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I gather the reaction produces Sodium Oxide and Potassium Oxide as well as Hydrogen. Would there be traces of Sodium Oxide and Potassium Oxide left of a cloth that was positioned near the reaction? Thank a ton.

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According to this site:-

NaK reacts explosively with water to form hydrogen, potassium hydroxide and sodium hydroxide.

This site says:-

NaK is a strong reducing agent. It is corrosive and will react violently with water. The reaction with water will liberate extremely flammable gases potentially causing a fire or explosion.

This youtube video confirms the above fact.

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The answer to the question is yes, there will be residue. Let's look at the chemical reaction/equation: $$\ce{2NaK + 2H2O -> Na2O + K2O + 2H2}$$

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