[madisonwi]

This is a question that has perplexed me for some time.  Allow me to outline.

The three basic Gas Laws appear below:

Boyle's Law
P₁V₁ = P₂V₂


Charles' Law
V₁/T₁ = V₂/T₂


Gay-Lussac's Law
P₁/T₁ = P₂/T₂

All of these laws were independently discovered and verified.  Subsequently, there was the notion to combine the law together - hence the "Combined Gas Law".


Combined Gas Law
P₁V₁/T₁ = P₂V₂/T₂


However, when going from the individual laws to the combined laws is where my problem lies.  I have attempted to combine the individual laws algebraically, and have been unable to do so.  Every time I end up with T₁ and T₂ inverted in the final equation, suggesting to me that it is algebraically impossible to do so.  

So my question is, am I simply needing a complete redo of 9^th grade algebra, or do the individual laws truly not mathematically compile into the combined gas law?  If the latter is true then, how is it that the individual laws coalesce together and still make mathematicians happy?

Thank you for your assistance.
Douglas Weittenhiller

[Tiger]

I forgot all of these laws. :)Thanks for reminding me.

If
P1 = 5, V1 = 8, T1 = 13
P2 = 10, V2 = 4, T2 = 13

Plug these numbers in, it works for Boyle's Law, Charles' Law and Combined Gas Law. But it doesn't work for Gay-Lussac's Law.

[Tiger]

Here is how you might write the laws out mathematically correctly:

Law 1: p1v1 = p2v2 only when t1 = t2
Law 2: v1/t1 = v2/t2 only when p1 = p2
Law 3: p1/t1 = p2/t2 only when v1 = v2

Now when you try to combine say Law 1 and Law 2, you get:

(from Law 1) v1 = p2v2/p1 only when t1 = t2
(substituting into Law 2) p2v2/p1t1 = v2/t2 only when t1 = t2 and p1 =
p2

Continue this as you probably did and get a formula with a t2 where you expected a t1.  But since t1 = t2 in the restrictions of the formula by then, this is true.

Bottom line is that you can not just algebraically combine these three laws given that they each have different restrictions.  If you do you will just get a combined formula where the restrictions that go with it make it meaningless.

If you want to get to the combined gas law I suggest looking at it in a more simple manner.

Note that Law 1 can be written as p1v1/t1 = p2v2/t2 when t1 = t2
Note that Law 2 can be written as p1v1/t1 = p2v2/t2 when p1 = p2
Note that Law 3 can be written as p1v1/t1 = p2v2/t2 when v1 = v2

So all three laws can be written as the combined gas law using their restrictions.  But if Law 2 shows that the temperature does not need to stay constant for the formula to be correct then we can take that restriction out.  You can similarly take out the other restrictions and get the combined gas law with no restrictions.

If you want to see an algebraic way to derive it go to
http://dbhs.wvusd.k12.ca.us/webdocs/GasLaw/Gas-Combined.html
but note that this is a gimmick method.  They do not take into account the restrictions, opting to just use methods that get the desired result.

[AWK]

All of these laws were independently discovered and verified.  Subsequently, there was the notion to combine the law together - hence the "Combined Gas Law".

All this laws are special cases of more general Ideal Gas Law. and can be derived from it

[Demotivator]

Here is how you might write the laws out mathematically correctly:

Law 1: p1v1 = p2v2 only when t1 = t2
Law 2: v1/t1 = v2/t2 only when p1 = p2
Law 3: p1/t1 = p2/t2 only when v1 = v2

Now when you try to combine say Law 1 and Law 2, you get:

(from Law 1) v1 = p2v2/p1 only when t1 = t2
(substituting into Law 2) p2v2/p1t1 = v2/t2 only when t1 = t2 and p1 =
p2

Continue this as you probably did and get a formula with a t2 where you expected a t1.  But since t1 = t2 in the restrictions of the formula by then, this is true.

Bottom line is that you can not just algebraically combine these three laws given that they each have different restrictions.  If you do you will just get a combined formula where the restrictions that go with it make it meaningless.

If you want to get to the combined gas law I suggest looking at it in a more simple manner.

Note that Law 1 can be written as p1v1/t1 = p2v2/t2 when t1 = t2
Note that Law 2 can be written as p1v1/t1 = p2v2/t2 when p1 = p2
Note that Law 3 can be written as p1v1/t1 = p2v2/t2 when v1 = v2

So all three laws can be written as the combined gas law using their restrictions.  But if Law 2 shows that the temperature does not need to stay constant for the formula to be correct then we can take that restriction out.  You can similarly take out the other restrictions and get the combined gas law with no restrictions.

You're on the right track but not quite there. They can be combined.
This restriction stuff is too confusing and probably does not lead to the general law.
This is a better approach:
 

For system A: P1V1/T1 = P2V1/T2  
(V is a const as V1 and I deliberately leave it in)
For System B: P3V3/T3 = P3V4/T4  
For System c: P5V5/T5 = P6V6/T5

It is algebraicly correct to combine to the following for 3 independent systems:
P1P3P5V1V3V5/T1T3T5 = P2P3P6V1V4V6/T2T4T5

Now, here is the trick. If the three systems are identical to each other (which they in fact are), then it can be taken as one system. To transform the three systems into one system, one need only state that the variables are equal, ie: P1=P3=P5, V1=V3=V5, etc.

Therefore, it reduces to
(PV/T)^3 init = (PV/T)^3 final
(PV/T)init = (PV/T)final

[madisonwi]

Thanks for the replies.  I've been concentrating on the isomers of late, but will digest this and reply with a confirmational email.

Thanks,
Douglas Weittenhiller