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If the volumes are in a 2:1 ratio and the particles are in a 2:1 ratio, then a powerful conclusion emerges: equal volumes of the two gases must contain equal numbers of particles, regardless of whether they are hydrogen or oxygen. This seems like quite a leap to make, since we are concluding something about the numbers of particles without ever having counted them. What is the basis for this leap of logic? The most important part of the reasoning is the uniqueness of the integers. It is hard to come up with a simple explanation for why gas volumes should only react in integer ratios. The only simple conclusion is the one we have come to and that was first stated by Avogadro:
Avogadro’s Law: Equal volumes of gas contain equal numbers of particles, if the volumes are measured at the same temperature and pressure.
One way to come to this conclusion is to imagine that it is not true. Assume that equal volumes of gases do not contain equal numbers of particles and instead contain unrelated numbers of particles. Let’s assume for example that 1 L of gas A contains 3.14 times as many particles as 1 L of gas B. Then to take equal numbers of A and B particles, we would need 3.14 L of gas B for every 1 L of gas A. For the gas particles of A and B to react in a simple integer ratio of particles, we would then need a non-integer ratio of volumes. But this is not what is observed in the Law of Combining Volumes: the volume of A and B that react are always observed to be a simple integer ratio. Our assumption that equal volumes contain unrelated numbers of particles leads us to a conclusion that is contradicted by experiments, so our assumption must be wrong. Therefore, equal volumes of gases contain equal numbers of particles. We can conclude that Avogadro’s Law follows logically from the Law of Combining Volumes.
There is a problem that we have to work out. Looking back at one piece of evidence that led us to the Law of Combining Volumes, we found that 1 L of hydrogen plus 1 L of chlorine yields 2 L of hydrogen chloride. Using the conclusion of Avogadro’s Law, the volume ratio and the particle ratio must be the same. This seems to say that 1 hydrogen atom plus 1 chlorine atom makes 2 hydrogen chloride molecules. But this can’t be! How could we make 2 identical molecules of hydrogen chloride from a single chlorine atom and a single hydrogen atom? This would require us to divide each hydrogen and chlorine atom, violating the postulates of the Atomic Molecular Theory.
There is one solution to this problem, as was recognized by Avogadro. We have to be able to divide a hydrogen gas particle into two identical pieces. This means that a hydrogen gas particle must contain an even number of hydrogen atoms, most simply, two. This says that hydrogen gas exists as hydrogen molecules, and each hydrogen molecule contains two hydrogen atoms. The same conclusions apply to chlorine: a chlorine gas molecule must contain two chlorine atoms. If these conclusions are correct, then one hydrogen molecule, H _{2} , can react with one chlorine molecule, Cl _{2} , to form two hydrogen chloride molecules, HCl. The ratio of the reactant particles and the product particles is then the same as the ratio of the reactant gas volumes and the product gas volumes.
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