# How does one prove Avogadro's law from Gay-Lussac's?

To understand Gay-Lussac's law, Amedeo Avogadro said

In equal volumes of air under constant temperature and pressure, there are equal number of atoms (or molecules).

Let's take a hydrogen and an oxygen molecule. As the latter is larger in size, when stored in two separate equally-volumed containers, doesn't oxygen increase the pressure inside that container, if the same number of molecules occupied each container? (Does this make Avogadro's statement of constant pressure go wrong?) Again,

$$2mvN\cos\theta = F,$$

where $$F$$ is the force acting on the walls of the container, $$N$$ is the number of collisions of the molecules with the walls, $$2mv$$ is the change of momentum of the molecules, and $$\theta$$ is the angle of collision of the molecules on the walls.

Therefore, if Avogadro was correct, could anyone give me any reason why $$N_\ce{H2} > N_\ce{O2}$$ or $$v_\ce{H2} > v_\ce{O2}?$$

If Avogadro was correct, the above conditions must hold, as $$m_\ce{H2} < m_\ce{O2},$$ right?

Could you please prove the above phenomena happen when Avogadro's statement above holds?

• The kinetic theory of the collisions has no connection with Avogadro's law. Jan 10, 2021 at 20:05
• @Maurice why do you say they have no connection? Jan 12, 2021 at 11:23
• Avogadro's law describes the state of a system, and not a change. Gay-Lussac's law describes what happens when a change happens, when the system temperature is modified Jan 12, 2021 at 12:01
• @Maurice yes but, 2mvNcosθ=F is not Gay-Lussac's theory, right? and what I'm asking is how to describe Avogadro's law when above mentioned conditions (in the question) satisfy? Jan 12, 2021 at 17:02

equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.

More exactly, it also includes single atoms, e.g. for a case of noble gases, due the way how IUPAC defines a molecule.

But remember the Avogadro law implies ideal gases, similarly as the Gay-Lussac's law,Boyle's law and Charles' law. For ideal gases, sizes of molecules play no role. Both the size and volume of ideal gas molecules are considered negligible.

Real gases have deviations from this ideal behavior. See van der Waals equation as the most used real gas state equation.

$$\left(p+a \frac{n^2}{V^2}\right)(V-n b)=nRT$$ $$\left(p+ \frac{a}{V_\mathrm{m}^2}\right)(V_\mathrm{m} - b)=RT$$

where parameters $$a$$ (addressing cohesive forces ) and $$b$$ (addresing non zero molecular volume) are specific for the particular gas.

Then the molar amount of a real gas for given $$V$$, $$p$$ and $$T$$ is the solution of the cubic equation for variable $$n$$ :

$$pV - n ( pb + RT ) + a \frac{n^2}{V} - ab \frac{n^3}{V^2} = 0$$

It clearly shows that the same volumes of different real gases, at the same temperature and pressure, contain ( usually slightly ) different counts of molecules/atoms.

The question lies at the heart of Avogadro's law and Gay-Lussac's.

The question inquires about the role of "mass" which as a physical unit or parameter does not show up neither in Avogadro's law nor in Guy-Lussac's.

In fact, the principle of Avogadro's lies in mass of the specific gaseous element (atomic mass) or molecule being of no importance, making no difference (it's only coherent not to mention mass in the formula).

How could it be if mass (cp. explanatory text of the question above) determines impulse, and impulse determines pressure?

The more mass a gaseous particle (specific element, molecule) has the slower it is, the lesser its mean velocity as it hits the wall.

Then,

"why NH2>NO2 or vH2>vO"

must, in my opinion, be anwered as follows:

Lesser velocity means lesser number of hits (N for H2 is larger than N for O2 as H2 is lighter, thust faster, hence hits more often).

Again, refering to the example: O2 is the gas that has more mass, as previously said, its velocity is lower which means that the N number of hits is lower - which places O2 at equal footing with H2 the mass of which is less, the velocity and number of hits of which is greater.

This results in the number of hits canceling out the amount of momentums determined by different masses.

At a given temperature, the average kinetic energy of gas particles is the same, no matter what particles make up the gas. That means that lighter particles have a higher average speed than heavier particles.

Average (translational) kinetic energy determines both temperature and pressure of ideal gases, which is why they are proportional for systems with equal volume and amount of substance.