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The term metamer is another word for isomer. Two chemical compounds are isomers of each other if they have the same formula but different structures. In other words, the compounds have the same types of elements present in the same ratios but the elements are connected together via their spatial arrangement differently.
We are looking at the different number of isomers available for EtC3H8O. Now, if we just had the formula C3H8O, this would imply that all of the carbon atoms are fully saturated (have the maximum number of hydrogens allowed) to make the structure fit the parameters. The oxygen atom would simply be located on different carbons as an alcohol to make isomers. With the addition of the Et (ethyl) group, this just means that the Et group must be located between carbon atoms or between a carbon and the oxygen atom. If the Et group is at the end of the carbon chain, we will only be able to get 7 hydrogens in the structure. This ultimately means that there are 5 different isomers possible for EtC3H8O. They are shown below:
Any other structure would simply be able to be converted to one of the ones above by flipping the structure around or otherwise manipulating it without having to break any chemical bonds. So the answer is 5.
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