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If I place distilled water in a glass beaker, how long is it expected to stay distilled? What are the causes of it's impurity through air or other sources?

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Gases like oxygen and nitrogen from air are absorbed quite easily. As you pour the water to the beaker it is most likely already saturated with those gases. Other air components and contaminants (carbon dioxide, sulfur dioxide, nitrogen oxides, ammonia) will also get quickly absorbed. How fast they will contaminate your sample depends on their concentration in the air, temperature of the water, size of the container and so on. Other possible contaminants are dust particles, impurities present inside the beaker, and even glass itself (depending of its type) is slightly soluble in water. The concentration of solid impurities will usually not change very quickly. Of course, as the water evaporates, solid impurities will concentrate.

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Air contains anorganic gasses (HCl, HNO3, HNO2, SO2) and particles (aerosols) that usually consists of the following anions: CL-, NO3-, SO42-, NH4+, K+, Ca2+, Mg2 and organic matter (bacteria etc...). The latter is not taken into consideration now but can influence the long term quality of water due to growth.

On the water surface absorption of these substances will take place. Through diffusion these substances will be transported further into the water.

Absorption speed and concentration of gasses and aerosols depend on the local temperature/humidity and air quality.

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