Question of Physical Chemistry

Question

The mobility of a divalent cation in water is 8*10^-8 m²v^-1s^-1. The effective radius of the ion is ( viscosity of water = 1cP; e=1.6*10^-19C)

1. 106pm
2. 212pm
3. 424pm
4. 318pm

Answer 1

The answer to this question uses the Stokes-Einstein radius of an ion moving through a viscous fluid:

The drift velocity s of an ion in fluid is:

s = uE (1)

where u is the ion mobility and E is the strength of the electric field acting on the ion.

Let's define the two forces acting on the ion:

F_r = 6πηrs (2)

Where F_r is the retardation force on the ion, η is the viscosity of the fluid, r is the effective ion radius, and s is the drift velocity.

F_f = zeE (3)

Where F_f is the force from the electric field and ze is the total charge of the ion.

When an ion is at its drift velocity s, both F_r and F_f are equal:- no acceleration is occurring from the two forces!

F_r = F_f

6πηrs = zeE (4)

Rearranging (4) we end up with this equation:

s/E = ze/6πηr (5)

Recall that if we rearrange (1) we get:

u = s/E

u = ze/6πηr (6)

We have solved for ion mobility u as it relates to the ionic charge ze, the viscosity of the fluid η and the effective ionic radius r.

Because we know the ion mobility and the valency, we can rearrange (6) to solve for effective ionic radius r:

r = ze/6πηu (7)

Plugging in our values for ze (2 × 1.6 × 10^-19 C), η (1 cP), and ion mobility u (8 × 10^-8 m²v^-1s^-1) into (7), we get an ionic radius of:

2.12*10^-10 meters or 212 pm