Charles' Law

Charles' Law Worksheet

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Discovered by Joseph Louis Gay-Lussac (the uppermost picture to the right) in 1802. He made reference in his paper to unpublished work done by Jacques Charles (the lower GIF picture to the right) about 1787. Charles had found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 degree interval.

The ChemTeam does admit that Charles looks like he's ready to fail a sobriety test, but you would do well to cut the guy some slack. After all, he did invent the hydrogen-filled balloon and on December 1, 1783, he ascended into the air and became possibly the first man in history to witness a double sunset.

Gay-Lussac was no slouch in the area of ballooning. On September 16, 1804, he ascended to an altitude of 7016 meters (just over 23,000 feet - about 4.3 miles). This remained the world altitude record for almost 50 years and then was broken by only a few meters. His image is from a stamp France issued in memory of the 100th anniversary of his death in 1950.



Because of Gay-Lussac's reference to Charles' work, many people have come to call the law by the name of Charles' Law. There are some books which call the temperature-volume relationship by the name of Gay-Lussac's Law and there are some which call it the Law of Charles and Gay-Lussac. Needless to say, there are some confused people out there. Most textbooks call it Charles' Law, so that's what the ChemTeam will use.

The same year a 23-year-old Gay-Lussac discovered this law, he had occasion to walk into a linen draper's shop in Paris and there he made a wonderous discovery. He found the 17-year-old shopgirl reading a chemistry textbook while waiting for customers. Needless to say, he was intrigued by this and made more visits to the shop. In 1808, he and Josephine were married and over the years, five little Gay-Lussac ankle-biters were added to the scene.

So ladies, while the ChemTeam cannot guarantee that the study of chemistry will land you the man of your dreams, you just never know what might happen through diligent study of your chemistry.

This law gives the relationship between volume and temperature if pressure and amount are held constant.

If the volume of a container is increased, the temperature increases.

If the volume of a container is decreased, the temperature decreases.

Why?

Suppose the temperature is increased. This means gas molecules will move faster and they will impact the container walls more often. This means the gas pressure inside the container will increase (but only for an instant. Think of a short span of time. The span of time the ChemTeam is referring to here is much, much shorter than that. So there.). The greater pressure on the inside of the container walls will push them outward, thus increasing the volume. When this happens, the gas molecules will now have farther to go, thereby lowering the number of impacts and dropping the pressure back to its constant value.

It is important to note that this momentary increase in pressure lasts for only a very, very small fraction of a second. You would need a very fast, accurate pressure sensing device to measure this momentary change.

Consider another case. Suppose the volume is suddenly increased. This will reduce the pressure, since molecules now have farther to go to impact the walls. However, this is not allowed by the law; the pressure must remain constant. Therefore, the temperature must go up, in order to get the moecules to the wals faster, thereby overcoming the longer distance and keeping the pressure constant.

Charles' Law is a direct mathematical relationship. This means there are two connected values and when one goes up, the other also increases.

The mathematical form of Charles' Law is: V ÷ T = k

This means that the volume-temperature fraction will always be the same value if the pressure and amount remain constant.

Let V1 and T1 be a volume-temperature pair of data at the start of an experiment. If the volume is changed to a new value called V2, then the temperature must change to T2.

The new volume-temperature data pair will preserve the value of k. The ChemTeam does not care what the actual value of k is, only that two different volume-temperature data pairs equal the same value and that value is called k.

So we know this: V1 ÷ T1 = k

And we know this: V2 ÷ T2 = k

Since k = k, we can conclude that V1 ÷ T1 = V2 ÷ T2.

This equation of V1 ÷ T1 = V2 ÷ T2 will be very helpful in solving Charles' Law problems.

This graphic simply restates the above in a way HTML cannot do.

Notice that the right-hand equation results from cross-multiplying the first one. Some people remember one better than the other, so both are provided.

Before going to some sample problems, let's be very clear:

EVERY TEMPERATURE USED IN A CALCULATION MUST BE IN KELVINS, NOT DEGREES CELSIUS.

The ChemTeam hopes you understand this very well. Repeating it does not hurt:

DON'T YOU DARE USE CELSIUS IN A NUMERICAL CALCULATION. USE KELVIN EVERY TIME.

Now, please don't send me e-mail asking me what I meant by that. Thanks.


Example #1: A gas is collected and found to fill 2.85 L at 25.0 °C. What will be its volume at standard temperature?

Solution:

1) Convert 25.0 °C to Kelvin and you get 298 K. Standard temperature is 273 K. We plug into our equation like this:

Remember that you have to plug into the equation in a very specific way. The temperatures and volumes come in connected pairs and you must put them in the proper place.

2) Cross-multiply and divide:

x = 2.61 L

Example #2: 4.40 L of a gas is collected at 50.0 °C. What will be its volume upon cooling to 25.0 °C?

Comment: 2.20 L is the wrong answer. Sometimes a student will look at the temperature being cut in half and reason that the volume must also be cut in half. That would be true if the temperature was in Kelvin. However, in this problem the Celsius is cut in half, not the Kelvin.

Solution:

1) Convert 50.0 °C to 323 K and 25.0 °C to 298 K. Then plug into the equation and solve for x, like this:

2) Cross-multiply and divide:

x = 4.06 L

Example #3: 5.00 L of a gas is collected at 100 K and then allowed to expand to 20.0 L. What must the new temperature be in order to maintain the same pressure (as required by Charles' Law)?

Solution:

x = 400. K

Be aware that a problem might ask you for the new temperature to be given in Celsius. Make sure to do the problem in Kelvin, get the new temperature in Kelvin and then convert to Celsius. Like this:

400. minus 273 = 127 °C

Example #4: A 2.5 liter sample of gas is at STP. When the temperature is raised to 273 °C and the pressure remains constant, what is the new volume?

Solution:

We know the gas starts at standard temperature, zero degrees Celsius. In Kelvins, this is 273 K. Now, note the ending temperature, 273 °C. In Kelvins, that is 546 K.

The absolute temperature has doubled! Since Charles' Law is a direct relationship, the volume also doubles, to 5.0 L and that is the answer.

Setting it up mathematically gives this:

5.00 L / 273 K = x / 546 K

On the Internet, you often see problems set up as the above one is. Be very careful reading it and deciding which values to multiply & divide when you cross-multiply and divide.

Note also, the pressure remains constant, so it simply drops from all consideration in the solving of this problem. After all, it remained constant during the entire problem.


Example #5: An ideal gas at 7.00 °C is in a spherical flexible container having a radius of 1.18 cm. The gas is heated at constant pressure to 88.0 °C. Determine the radius of the spherical container after the gas is heated. [Volume of a sphere = (4/3)πr3]

Solution:

1) Convert Celsius to Kelvin:

7.00 °C = 280.0 K
88.0 °C = 361.0 K

2) Set up Charles' law equation:

[(4/3) (3.14159) (1.18)3] / 280 = [(4/3) (3.14159) (x3)] / 361

(1.18)3 / 280 = x3 / 361

x = 1.28 cm

Note that the (4/3)π factor drops out. This is because it remains constant during the entire problem, so it can be divided out. If the problem had asked for the new volume (as opposed to the new radius), we could have include it at the end of the problem. Like this:

V = (4/3) (3.14159) (1.28)3

Also, note that the cube remains in step #2 above. This is because the volume change is proportional to the cube of the radius, not the radius itself.


Charles' Law Worksheet

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