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Calculation of Mobility of Ions

Consider a dilute solution of an electrolyte containing m gram equivalents of completely dissociated molecules per unit volume. Then, in a unit volume of the electrolyte, there are m gram equivalents of the cations and m gram equivalent of the cations and m gram equivalents of the anions.

Let Q coulomb be the charge carried by a gram equivalent of the ions.
Let u and v be the velocities of the cation and the anion respectively.
The charge passing across unit area of cross-section of the electrolyte per second due to cations is muQ. Similarly, the charge passing across unit area per second due to anions is mv Q (in the opposite direction). Hence the current density of the current passing through the unit cross-section of the electrolyte is
            C = muQ – [-mvQ] = mQ[u + v]                        … (1)

Let σ be the conductivity of the electrolyte, and dV/dx the potential gradient in it. Then the current density through the electrolyte is given by
            C = σdV/dx                                … (2)
Comparing Eqs. (1) and (),
        mQ(u + v) = σdV/dx
or        (u + v) = σ/m x 1/Q x dV/dx.
The ratio σ/m = conductivity/Concentration = Equivalent conductivity.
Since the electrolyte is considered to contain completely dissociated molecules, σ/m = λ∞, the equivalent conductivity at infinite dilution.
. :        (u + v) = λ∞ x 1/Q x dV/dx                            … (3)

dV/dx can be found from the potential difference between the electrodes and the distance between them. Q = 96500 coulombs. λ∞ can be found experimentally. Therefore (u + v) is known.
The transport numbers nc and na are known from Hittorf’s method.
        nc = u / u + v and na = v / u + v                            … (4)
u and v may be separately calculated by substituting for (u + v)  Eq. (4).
Thus the values of the mobility of ions can be calculated.

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