Part I: Introduction to Variation
Maybe when you were a child, you and your sibling, or maybe a friend invented your own language. Once you got the hang of it, it probably became fairly easy to converse in this specialized language. Math is similar as it has its vocabulary, and variation problems are one example of a type of math problem that has its own lingo. Once you become familiar with the lingo, problems are easier to understand. In variation problems, the lingo describes a relationship between variables.
For example, what is the area of a square? To find the area we multiply the length of one side by four. But how to put this into an equation? We write: A = 4s (A stands for the Area and s represents the length of one side.) The 4 in this equation is constant, it will never change.
To express this as a variation problem, we would say that the area, A, of a square is directly proportional to the length of its side, s. In variation terms we would write: A = ks, where k is a constant. For the area of a square, the constant of variation is 4. We could find the area of any square with the variation equation using 4 in place of the constant, k.
Part II: Boyle’s Law and Charles’s Law
A. What do you think will happen to a syringe with sudsy water in it when you start squeezing one end and the other end is plugged?
B. What do you think will happen to a balloon if it’s put into cold water? In hot water?
>>Your teacher will demonstrate both of these cases in class. Is this what you hypothesized?
Based on experiments testing these hypotheses, Boyle’s Law and Charles’s Law were found. In science books, one can find their laws written in variety of different ways. In this lesson, we are going to write these gas laws in terms of variation.
In Boyle’s Law the pressure of an object at a constant temperature varies inversely to its volume. In
other words: P = . In other words, as the pressure increases on an object, the volume decreases.
In Charles’s Law (also known as the Law of Volumes), the volume of an object at a constant pressure varies directly to its temperature: V = k2T. In other words, as the temperature increases in an object, the volume increases.
Note on Units:
Examples of Problems:
Example 1: At a constant temperature, 2 L of a certain gas would obtain a pressure of 1 atm. What would the volume of a gas be if the pressure was increased to 60 atm?
Solution:
1. Substitute what we know into the variation formula for Boyle’s Law (do you see why it’s Boyle’s Law?):
P = so we have 1 atm =
2. Solve for the constant,
3. Substitute our known and the final pressure back into the variation formula and solve.
So we have 60 atm =
Or V = = 0.033 L Final Answer
Example 2: A man heats a balloon in the oven. If the balloon initially has a volume of 4 liters and a temperature of 20 0C, what will the volume of the balloon be after he heats it to a temperature of 250 0C?
Solution:
1. Change 20 0C into kelvins. K = 20 0C + 273 = 293 K
2. Substitute what we know into the variation formula for Charles’s Law (do you see why it’s Charles’s Law?):
V = k2T so we have 4 L= k2· (293 K)
So we have V = 0.0134 · (250 + 273 K)
Or V = · (523 K) = 7 L Final Answer
Sources Cited: http://misterguch.brinkster.net/gaslawworksheets.html
Part III: Exercises
1. A balloon contains 30 L of helium gas at 100kPa. What is the volume when the balloon rises to an altitude where the pressure is only 25 kPa? (Assume the temperature remains constant.) 120 L
2. In a thermonuclear device, the pressure of 0.050 liters of gas within the bomb casing reaches 4.0 x 106 atm. When the bomb casing is destroyed by the explosion, the gas is released into the atmosphere where it reaches a pressure of 1.00 atm. What is the volume of the gas after the explosion? 600 L
3. A gas with a volume of 4.0 L has pressure of 90 kPa. What is the new volume if the pressure drops to 20kPa? (18L)
5. Submarines need to be extremely strong to withstand the extremely high pressure of water pushing down on them. An experimental research submarine with a volume of 15,000 liters has an internal pressure of 1.2 atm. If the pressure of the ocean breaks the submarine forming a bubble with a pressure of 250 atm pushing on it, how big will that bubble be? 72L
6. Divers get “the bends” if they come up too fast because gas in their blood expands, forming bubbles in their blood. If a diver has 0.05 L of gas in his blood under a pressure of 250 atm, then rises
instantaneously to a depth where his blood has a pressure of 50.0 atm, what will the volume of gas in his blood be? Do you think this will harm the diver? V = 0.25 L, yes
7. If I place a balloon with a temperature of 220 C and a volume of 0.5 liters into my refrigerator, what will be the volume of the balloon when it is fully cooled by my refrigerator to 40 C? .47L
8. A man heats a balloon in the oven. If the balloon initially has a volume of 0.4 liters and a temperature of 20 0C, what will the volume of the balloon be after he heats it to a temperature of 250 0C? .71 L
you have an empty soda bottle (volume of 2 L) at room temperature (25 C), what will the new volume be if you put it in your freezer (-4 0C)? 1.81 L
Name_________________________________________________________ V A R I A T I O N: C O N T I N U E D
1. The number of hours, h, it takes for a block of ice to melt varies inversely as the temperature, t. If it takes 2 hours for a square inch of ice to melt at 65º, how long would it take to melt the ice if it was 85°?
2. In kick boxing, it is found that the force, f, needed to break a board, varies inversely with the length, l, of the board. If it takes 3 lbs of pressure to break a board 2 feet long, how many pounds of pressure will it take to break a board that is 5 feet long?
3. Your weight on Mars varies directly with your weight on Earth. A person weighing 140 lb on Earth weighs 23.2 lb on Mars, since Mars has less gravity. If you weigh 180 lb on Earth, how much will you weigh on Mars?
The following problems use the Combined Gas Law. The Combined Gas Law uses both Boyle’s and Charles’s Laws. It states that the pressure of an object is directly proportional to the temperature and
inversely proportional to its volume.
4a. Write an equation which shows the relationship between pressure (P), temperature (T) and volume (V) in the Combined Gas Law. Use k as the constant.
What volume will the balloon have at standard temperature (273 K) and standard pressure (100 kPa)? 39.2 L. Note: Standard temperature and standard pressure are always at 273K and 100kPa, it represents the ideal conditions to measure the volume of a gas.
6. A container with an initial volume of 1.0 L is occupied by a gas at a pressure of 150kPa at 25° C. By changing the volume, the pressure of the gas increases to 600kPa as the temperature is raised to 100° C. What is the new volume? .31L
7. A 5.0 L air sample at a termpearture of -50° C has a pressure of 108 kPa. What will the new pressure be if the temperature is reaised to 100° C and the volume expands to 7.0 L? 128 kPa
