How can rain deplete soil from nutrients?

If you mean "how can rain deplete nutrients from soil" then there are a number of processes by which this can happen. As well as physically eroding soil, rainwater can "leach" nutrients from soil and move then downward in the soils profile or laterally in groundwater. The composition of the rainwater can affect this process.E.g.

When acid rain falls on soils, cation exchange can occur and minerals such as calcium, magnesium and potassium bonded to soil particles are released by the acid and "washed" out of the soil into deeper strata where they are no use to plants. Under natural circumstances a balance may be achieved and as plants decay they replace nutrients that are used by plants and leached. If an area of vegetation is suddenly cleared however, thus exposing it to increased rainfall and lack of organic matter input in the system, the nutrient levels may decrease over time as they are leached but not replaced.

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