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To test for the presence of sulfate ions in the crude potassium alum, take a small amount of solid and put it in a glass tube. Add a few drops of water and flick the glass tube to dissolve. You do not need to measure exact quantities of alum and water as this test is qualitative, meaning it will only indicate whether the ion is present but not in what quantity.
Take the dropper bottle containing the barium chloride solution and add two drops into the glass tube. If the sulfate ions are present, you will see a white milky precipitate form, which is barium sulfate, a compound insoluble in water.
For comparison, let's see what a negative test looks like. In a second glass tube, add only deionized water. Then add two drops of barium chloride solution. No precipitate form meaning there is no sulfate in the sample.