### The Named Gas Laws"derived"from the Ideal Gas Law

I propose to "derive" each of the named gas laws from the starting point of the Ideal Gas Law. I will also discuss the Combined Gas Law, starting at Example #5.

The named gas laws are these:

Boyle
Charles
Gay-Lussac

These laws were discovered before the ideal gas law (disc. 1834) and named for their discoverers (more or less). All four of these laws can be "derived" from the Ideal Gas Law when two of the four gas variables are held constant.

The other "two-constant" gas laws (I call them Diver's Law and The Gas Law with No Name) will not be discussed. Although, nothing is to stop you from "deriving" them following your examination of the four examples I give below.

As a reminder, these are the four gas variables:

pressure
volume
temperature
amount

And remember, the Ideal Gas Law (PV = nRT) incorporates all four variables into one equation.

Example #1 - Charles' Law: If the gas volume decreases, but its pressure doesn't change, what happens to the temperature?

Solution:

Note that the amount of gas is not mentioned. This happens often and what you should do is assume the amount of gas is constant.

1) The Ideal Gas Law is true for the beginning set of conditions for volume and temperature:

PV1 = nRT1

Note that P, n, and R are all constants.

2) We decrease the volume of the gas to some new value, keeping P, n, and R constant. The temperature will change to some new value in such a manner as to make the following true:

PV2 = nRT2

3) Divide one by the other:

 PV1 nRT1 –––– = ––––– PV2 nRT2

4) Cancel out all the constant values:

 V1 T1 ––– = ––– V2 T2

5) As the volume decreases, the temperature must also decrease. This relationship of V and T is known as Charles' Law.

Example #2 - Boyle's Law: If the gas volume decreases, but its temperature doesn't change, what happens to the pressure?

Solution:

Once again, no mention of the amount of gas changing, so we assume it to not change.

1) The Ideal Gas Law is true at the first set of conditions for volume and pressure:

P1V1 = nRT

2) We decrease the volume of the gas to some new value, keeping T, n, and R constant. The pressure will change to some new value in such a manner as to make the following true:

P2V2 = nRT

3) Examining both equations above, we take note of this:

nRT = nRT

4) This leads immediately to what we know as Boyle's Law:

P1V1 = P2V2

Example #3 - Avogadro's Law: This involves volume and amount.

Solution:

1) First set of conditions:

PV1 = n1RT

2) Second set of conditions:

PV2 = n2RT

3) Rearrange each:

V1 / n1 = RT / P

V2 / n2 = RT / P

4) Therefore:

V1 / n1 = V2 / n2

Example #4 - Gay-Lussac's Law: Pressure and temperature.

Solution:

P1V = nRT1

and

P2V = nRT2

P1 / T1 = nR / V = P2 / T2

and finally

P1 / T1 = P2 / T2 <--- Gay-Lussac's Law

Example #5 - Combined Gas Law - Three-Variable Examples:

Solution:

1) This is the usual form of the Combined Gas Law:

 P1V1 P2V2 ––––––– = ––––––– T1 T2

2) We use the Ideal Gas Law as in all the previous examples, but we allow P, V, and T to change, keeping only n (and R) constant:

P1V1 = nRT1

and

P2V2 = nRT2

3) It is seen that nR is constant. We move T1 and T2 to the other side. Since this is true:

nR = nR

by substitution, we recover the form of the Combined Gas Law given in step 1.

4) Three other forms of the Combined Gas Law are possible. Here is one:

 V1 V2 ––––––– = ––––––– n1T1 n2T2

5) The most common one taught, by far, is the one I gave first. This is because there are many problems that ask you to convert a volume to what it would be at the conditions of STP. I suppose there are problems out there that use other forms of the Combined Gas Law, but they are rare.

Example #6 - Combined Gas Law - Four-Variable Example:

Solution:

1) This is the form of the Combined Gas Law I'm talking about when I say four-variable:

 P1V1 P2V2 ––––––– = ––––––– n1T1 n2T2

2)We start with the Ideal Gas Law in the usual manner. It's just that all four variables are allowed to change, resulting in this:

 P1V1 ––––– = R n1T1

and

 P2V2 ––––– =  R n2T2

3) Since R = R, we substitute and get the four-variable Combined Gas Law at the top of the example.

4) By the way, you can recover the four forms of the three-variable Combined Gas Law by using the four-variable form. In successions, simply hold each of the four variables constant. The one being held constant would drop out of the equation, leaving you with the four different three-variable forms.