What is the difference between the Law of definite proportions and the Law of multiple proportions?

Question:  What's the difference between the Law of definite proportions and the Law of multiple proportions?  They sound like they're the same thing.

The Law of definite proportions states that no matter how you make a chemical compound, it's got the same ratio of elements.  An example:  Whether you make water by combining hydrogen and oxygen or by decomposing hydrogen peroxide, the resulting water will still be 1 part by mass of hydrogen to eight parts by mass of oxygen.  This seems obvious to us, but was pretty revolutionary back in Dalton's day.  Back in the old days, chemical compounds frequently appeared to have different properties when made by different methods.  We now know that this isn't because the compounds actually have different properties, but because when you use different methods to make chemical compounds and have low tech equipment, the different methods used cause different contaminants to be in the compound formed, making it appear as if it's a different compound with related, but different properties.

The law of multiple proportions refers to different chemical compounds that can be formed when two elements react with each other.  For example, hydrogen and oxygen can react to form water (H2O) and hydrogen peroxide (H2O2).  In the first compound, one gram of hydrogen combines with 8 grams of oxygen.  In the second, one gram of hydrogen reacts with 16 grams of oxygen.  If we look at the ratio of these to each other, 8/16 =2, which is a whole number.  The law of multiple proportions says that whenever oxygen and hydrogen (or any two elements react to form more than one compound), the ratios of the elements to each other will be some whole number multiple of the ratios of the elements in the other compounds.

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