. 3.30 in the textbook), (3.4)
91
ION-ION INTERACTIONS
Substituting this value into Eq. (3.3),
w=
(2 )(1.602XlO -19
e)
2
(0.51 v)x 2 1ve
= 8.18xlO -20 J
(3.5)
3.3 (a) The excess charge of an ionic atmosphere varies with distance out from the central ion. Starting from the expression for the excess charge density, show that the net change in a spherical shell of thickness dr is dq = -zieOe
-/(J"
K
2
rdr
(3.6)
(b) At a certain distance from a central ion, there will be a ring with a maximum charge. Find the distance of this ring from the central ion. (Cf. Problem 3.7 in the textbook) (Bockris-GamboaAldeco) Answer: (a) The excess charge density is given by (cf. Eq. 3.35 in the textbook),
e
-/(J"
(3.7)
r
Therfore, the total charge in a spherical shell of thickness dr at a distance r from the reference ion is given by p, times the volume in this shell, i.e., 4n?dr, or, (3.8) Substituting p, from Eq. (3.7) into Eq. (3.8) gives the equation we are looking for, i.e., 2
dq = -ZjeOK re
-/(J"
dr
(3.9)
92
CHAPTER 3
(b) The excess c
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