Sand equation

The Sand equation relates to chrono potentiometric experiments:

${\displaystyle J\cdot {\sqrt {\tau }}={\frac {1}{2}}\cdot z\cdot F\cdot {\sqrt {\pi \cdot D}}\cdot c_{0}}$

With

• Current density ${\displaystyle J}$
• Transition time (transition time, sand time) ${\displaystyle \tau }$
• Diffusion coefficient ${\displaystyle D}$
• Concentration of reactants . ${\displaystyle c_{0}}$

One of the most important applications of the sand equation is the determination of the diffusion parameter D using a given current density from the experimentally determined sand time τ:

${\displaystyle D={\frac {\tau }{\pi }}\left({\frac {2J}{z\cdot F\cdot c_{0}}}\right)^{2}}$.

The equation can be derived from Fick's second law under the assumption of linear diffusion . For this assumption, the current must be in the limit range, i.e. H. diffusion will be the rate-limiting step in implementation.

The equation was published together with its derivation in 1899, 1900 and 1901 by the electrochemist Henry Julius Salomon Sand (1873-1944) and is therefore named after him.

background

In electrolysis , the molecules or ions reacting at the electrode must reach the electrode surface by diffusing through the solution . In doing so, they may be supported by the electric field of the externally applied voltage ( electromigration , i.e. by transport stimulated by the electric field). The Sand equation describes a relationship that can be important for electrolysis with direct current if it is carried out at a constant current intensity, i.e. H. with a galvanostat . It is relevant when the constant current strength is so great that the ions or molecules reacting at the electrode are consumed faster by the electrochemical reaction than can be replenished from the electrolyte solution by transport (diffusion and migration). The current strength is then greater than the so-called limiting current density . The solution in front of the electrode is therefore increasingly depleted in reaction partners: The concentration of the reacting substances continues to decrease and finally becomes zero after a certain time. This time is called the transition time or sand time τ. The electrolysis reactions that have taken place up to this point in time are then no longer possible. The galvanostat must therefore increase the voltage in order to keep the current strength constant. In the simplest case, the time until the electrolysis voltage increases significantly is the sand time τ.

requirements

The equation is only valid if a macro electrode (with a planar diffusion field, i.e. greater than 25  μm in length and width) is used. In addition, it is a prerequisite that the solution is not stirred and that no convection currents occur. Sand has proposed to arrange the electrodes horizontally and to arrange that electrode at the top where the density of the electrolyte decreases during electrolysis.

If the equation is not to apply only approximately, the current flow that results from the charging of the electrochemical double layer must be negligible . If this is not the case, e.g. B. the concentration of the reactant is very small, a correction should be made. The influence of the double-layer capacitance can be determined experimentally by taking a chronopotentiogram of the electrolyte without reactant, i.e. H. for , is recorded. ${\displaystyle c_{0}}$ ${\displaystyle c_{0}=0}$

About the meaning of the equation

Sand's 1901 publication is among Philosophical Magazine's 100 most cited papers for the period 1945-2002.

Individual evidence

1. ↑ G. Schwedt, Analytische Chemie , 2nd edition, Wiley-VCH, Weinheim, 2008, pp. 184ff.
2. ↑ ^{a b} Henry Julius Salomon Sand : About the concentration at the electrodes in a solution, with special consideration of the hydrogen evolution through electrolysis of a mixture of copper sulfate and sulfuric acid . In: Journal of Physical Chemistry . tape  35 , no. 1 , October 1900, p. 641-651 , doi : 10.1515 / zpch-1900-0143 .
3. ^ Henry JS Sand: On the Concentration at the Electrodes in a Solution, with special reference to the Liberation of Hydrogen by Electrolysis of a Mixture of Copper Sulphate and Sulfuric Acid . In: Proceedings of the Physical Society of London . tape 17 , no. 1 , 1899, p. 496-534 , doi : 10.1088 / 1478-7814 / 17/1/332 .
4. ↑ ^{a b} Henry JS Sand: On the concentration at the electrodes in a solution, with special reference to the liberation of hydrogen by electrolysis of a mixture of copper sulphate and sulphuric acid . In: Philosophical Magazine Series 6 . tape 1 , no. 1 , 1901, p. 45–79 , doi : 10.1080 / 14786440109462590 (Sand names the diffusion parameter K and not D, as is common today).
5. ↑ J. Wang: Analytical Electrochemistry , 3rd Ed. Wiley-VCH, New Jersey, 2006, p. 149.
6. ^ Robert S. Rodgers, Louis Meites: Corrections for double-layer charging in chronopotentiometry . In: Journal of Electroanalytical Chemistry and Interfacial Electrochemistry . tape 16 , no. 1 , January 1968, p. 1-11 , doi : 10.1016 / S0022-0728 (68) 80271-2 .
7. ^ Philosophical Magazine 100 Most Cited Articles. Retrieved February 15, 2017 .