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2.11.01 : Please note - The volume V is inversely proportional to the density (ro). Therefore the expression of the pressure divided by the density which we used in class, can be replaced by an expression where the pressure is multiplied by the specific volume. This will be further discussed in class. Boyle's LawDiscovered by Robert Boyle in 1662. On the continent of Europe, this law is attributed to Edme Mariotte, therefore those counties tend to call this law by his name. Mariotte, however, did not publish his work until 1676. His law gives the relationship between pressure and volume if temperature and amount are held constant. If the volume of a container is increased, the pressure decreases. If the volume of a container is decreased, the pressure increases. Why? Suppose the volume is increased. This means gas molecules have farther to go and they will impact the container walls less often per unit time. This means the gas pressure will be less because there are less molecule impacts per unit time. If the volume is decreased, the gas molecules have a shorter distance to go, thus striking the walls more often per unit time. This results in pressure being increased because there are more molecule impacts per unit time. The mathematical form of Boyle's Law is: PV = k This means that the pressure-volume product will always be the same value if the temperature and amount remain constant. This relationship was what Boyle discovered. This is an inverse mathematical relationship. As one quantity goes up in the value, the other goes down. In the classroom, a student will occasionally ask "What is k?" Suppose P1 and V1 are a pressure-volume pair of data at the start of an experiment. In other words, some container of gas is created and the volume and pressure of that container is measured. Keep in mind that the amount of gas and the temperature DOES NOT CHANGE. We do not care what the exact numbers are, just that there are two numbers. When you multiply P and V together, you get a number that is called k. We don't care what the exact value is. Now, if the volume is changed to a new value called V2, then the pressure will spontaneously change to P2. It will do so because the PV product must always equal k. The PV product CANNOT just change to any old value, it MUST go to k. (If the temperature and amount remain the same.) Of course, you now want to ask "Why does it have to stay at k?" We believe it is best right now to ignore that question even though it is a perfectly valid one. So we know this: P1V1 = k And we know that the second data pair equals the same constant: P2V2 = k Since k = k, we can conclude that P1V1 = P2V2. This equation of P1V1 = P2V2 will be very helpful in solving Boyle's Law problems.
Example #1: 2.00 L of a gas is at 740.0 mm Hg pressure. What is its volume at standard pressure? Answer: this problem is solved by inserting values into P1V1 = P2V2. (2.00 L) (740.0 mm Hg) = (x) (760.0 mm Hg) Cross-multiply and divide to solve for x. Note that the units of mm Hg will cancel. x is a symbol for an unknown and, technically, does not carry units. So do not write x L for x liters. Just keep checking to see you are using the proper equation and you have all the right values and units. Don't put a unit on the unknown. Example #2: 5.00 L of a gas is at 1.08 atm. What pressure is obtained when the volume is 10.0 L? Answer: use the same technique. (5.00 L) (1.08 atm) = (10.0 L) (x) Example #3: 2.50 L of a gas was at an unknown pressure. However, at standard pressure, its volume was measured to be 8.00 L. What was the unknown pressure? Answer: notice the units of the pressure were not specified, so any can be used. If this were a test question, you might want to inquire of the teacher as to a possible omission of desired units. Let's use kPa since the other two units were used above. Once again, insert into P1V1 = P2V2 for the solution. (2.50 L) (x) = (8.00 L) (101.325 kPa)
Hopefully you can see that Boyle's Law problems all use the same solution technique. It's just a question of where the x is located. Two problems will arise during the gas laws unit in your classroom:
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