### Combined Gas LawDerived from Byle's Law and Charles' Law

Discussion: (a) Charles' Law is the change in volume with respect to temperature for constant pressure while (b) Boyle's law is the change in volume with respect to pressure for constant temperature. When you combine the two laws, you must account for these two points.

Solution:

1) Assume we have a gas sample present at the following values:

pressure = P1
volume = V1
temperature = T1

2) We now allow the gas to change, at constant temperature, to the following state:

pressure = P2
volume = V*
temperature = T1

3) By Boyle's Law, we know the following to then be true:

P1V1 = P2V* <--- no T value since it is constant

4) Keeping the pressure constant, we allow the gas to move to the following state:

pressure = P2
volume = V2
temperature = T2

5) By Charles' Law, we know the following to be true:

 V* V2 –––– = –––– <--- no P value since pressure is constant T1 T2

6) Isolate V* in each expression:

 P1V1 V*  = –––– P2

 T1V2 V*  = –––– T2

7) Since, V* = V*, we can write this:

 P1V1 T1V2 –––– = –––– P2 T2

8) Multiplying each side by the factor P2 / T1 yields the most-widely used form of the combined gas law:

 P1V1 P2V2 –––– = –––– T1 T2