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Say you spilled elemental mercury from a thermometer and used household bleach on it, what would be the reaction if any?

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    $\begingroup$ More importantly, this is a very poor way to decontaminate anything where mercury has spilled. $\endgroup$ – matt_black Apr 14 at 23:42
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There are several redox reactions can be taken place when you add household beach on elemental mercury metal. However, this question is more like a homework question. Thus, I give you insight to the answer and you may read a bit and find out what's happening. This reaction is studies and results have been published (Reference 1), the abstract of which states that:

The absorption of elemental $\ce{Hg}$ vapor into aqueous hypochlorite was measured in a stirred tank reactor at $25$ and $\pu{55 ^{\circ}C}$. $\ce{NaOCl}$ strongly absorbs $\ce{Hg}$ even at high pH. Low $\mathrm{pH}$, high $\ce{Cl-}$ and high-temperature favor mercury absorption. Aqueous free $\ce{Cl2}$ was the active species that reacted with mercury. However, chlorine desorption was evident at high $\ce{Cl-}$ and $\mathrm{pH}<9$. Overall second-order reaction was observed between mercury and chlorine with an apparent second-order rate constant of $\pu{1.7\times 10^{15} M^{-1} s^{-1}}$ at $\pu{25 ^{\circ}C}$ and $\pu{1.4\times 10^{17} M^{-1} s^{-1}}$ at $\pu{55 ^{\circ}C}$. Gas-phase reaction was observed between $\ce{Hg}$ and $\ce{Cl2}$ on apparatus surfaces. Strong mercury absorption in water was also detected with $\ce{Cl2}$ present. Results indicate that the chlorine concentration, moisture, and surface area contribute positively to mercury removal.

I hope you will figured it out when you consider the standard reduction potentials of $\ce{Hg}$ and $\ce{OCl-}$ in any electrochemical series. For example, see Reference 2.

References:

  1. L. L. Zhao, G. T. Rochelle, “Mercury absorption in aqueous hypochlorite,” Chemical Engineering Science 1999, 54(5), 655-662 (https://doi.org/10.1016/S0009-2509(98)00263-2).
  2. S. G. Bratsch, “Standard Electrode Potentials and Temperature Coefficients in Water at $\pu{298.15 K}$,” Journal of Physical and Chemical Reference Data 1989, 18(1), 1-21 (https://doi.org/10.1063/1.555839).
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As noted by @Mathew, sodium hypochlorite can be used to remove elemental mercury. It is a redox reaction involving oxidation of mercury to $\ce{Hg^2+}$. There are various papers which discussed about the reaction of sodium hypochlorite and mercury and its role in removing of mercury. This paper1 discusses about the dissolution of mercury in aqueous solution of sodium hypochlorite at pH of 5.9–8.5 and the mechanism of oxidation of mercury to $\ce{Hg^2+}$. There is also another paper[2] which discuss about the role of sodium hypochlorite in the removal of mercury by wet-scrubbing method.

The important step for increasing gaseous elemental mercury removal in wet scrubber systems is altering the chemical form of the Hg to a water‐soluble oxidized species. This work focuses on the removal of elemental mercury from simulated flue gas by aqueous sodium chlorite in a bubble reactor. The effects of initial oxidizing solution concentration, reaction temperature, pH and mercury concentration in the inlet of the reactor on mercury oxidative absorption in sodium chlorite were investigated. The results indicate that higher concentrations of sodium chlorite favor Hg removal, with a greater efficiency observed in acidic than in alkaline solution. High temperature inhibits Hg absorption in aqueous sorbent when the reaction temperature is lower than ca. 40 °C, and the removal efficiency increases when the temperature is higher than that value. In conclusion, the major influencing factors on the levels of Hg removal are pH and chlorite concentration in solution.

References

  1. I. P. SizenevaV. A. Val’tsiferV. N. Strel’nikov "A Study of Mercury Dissolution in Aqueous Solutions of Sodium Hypochlorite" Russian Journal of Applied Chemistry April 2005, Volume 78, Issue 4, pp 546–548 (link)
  2. Y. Zhao S.T. Liu C.M. Chen X.Y. Ma "Removal of Elemental Mercury by Sodium Chlorite Solution" 28 February 2008 (https://doi.org/10.1002/ceat.200700381)
  3. https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/98JD02304
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