## Wednesday, October 5, 2016

### Gas Laws and Number of Molecules

Applying the Kinetic Theory of Matter to can explain many behaviors of gases with relationship to volume, temperature, and pressure. Extending the exploration of gas laws by including consideration of the number of molecules gives us the ability to describe what is happening even more clearly.

Gases, because the expand or compress to fill the container in which they are kept, can be understood within the context of kinetic theory with a few principles. With changing temperature:
• 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.

If we change the volume:
• 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.
• 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.
Changing the pressure, likewise, has an affect on either the volume or the temperature.

But what about if we change the number of molecules? Clearly, changing the number of molecules would have an affect on the gas's behavior within its container.

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).

Naturally, there is a way to view all of this quantitatively. In order to do that, it is necessary to reprise our understanding of how to count molecules.

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

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:

$PV=nRT$

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.

What does it all mean?

Besides pressure, volume, and temperature, the number of molecules is equally important. Changing the number of molecules has an affect as follows:
• 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.

EXAMPLE PROBLEM
A container of gas at 1 atm pressure holds 2 moles of gas, and fills 2 liters at 300K. If the number of moles is doubled and temperature and pressure stay the same, what is the new volume?

Pressure and temperature will cancel out leaving:

Multiple both sides by 4 moles…

and thus

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Definitions and content from:

New Oxford American Dictionary
Physical Science Concepts in Action, Pearson
http://en.wikipedia.org
http://www.chemguide.co.uk/physical/kt/idealgases.html