Dropping-mercury electrodes have been used extensively in analytical chemistry and electrochemical research. As shown in Fig. P8.11, a small mercury drop of radius a is suspended from a tube in an aqueous electrolyte solution. A constant volume flow rate Q causes the drop to grow until surface tension can no longer support it and it detaches. The time-dependent current caused by an electrochemical reaction at the mercury–water interface is measured up to the moment of detachment. Knowledge of the velocity in the surrounding solution is needed to fully interpret such data.

(a) Assuming that Stokes’ equation applies in the electrolyte solution and that v_θ and v_ϕ inside the mercury drop are each negligible, show that the external flow is time-independent and purely radial. Determine v_r(r) and ℘(r) in the electrolyte solution.

(b) A reasonable criterion for the use of Stokes’ equation is 

Will Stokes’ equation become less or more accurate as the drop grows? In a typical application, a mercury drop has a radius a_max = 0.5 mm just before detachment and a lifetime t_max = 3 s. Over approximately what fraction of that lifetime will Stokes’ equation apply?

Transcribed Image Text:

1. 6 Q [₁] Mercury a(t) Water Figure P8.11 A growing mercury drop suspended from a tube in an aqueous solution.