Well there is not because the amount of chloride ions present for combining with the gold is not constant, but ever decreasing down to zero.
For there to be a timing constant, the mix would have to obey Faraday's law but it does not.
Any electron flowing through the circuit might create a gold chloride, but they might just produce oxygen and hydrogen from the electrolysis of water.
Even a chloride ion reacting with a gold ion on the surface of the anode might or might not produce a gold chloride simply because a gold ion requires 3 chlorides to be a soluble gold compound. Gold ions are not released from the anode unless they are gold tri-chloride.
There is not even a solid relationship between the amount of chloride present and the amount of gold released for the reason that some of the chloride is simply released as chlorine gas without generating a gold chloride molecule.