The easiest, non-destructive way to test the purity of a silver coin would be to measure the density of the material. First, you find the mass of the silver coin. Then, you need to determine the volume of the coin, since density = mass / volume. To determine the volume of the coin, you could simply place the coin in water, and then measure how much water is displaced. For example, you could fill a graduated cylinder with water and measure it out very well. Then, put the coin into the water and determine the volume again. The difference in volume would then be the volume of the coin.
The density is not necessarily a unique number for each element. For example Cobalt and Nickel both around 8.9 g / cc. If you are trying to distinguish two different and known elements it's probably okay as long as the densities aren't too similar. However, if you were trying to identify something and only had its density, you might need more information. In testing the purity of a silver coin, you should measure a density of 10.5 g/ cc. If the coin has, for example, Nickel in it then it would probably have a lower density.
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