Derive the ideal gas law from Charles', Boyle's, and Avogadro's, how?

[Try Hard0]

What is the trick to derive the ideal gas law and gas constant from the "combination" of Charles', Boyle's, and Avogado's laws? The general chemistry books I have seen tell me this is how they derived the ideal gas law and constant but they do not show how that is achieved. I tried to go about the process and failed. Let me show you what I did and maybe you can help me out.

The book says:
Boyle's Law => V = k/P (the constant is k, and the units seem to be L*atm)
Charle's Law => V = bT (b is the constant, and the units seem to be L/Kelvin)
Avogadro's Law => V = an (a is the constant, and the units seem to be L/mol)

But if you try to combine these three laws in a straightforward fashion I think you will fail as I did. What I tried was to multiply Boyle's and Charle's equations and then divide that result by Avogadro's equation to get V = (k/P)(bT)/(an) . . . but that is not right! (Note: n needs to be in the numerator on the right side of the equation). (Additionally, the units won't work-out for the ideal gas constant R, where kb/a = R has units of (L*atm*mol)/Kelvin which is wrong, i.e. should be (L*atm)/(mol*Kelvin).)

So, what is the trick I am missing here? Is there a simple and straightforward way to derive the ideal gas law and gas constant from the "combination" of Boyle's, Charles', and Avogadro's laws? Or is the process actually quite involved, and for that reason most general chemistry texts (i.e. all that I have seen, including physical chemistry texts) avoid providing the "derivation"? Alternatively, does someone know of a book or a webpage or someplace I can look to find this derivation of the ideal gas law and constant from Charles', Boyle's, and Avogadro's laws?

Thanks!

 

[Borek]

Why do you think "an" needs to be in numerator?

 

[Ygggdrasil]

It's important to remember that the Boyle's law "constant" k depends on temperature and the number of moles, the Charles' law "constant" depends on pressure and the number of moles, and the Avogadro's law "constant" depends on temperature and pressure. You may also need the Gay-Lussac law which states that, for constant volume and number of moles, temperature is directly proportional to pressure.

Perhaps it might be useful exercise to first derive these laws from the ideal gas law.

Finally, I will note that while the ideal gas law was originally discovered by combining these laws, which were derived from experiments, modern chemists can now derive the ideal gas law from first principles (i.e. using only the theoretical assumptions of the kinetic theory of gases). The methods used for this theoretical derivation are somewhat complex and would be taught in advanced university chemistry courses (specifically a course called statistical mechanics).

 

[Try Hard0]

Borek and Ygggdrasil;
What you two have inspired me to do was go to Wikipedia and look up the Gay-Lussac law, which did not provide any kind of satisfaction. However, I ran into something called the "combined gas law" which does seem to lead me in the right direction. Check this out:

The combined gas law says: PV/T = k (a constant)

They did give the derivation (I think) and it looks like they got this "combined gas law" from the Boyle's, Charles', and Gay-Lussac laws, but not from Avogadro's law.

Anyway, my thinking is that I can take the combined gas law as derived and substitute R*n for k.

Look like I'm on the right track?

I assume the units of the constant k are atm*L/Kelvin, and just multiplying that by 1/mol would give the correct units of R and everything unit-wise cancels out nicely between the factors P, V, T, and n with the units of R.

But the n just came out of nowhere. But maybe that is okay here? I mean, PV/T are dependent upon the number of moles (n) too, right? I'm a little unsure of myself here though. Any words of wisdom for me?

 

[Borek]

You are on the right track. If k=n*R that's the place to apply Avogadro's.

Look at it from this perspective: long ago there were several laws, and a good physicist should be able to see they can be all parts of something more universal. It is possible to combine them and experimentally check if that combined equation doesn't hold. And lo, it does! That's not the only moment in the history of science when partial results were combined into general theory.

 

[Ygggdrasil]

Here's my attempt at a somewhat more clear explanation of the derivation. Let's start with Charles's Law. This law states that at constant pressure (P) and number of moles (n), volume (V) and temperature (T) are directly proportional. Put mathematically, this means V = cT, where c is a constant. Now, experiments show that this relationship breaks down if you add or remove gas from the system or change the pressure, indicating that c is a function of n and P. Therefore, we can write:

V = c(n,P)*T

How does the function c depend on n and P? For this, we can turn to Boyle's law and Avogadro's law. For Boyle's law, V = b/P when n and T are constant. If we look at our previous expression from Charles's law:

V = c(n,P)*T

We can see that this will fit Boyle's law if the function c is of the form c(n,P) = a(n)/P, where a is some function dependent on n. This gives us the expression:

V = a(n)*T/P

If we treat T and n as constants, we can recover Boyle's law where b = a(n)*T (since n and T are constants, so is b).

Rearranging this expression, we can get the combined gas law:

PV/T = a(n)

That is, at a constant number of moles of gas, PV/T equals a constant. Similar reasoning using Avogadro's law can show how the function a(n) depends on n. Experiments are required to determine the constant of proportionality that pops our (R, the ideal gas constant).

 

[edward levay]

Gentlemen:

Is there any record of Avogadro's laboratory notes, that he may at that 'time' of this steps- procedure in developing his law and the R factor?

 

[Ygggdrasil]

The R factor (Avogadro's constant) was not actually measured by Avogadro, but named in his honor. For a history of Avogadro's constant, see the wikipedia page:

https://en.wikipedia.org/wiki/Avogadro_constant#History