To derive the Combined Gas Law, do the following:
Step 1: Write Boyle's Law:
Step 2: Multiply by Charles Law:
Step 3: Multiply by Gay-Lussac's Law:
Step 4: Take the square root to get the combined gas law:
If all six gas laws are included (the three above as well as Avogadro, Diver, and "no-name"), we would get the following:
However, this more complete combined gas law is rarely, if ever, discussed. Consequently, we will ignore it in future discussions and use only the law given in step 4 above.
This type of combined gas law problem (where everything goes to STP) is VERY common:
2.00 L of a gas is collected at 25.0°C and 745.0 mmHg. What is the volume at STP?
STP is a common abbreviation for "standard temperature and pressure."
You have to recognize that five values are given in the problem and the sixth is an x. Also, remember to change the Celsius temperatures to Kelvin.
When problems like this are solved in the ChemTeam classroom, I write a solution matrix, like this:
and fill it in with data from the problem.
Here is the right-hand side filled in with the STP values:
You can be pretty sure that the term "STP" will appear in the wording of at least one test question in your classroom. The ChemTeam recommends you memorize the various standards conditions. If your teacher allows a "cheat sheet" to be used on the test, MAKE CERTAIN those values are there. Pretty obvious, don't you think. Believe me, there are a lot of dunderheads out there! Don't be one
Here's the solution matrix completely filled in:
Insert the values in their proper places in the combined gas law equation:
and solve for x.
This next problem uses two gas laws in sequence. It involves using Dalton's Law of Partial Pressures first, then use of the Combined Gas Law. The explanation will assume you understand Dalton's Law. These two laws occuring together in a problem is VERY COMMON.
1.85 L of a gas is collected over water at 98.0 kPa and 22.0°C. What is the volume of the dry gas at STP?
The key phrase is "over water." Another phrase to look for is "wet gas." This means the gas was collected by bubbling it into an inverted bottle filled with water which is sitting in a water bath. The gas bubbles in and is trapped. It displaces the water which flows out into the water bath.
The problem is that the trapped gas now has water vapor mixed in with it. This is a consequence of the technique and cannot be avoided. However, there is a calculation technique (Dalton's Law) that allows use to remove the effect of the water vapor and treat the gas as "dry." For this example, we write Dalton's Law like this:
We need to know the vapor pressure of water at 22.0°C and to do this we must look it up in a reference source.
It is important to recognize the Ptot is the 98.0 value. Ptot is the combined pressure of the dry gas AND the water vapor. We want the water vapor's pressure OUT.
We solve the problem for Pgas and get 95.3553 kPa. Notice that it is not rounded off. The only rounding off done is at the FINAL answer, which this is not.
Placing all the values into the solution matrix yields this:
Solve for x in the usual manner of cross-multiplying and dividing.