# Is Avogadro's law applicable for atoms or just for molecules?

I notice that online definitions of this experimental law always say, molecules or atoms.

$${\frac {V_{1}}{n_{1}}}={\frac {V_{2}}{n_{2}}}$$ The equation shows that, as the number of moles of gas increases, the volume of the gas also increases in proportion. Similarly, if the number of moles of gas is decreased, then the volume also decreases. Thus, the number of molecules or atoms in a specific volume of ideal gas is independent of their size or the molar mass of the gas.

### Key Points

• The number of molecules or atoms in a specific volume of ideal gas is independent of size or the gas’ molar mass.

This made me wonder if $n$ in the $PV = nRT$ can also be the number of atoms in that volume of gas. Taking a practical example, what is the answer to the following question?

1. Statement (I):
Atoms can neither be created nor destroyed.

Statement (II):
Under similar conditions of temperature and pressure, equal volumes of gases do not contain an equal number of atoms.

My question is, if $P$, $V$ and $T$ are equal, can we say $n$ (number of atoms) are equal?

The answer given is that, no they need not be equal since only number of molecules will be equal. The gas can consist of a mixture of diatomic and triatomic molecules, we can have the same number of molecules but different number of atoms.

From what I read on Kinetic Molecular Theory, the volume occupied by the molecules of the gas is negligible compared to the volume of the gas itself. This is the central assumption. So I guess the law applied only to molecules and not atoms or the generic "particles" as how some sites define it.

• Not clear what you are asking, since you have answered your own question in the paragraph "The answer given is that..." – sammy gerbil May 21 '18 at 16:50
• I was wondering if the given answer was wrong. Since when they derive this law they assume each gas "molecule" is point like, so a possibility I thought of was, if there is more than one atom, does this cause a problem since some molecules are now larger than others? Or since we are counting the pressure exerted by these molecules, does a molecule with more atoms exert more pressure. I wondered this calculations were done in Kinetic theory. Why they broke the gas down to the level of the molecule and made that assumption. – Aditya P May 21 '18 at 18:10
• Further, since this is a approximation, an experimental law and not a fundamental law, I wondered what the error rates would be like. Would the number of molecules be be same for both volumes to a similar degree that the atoms would be? – Aditya P May 21 '18 at 18:13
• "Thingies that bounce around" - atoms within a molecule, electrons within an atom, nucleons within a nucleus, ... do not bounce around by themselves. Instead, they participate in the bouncing jointly with their "master", and that is what counts for the statistics. For the detailed view, however, do note that they contribute to inner structure and degrees of freedom that can exchange energy in a bounce. Also, it is of course possible that molecules etc. fall apart or reassemble ... – Hagen von Eitzen May 21 '18 at 20:40

I notice that online definitions of this experimental law always say, molecules or atoms.

The problem with just calling them all "molecules" and being done with it is some are uncomfortable with using that term for unbound atoms. If you have a container of He, there are no "molecules" in it.

So when it says "molecules or atoms", it means "molecules or unbound atoms". It's not trying to say that the total number of atoms within the different molecular species matter.

• The expression we need here is, I think, "particles". – StephenG May 21 '18 at 16:52
• I would have gone with "stuff". – Anthony May 21 '18 at 19:53
• @StephenG it'd be good, but I remember that when I read similar definitions at school where "particles" term was used, I felt that my understanding of the definition was incomplete: electrons are also particles, so are neutrons etc., but for some (unstated) reason they aren't taken into account. – Ruslan May 21 '18 at 19:58
• The Wikipedia aptly says: "In the kinetic theory of gases, the term molecule is often used for any gaseous particle regardless of its composition. According to this definition, noble gas atoms are considered molecules as they are monatomic molecules" – leonbloy May 21 '18 at 20:49
• @StephenG "particle" has its own load of unrelated meanings, so it will be even more confusing. – ivan_pozdeev May 21 '18 at 21:13

The number n in the Boyle-Mariotte-Gay-Lussac gas law represents the number of moles of the considered gas. Mole is a measure of the number of distinct particles (molecules or atoms) of a substance. Avogadro's law states that the number of gas particles in a given volume of an ideal gas is the same for different ideal gases at the same pressure and temperature. It is related to the mean kinetic energy of distinct gas particles viewed as mass points. Thus Avogadro's law holds for gases consisting of molecules as well as of atoms. Examples for gases consisting of atoms are the noble gases, e.g., helium and argon.

• So if the gas is made of unbound atoms. That is if the molecule so to speak is a single atom, then you are saying it applies to atoms? – Aditya P May 21 '18 at 18:05
• @Aditya - You are right. Avogadro's law applies to any gas composed of separately moving particles, atoms or molecules. – freecharly May 21 '18 at 19:18

Your problem relates to the Ideal Gas law and Kinetic Theory rather than to Avogadro's Law alone, which is a deduction from the Ideal Gas Law.

In the Ideal Gas Law $pV=NkT$ the variable $N$ refers to the number of separate particles in the sample of gas. These particles can be individual atoms (eg atoms of the gas helium $He$) or diatomic molecules (eg molecular hydrogen $H_2$) or polyatomic molecules (eg ammonia $NH_3$) or even a mixture of different types of particles (eg air which is a mixture of $N_2, O_2, Ar, CO_2$ and smaller amounts of other gases).

If the ratio $pV/T$ is a constant for two samples of gas (which defines what it means to be an ideal gas) then it is the same constant, and the two samples contain the same number of particles regardless of their composition.

In Kinetic Theory the particles are assumed to be point masses or hard spheres. Their structure does not matter, neither does their mass, as far as this equation is concerned. The key assumption (which is justified by the accuracy with which the theory applies in experiments) is that the particles exchange energy with each other indirectly, via collisions with the walls of the container, and thereby reach an equilibrium in which each particle has the same average translational kinetic energy, regardless of its mass or its internal structure.

The structure of the particles and the composition of the gas mixture do matter when you are asking about heat capacity of gases, but the Ideal Gas equation tells you nothing about that. For that you need to know about other forms beside translational KE in which energy can be stored inside the particles of gas, such as rotational and vibrational energy.

You ask about departures (error rates) from Avogadro's Law. More generally, gases depart further from the Ideal Gas law as the size of the particles increase. The 2 major corrections to the Ideal Gas law relate to the amount of space occupied by the particles, and the forces of attraction between particles. These are expressed in the parameters $b$ and $a$ respectively in the Van der Waals equation of state for real gases
$$(p+\frac{a}{V_m^2})(V_m-b)=RT$$ where $V_m$ is the volume of one mole (Avogadro's Number) of gas particles. Both parameters $b, a$ increase as the size of the particles increase, and the bigger these parameters the greater the departure from the Ideal Gas law $pV_m=RT$ and consequently also from Avogadro's Law.

No, it is particles, i.e molecules, not atoms.

Imagine two identical containers (the same volume) with different gasses added to each and allowed to settle at the same surrounding temperature. Add each gas until the pressures are the same.

Suppose one has O2 and the other He in it.

Since P,V,and T are the same, and R is a constant, both containers have the same n - number of particles.

But the O2 container has twice as many atoms at the He container, since each molecule is two atoms.

It refers to molecules. If the molecules are monatomic (such as He) instead of containing multiple atoms (such as H2 or O2) it's the same thing. When there's more than one atom in a molecule, count molecules.