Desperate Deriving Ideal Gas Formula

[ProPM]

I know how temperature, volume and pressure are related and I think (think because I haven't tried any exercise alone yet) I can apply PV = nRT with no problems. But usually I am not able to apply a formula until I understand how I got to it, what it means.

I'd like to start from the beginning: These are my thoughts:

Boyle's Law: P₁V₁ = k, since k = PV, we can replace k by PV:

P₁V₁ = P₂V₂​

Applying the same logic, we get the two other rules in the same format:

V₁/ T₁=V₂/ T₂​

and:

P₁/ T₁=P₂/ T₂​

If we multiply all of them together: (PV / 1) x (V / T) x (P / T), we get:

P²₁V²₁/T²₁= P²₂V²₂/T²₂​

And taking the square root of both sides, we get the Universal Gas Law

Now, (provided the above is correct ) Is where I get confused.

I have to go from there to the famous: PV = nRT

From what I remember my teacher saying,

P₁V₁/ T₁ is equal to a constant, as well as equal to P₂V₂/ T₂​

We could then replace constant with the letter R and to get the n apply Avogadro's Law I guess?

Well, in summary I think. My doubt is how, or to what, P₁V₁/ T₁ is proportional to - where is the constant? Did it get "carried over" from the initial equations that have constants?" And then, where the n appears.

I hope I didn't make things VERY confusing...

Thanks a lot,
ProPM

 

[ehild]

PV must depend on the amount of gas, too. Imagine that you have two identical containers filled with the same amount of gas at the same pressure, same temperature, and the two containers are brought together, and the wall between them removed: You get a a single container with the same pressure and temperature as before, twice the original volume and twice the amount of gas. PV'/T = 2 PV/T. If you measure the amount of gas in moles (n) pV/T should be proportional to n, so PV/T=nR.

There is a theoretical derivation of the ideal gas low assuming elastic collision of the molecules with the walls of the container. http://en.wikipedia.org/wiki/Kinetic_theory.

ehild

 

[ProPM]

Cool, I understand the relation with the amount of gas. I feel comfortable adding a (n) to the equation. But what is the R?

Check if this makes sense:

PV = k₁
V / T = k₂
P / T = k₃

If we multiply all together:

P₁V₁/ T = n(k₁k₂k₃)​

Then we can replace the k's for R.

Now, IF that's right, I understand where the constant comes from, but, looking at PV = nRT I can't picture what I do when I look at V / T = k, for example: If V increases, for K to remain the same, T must also increase.

 

[Ygggdrasil]

The reasoning for including n in the ideal gas law comes from another empirically derived gas law called Avogadro's law, which states that the volume of a gas and the number of moles of gas particles are directly proportional if pressure and temperature are kept constant.

For more on the informal derivation of the ideal gas law, see my post at https://www.physicsforums.com/showthread.php?p=3007690#post3007690

As ehild mentioned, the most rigorous derivations comes from applying assumptions of the kinetic theory of gases and you can find the derivation in most statistical mechanics textbooks.

 

[ehild]

Cool, I understand the relation with the amount of gas. I feel comfortable adding a (n) to the equation. But what is the R?

R is an empirical constant. You can get R, by measuring the volume of 1 mol gas at standard condition (T=273 K and P=10⁵Pa, for example. You would get V=0.0227 m³. What is the value of R then?

ehild

 

[ProPM]

8.31! So, for any 1 mole of gas, the product of the pressure and volume divided by the temperature will always equal 8.31.

The last thing (I hope/promise)

Is it correct to say:
That the Combined Gas Law is achieved by doing the multiplication I did in my first post of the three laws. (I ask so because many places say it is a result of combining two gas laws while my book says three)

Or that: If P is inversely proportional to V and directly proportional to T, then: PV / T

Or do both work?

And finally, if it is said: a / b = a /b it is the same thing as saying: a / b is constant?

I think that's all!

 

[ehild]

Is it correct to say:
That the Combined Gas Law is achieved by doing the multiplication I did in my first post of the three laws.

Or that: If P is inversely proportional to V and directly proportional to T, then: PV / T

Or do both work?

Both work. If P is proportional to T and inversely proportional to V then it can be written as P=c*T/V (c is a constant).

And finally, if it is said: a / b = a /b it is the same thing as saying: a / b is constant?

a/b =a/b is an identity and it is always true. But in case a and b are functions of some variable(s) and a/b is the same for any value of the variable(s) than you can write that a/b =constant.

ehild

 

[ProPM]

Ok, so I guess we can say it is constant in this case.

And the last little thing, I promise

Do you get 8.31 for one mole of any gas only at the specific conditions (1 atm, 22.4 L, 273 K) or if, for example, you change one variable, allow another to change accordingly and keep one constant it will also work?

Thanks

 

[ehild]

Do you get 8.31 for one mole of any gas only at the specific conditions (1 atm, 22.4 L, 273 K) or if, for example, you change one variable, allow another to change accordingly and keep one constant it will also work?

R is an universal constant so it must be the same for any conditions for an ideal gas: R=8.314472 J/(molK). At very low temperatures, at very high pressures the gases can deviate from the ideal behaviour and in such cases PV/(nT) will differ from 8.314472.

ehild