## Sunday, September 25, 2016

### The Gas Laws

The Kinetic Theory of Matter models many observed phenomenon related to the way matter behaves in relationship to temperature. It is especially useful to explain how gases behave and serves well to help conceptualize The Gas Laws.

Because gases expand to fill the container they are in, when conditions change, the gases respond. How they respond depends on what changes. In order to work with gases, it is necessary to understand four concepts within the context of the Kinetic Theory of Matter.

Two of the concepts are not new: Volume and Temperature.

Volume
The total space something occupies.

The amount of space that a substance or object occupies, or that is enclosed within a container.

Volume is often measured in some unit cubed. For example, it may be in cm3 or mm3 or cubic inches or cubic meters which is m3Volume can also be measured in liters or millimeters (l or ml). In some of the models, using cubic meters is necessary, so becoming familiar with gases in that unit is important.

Temperature
The average kinetic energy of the molecules within a substance. An indication of the degree of warmth.
Within the context of the kinetic theory of matter, temperature is the measure of how much energy the molecules have, on average. Another way to think about temperature is (because KE = 1/2MV2) that it tells us how fast the molecules of a substance are moving.
In science, we will use Celsius or Kelvin temperature scales to describe temperature.
To convert:
Celsius = Kelvin - 273.15
Kelvin + 273.15 = Celsius

The third concept related to gas within the kinetic theory of matter is pressure.

Pressure
The continuous physical force exerted on or against an object by something in contact with it. The force exerted per unit area.

Pressure is the result of a force distributed over an area. There are several ways to measure pressure.

Pressure is measured in pascals, Pa - sometimes expressed as newtons per square meter, N/m2. These mean exactly the same thing. Pascals is the SI unit for pressure.

Be careful if you are given pressures in kPa (kilopascals). For example, 150 kPa is 150,000 Pa. You must make that conversion before you use the ideal gas equation.

Should you want to convert from other pressure measurements:

1 atmosphere = 101,325 Pa

1 bar = 100 kPa = 100,000 Pa

http://www.chemguide.co.uk/physical/kt/idealgases.html

Another way to measure pressure in in millimeters of mercury, which is a measure used in weather and comes from the use of mercury barometers.

Within the framework of the kinetic theory of matter, pressure can be understood as the sum of all the forces of all the molecules of a gas colliding with the sides of the container. Each molecule is moving quickly and sometimes the collide with the container. When the molecules of the gas strike the molecules of the container, kinetic energy is transferred, and the effect is noted as pressure. The more often and more energetically molecules strike the container, the higher the pressure will be.

Number of Molecules

One last concept that must be mentioned is the number of molecules being observed. This is rather intuitive: the number of molecules is… the number of molecules in the container.

Counting molecules is not easy. They are… small and do not take up much room. Any sample would have a bazillion molecules in it!

bazillion |bəˈzilyən|
cardinal number informal, chiefly N. Amer.
a very large exaggerated number.

Numbering molecules is usually done by saying how many moles are present.

The mole is the unit of measurement in the International System of Units (SI) for amount of substance. It is defined as the amount of a chemical substance that contains as many elementary entities, e.g., atoms, molecules, ions, electrons, or photons, as there are atoms in 12 grams of carbon-12 (12C), the isotope of carbon with relative atomic mass 12 by definition. This number is expressed by the Avogadro constant, which has a value of 6.022140857(74)×1023/mol. The mole is one of the base units of the SI, and has the unit symbol mol.

Avogadro's Number: 6.0221409e+23 = 6.0221409 X 1023

= 602,214,090,000,000,000,000,000

For reference and further understanding, one mole of a gas at standard temperature and pressure (273.15 K and 1 ATM) occupies a volume of 22.4 L.

The Gas Laws In Words

With an understanding the above concepts, it is very easy to make sense of what happens with gases as changes occur.

Changing Temperature
When the temperature of a sample of gas in a container goes up…

By definition, this means that the molecules have higher kinetic energy. Therefore:
• if the volume of the container cannot change, the molecules (moving faster) will hit the container more often and with more kinetic energy and raise the pressure.
• if the volume of the container CAN change, the molecules (moving faster) will take up more space (spread out) and increase the volume.
Raising the temperature of a gas will increase its pressure if the volume of the gas and number of molecules are constant.

Changing Volume
When the size of the container decreases…

If the volume changes, the molecules will strike the walls of the container more often (because they keep moving, but have less space to move in). Therefore:
• if the volume of the container decreases and the temperature stays the same, the molecules will strike the walls of the container more often and, thus, increase pressure.
• if the volume of the container decreases and the pressure stays the same, the temperature will decrease.
Reducing the volume of a gas increases its pressure if the temperature of the gas and the number of particles are constant.

Changing the Number of Molecules
When the number of molecules goes up…
• more molecules in the container will result in more frequent collisions with the container.
• Increasing the number of molecules will increase the pressure if the temperature and volume are kept constant.
• Increasing the number of molecules will increase the volume if the pressure and temperature are kept constant.

The most obvious example is adding molecules of air to a balloon. Blowing into a balloon obviously changes its volume. It is easy to see that one affect adding molecules can have is to increase volume. The opposite is also true.

If the volume cannot change, such as in a basketball or volleyball that "isn't flat" something else must happen. Pumping more air into a basketball or volleyball does not change the size (unless it was flat to start with). It makes the ball harder. Why? The more molecules inside the set, fixed volume, the more collisions with the inside wall of the ball, and therefore, the higher the pressure. So a second affect adding molecules can have is to increase pressure.

Understanding the affect adding molecules has on temperature is not as intuitive. To examine this, we have to imagine a situation where molecules are added but that neither the volume, nor pressure changes. In this case we have more molecules in the same space, which means more collisions with the unmovable wall. For there to be more collisions but to NOT have a pressure change must mean that the molecules are moving slower. Therefore, if molecules are added and if pressure and volume do not change THEN temperature must go down (it must be lowered on purpose).

Changing the Pressure
When the pressure goes up…

An example of changing pressure could be rising from one altitude to a higher one. In such a case, the pressure would go down. Or going down deeper in water would cause the pressure to go up.

Reducing the pressure of a gas increases its volume if the temperature of the gas and the number of particles are constant.

Opposite: Increasing the pressure of a gas decreases its volume if the temperature of the gas and the number of particles are constant.

More on this…

The Gas Law Equations

Beyond just understanding what happens, the gas laws turn out to perfectly fit an equation that (although it looks crazy) is actually easy to use. There are some lead-up equations to understand first.

Charles's Law
The volume of a gas is directly proportional to its temperature in kelvins if the pressure and number of molecules are constant.

Boyle's Law
The volume of a gas is inversely proportional to its pressure if the temperature and the number of molecules are constant.

The behavior of gas, when the number of molecules is a constant, can be described by combining Boyle's Law and Charles's Law into a single equation. This is the Combined Gas Law

Combined Gas Law
Pressure is inversely proportional to volume, or higher volume equals lower pressure. Pressure is directly proportional to temperature, or higher temperature equals higher pressure.

Using these three models, we can explain and predict how gases will behave under many different circumstances.

Universal Gas Law

When the number of molecules are included in calculations, the following formula can be used:

PV = nRT

where P is pressure, V is volume, n is number of moles, R is a number called the "gas constant", and T is temperature in Kelvin.

Just so you know… the gas constant changes depending on what units you are using. It can be Googled, such as: 82.05746  cm^3 atm / (K • mol)

Looking at the equation allows predicting how gases will behave in relationship to changing the number of molecules:

PV = nRT
(Remember, R cannot change because it is just a number like π (pi)
• if n goes up, then P and/or V have to go up unless T goes down
• if n goes down, the P and/or V have to go down unless T goes up.

To look at this another way, the equation can be modified:

Dividing both sides by nRT…

Therefore, setting up an equation for a second state, should something from a first state change, would result in this:

Multiplying both sides by R would result in:

Now, using this equation to examine changes can result in deeper understanding. As n changes from state 1 to state 2, P, V and T must also respond in the second state as well. If n increases and T stays the same, then (as was noted before) PV must go up.  If n goes down, then PV must go down, as well.

NOTE: When working with these formulas, you can use a variety of units for pressure and volume and can, should you wish to do so, convert from one to the other. However, the temperature must be expressed in Kelvin.

SUMMARY:

Applying the Kinetic Theory of Matter can explain many behaviors of gases with relationship to volume, temperature, number of molecules and pressure.

Gases, because they expand or compress to fill the container in which they are kept, can be understood within the context of kinetic theory.

With changing temperature (where the number of molecules stays the same):
• higher temperatures means more molecular motion.
• more motion means the molecules take up more space.
• more motion means more frequent and more vigorous collisions with the walls of the container.
• taking up more space means higher volume.
• more frequent and/or more vigorous collisions means higher pressure.
Lower temperature means the opposite.

With changing volume (where the number of molecules stays the same):

• smaller volume means that the molecules are more tightly packed.
• if they have the same temperature, they will strike the walls of the container more often.
• more frequent collisions means higher pressure.
• OR in order to keep the pressure the same within a smaller space, the molecules would have to strike the container with less velocity, meaning they would have to have lower temperature.
With changing number of molecules (if volume and temperature stay the same):

• more molecules means that the molecules are more tightly packed.
• if they have the same temperature, they will strike the walls of the container more often.
• more frequent collisions means higher pressure.

Remember, there is no way to directly change the pressure. Pressure changes as a result of a change in one or more of the other quantities.

The formula…

P1V1n1T1=P2V2n2T2

…for any values NOT given, use the value of 1.

BONUS!
How fast are molecules moving? Really fast! Molecules move at hundreds of meters per second, which is even more hundreds of miles per hour!

http://physics.stackexchange.com/questions/128090/how-fast-do-molecules-move-in-objects

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Definitions and content from:
New Oxford American Dictionary
Physical Science Concepts in Action, Pearson
http://www.chemguide.co.uk/physical/kt/idealgases.html