First, let's look at the two concepts individually.
Avogadro's hypothesis (also referred to as Avogadro's Law), is a gas law, experimentally derived by Amedeo Avogadro. It says that the volume of a gas is proportional to the number of moles of the gas (at constant pressure and temperature). It tells us that increasing the number of moles at a given temperature and pressure would result to an increase in volume, and a decrase in number of moles, a decrease in volume. Stated in a different way, it can be interpreted as two gases occupying the same amount of space (volume) have the same number of moles. We will use this latter interpretation to relate it to Gay-Lussac's Law.
Gay-Lussac's Law is another law the observes propety of gases (by Joseph Louis Gay-Lussac) -- it is sometimes referred to as the Law of Combining Volumes (of Gases). It tells us that the ratio between the volumes of gases of the reactant and the products can be expressed in small whole numbers. For instance, in the following reaction:
`C + O_2 rarr CO2`
a liter of carbon combined with a liter of oxygen would produce a liter of carbon dioxide (ratio is 1 L : 1 L).
At this point, we can easily see how Avogadro's Law explains the observation by Gay-Lussac.
Let's look at the reaction of hydrogen and oxygen to form water.
`2H_2 + O_2 rarr 2H_2O`
This chemical reaction tells us the following:
- 2 moles of hydrogen combines with a mole of oxygen to form 2 moles of water [interpreting it the usual way we interpret chemical equations using stoichiometry]
- 2 liters of hydrogen gas reacts with a liter of oxygen to form 2 liters of water (steam or gaseous). [interpreting it as Gay-Lussac did in his experiments]
This means that for reaction of gases, interpreting the chemical equation in terms of moles of gases (and consequently, in terms of molecules -- since a mole is ALWAYS equal to `N_A` number of species, where `N_A = 6.022 \times 10^(23)` , Avogadro's number) or interpreting them in terms of the volume of gases we have at a given temperature and pressure would mean the same thing.``
--That is, stoichiometrically, if we want to create 2 moles of water, we must have 2 moles of hydrogen and a mole of oxygen --- OR if we want to create 2 liters of oxygen, we must have 2 liters of hydrogen and a liter of oxygen.
In the end, we see here a relationship between the number of moles (and species, consequently) and the volume of a gas. And this relationship is ultimately explained by Avogadro's Law. Hence, Gay-Lussac's observation that a chemical equation can be interpreted in terms of volumes is (or can be seen as) an extension of Avogadro's observation that moles is directly related to volume at a given temperature and pressure.