Apparently easy but super frustrating question about Boyle's Law

[darussiaman]

Homework Statement

Okay... everywhere i look, it keeps talking about Boyle's Law applied to THE SAME gas! It's so frustrating and retarded! I get it, p1V1 = p2V2. Yay... I didn't even need to look THAT up!

But my question is, how do you compare 2 different (ideal) gases?? It seems like such an obvious problem to address, but google searching just yields pages and pages of what I said above, rather than addressing this... Grr. Anyway, if I have 2 different containers of gasses, and they're in thermal contact, how do I work with Boyle's Law? (This is for my thermodynamics course.)

My book says that Boyle's Law together with Avagdro's Law implies that p/n is the same for all ideal gases (where n is particle density, N/V). But this is completely NOT obvious to me. How do I prove this? Is it supposed to be intuitive??

Specifically, my example problem asks this... you have nitrogen at 1.26 kg/m^3 at pressure 1 bar and helium at 0.36 kg/m^3 at pressure 2 bar. Molecular weight of nitrogen = 28 and of helium = 4. Are the two gases in thermal equilibrium?

Homework Equations

pV = k

The Attempt at a Solution

Well, the solution is already given in the book (since it's a sample problem), but it doesn't explain why p/n is the same for all gases at same temperature -- it just states it as if it's obvious or something -- and I don't see why that necessarily is obvious, so... how do I prove that it is true? And am I the only one to whom this is not intuitive??


Thanks a lot!

 

[Borek]

My book says that Boyle's Law together with Avagdro's Law implies that p/n is the same for all ideal gases (where n is particle density, N/V). But this is completely NOT obvious to me. How do I prove this? Is it supposed to be intuitive??

What is characteristic of ideal gas? Or what are assumptions of kinetic theory of gases?

 

[tomcue]

Dear frustrated student,

I can understand your frustration with trying to compare two different ideal gases using Boyle's Law. It is a common misconception that Boyle's Law only applies to the same gas, but in fact, it can be applied to any ideal gas as long as the temperature and number of moles remain constant.

To prove that p/n is the same for all ideal gases at the same temperature, we can use the Ideal Gas Law, which states that PV = nRT. From this equation, we can see that at constant temperature, p/n is directly proportional to V. This means that for any ideal gas, the ratio of p/n will be the same as the ratio of V.

In your example problem, we can see that the ratio of p/n for nitrogen is equal to 1 bar / 1.26 kg/m^3 = 0.79 bar/kg/m^3. Similarly, the ratio of p/n for helium is equal to 2 bar / 0.36 kg/m^3 = 5.56 bar/kg/m^3. These two ratios are not equal, which means that the two gases are not in thermal equilibrium.

I hope this helps to clarify why p/n is the same for all ideal gases at the same temperature. Keep up the good work in your thermodynamics course!