# GasLaws X Mathematical and Chemical Concepts and Proofs

In this file I list all the mathematical and chemical concepts and proofs that are behind the inner workings of the GasLaws X program, and the gas laws themselves.  It may not be necessary to understand every last detail, but it is important to have a concept of why these things hold true, and why the program is written to function as it does.

## Ideal Gases vs. Real Gases and Absolute Zero

In all the following concepts and proofs, the gases taken are what is considered ideal gases, that is, if the gas is lowered to absolute zero it will have zero volume.  Obviously this doesn't work for real gases (i.e. the hydrogen that the sun burns, or the helium in a birthday balloon), since zero volume would be zero mass and that would be the destruction of matter, which is in violation of the Law of Conservation of Matter and Energy.  For the most part, ideal gases and real gases are much the same except for temperatures that approach absolute zero.  Absolute Zero (0 K) is actually found by extrapolating trends based on behavioral data of gases.  This will be proved later in the proofs section.

In mathematical terms, Ideal Gases have the properties of Pressure, Volume, and Temperature.  These ideas are at the base of the defined classes in the GasLaws series of programs.  These three properties are interrelated, and used in conjunction with one another, they form the three separate gas laws, from which the combined law is derived.  Ideal Gases are of a fixed mass when used in the gas laws, this is very important to remember, so remember it.  If you wish to make correlations between solids or liquids to gases, you need to explore the mole relationships of Gases.  Ideal Gases' mole relationships, which are not implemented in this release of GasLaws, will not be discussed in this help file.  If you want, the mole relationships are much discussed in any High School or College Chemistry textbook.

This description of Ideal Gases and Real Gases and Absolute Zero are a gross over-simplification and ignore several of their other properties, but they work in the context of this version of the program.

Also this is very important, make sure your units are consistent in all your calculations, meaning if you start using mL for volume, finish using mL for volume.

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## Boyle's Law:

### The Concept

In the age of scientific expansion, Robert Boyle did several experiments relating the pressure and volumes of gases.  He noted that for a fixed mass of gas that volume varied inversely in relation to its pressure when the temperature remained constant.  This inverse relation ship is very obvious if you blow up a bike tire.  When you are blowing up a bike tire, you are increasing the mass of a gas into a fixed volume, and as the tire fills, the pressure increases.  This is a bit of a mental stretch, but follow with me if you can.  Mass and volume behave in a direct relationship, that is, if you increase the mass of an object under a constant pressure, its volume will increase at the same time.  So logically, if the volume is fixed, such as in a container, and you added and increasing mass of gas to the container, it would be the same as if you were changing the volume of a fixed mass of gas.  If I lost you there, don't worry, all you need to know is that if you increase the pressure on a gas, its volume will decrease.

### Mathematical Proof

At this point I will be taking the words used above and translating them into the language of science.  I will use a set of made up data to show how these values are reached.

Problem:

• V1: 150 mL
• P1: 101.325 kPa
• P2: 202.650 kPa
• Find V2

To first solve this problem you have to make the correlation between words and symbols.  If there is an inverse relationship between volume and pressure the following formulas are true:

and      and

Also because of this relationship you can set two states of a gas equal to each other.  This is allowable since it is a fixed mass of gas you are working with.  So the following formula holds true, along with all its algebraic permutations:

So by this experimentally proven formula, one can use one of its permutations to find the second volume.  Carefully note the change of units and check the algebra and the label change.

V2 = 75.0 mL

This problem was obviously chosen for its simplicity, but it clearly illustrates the inverse relationship between volume and pressure.  Above, when the second state pressure was double of the first state pressure, the second state volume was one-half the first state volume.  It is fairly sound proof that is based on large amounts of scientific data, and is based on stronger proofs of better scientists than I.  It will always work, except when you approach absolute zero, but the assumed temperature is a logic constant such as Standard Temperature (273.15 K).

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## Charles' Law

### The Concept

Jacques Charles noted in several of his experiments with gases, that when a fixed mass of gas was placed in a container with a variable volume (a container with a fixed pressure) the volume changed very regularly as its temperature changed, as the gas gained heat and also the gas lost heat.  Also the note was made that the temperature would change as the volume changed, in the same exact relationship.  There is a direct-proportion relationship between the volume and temperature of every gas.  A very good example of this is the filling of a hot-air balloon, meaning the volume of the balloon increases as the temperature increase, which in turn makes the heated air less dense than the regular air.  Again, it is incredibly important in this law that the temperature input be in Kelvins.  Also it is important to note again that most real gases will become liquid or solid long before they reach 0 K.

### Mathematical Proof

Again, I will step by step take the words above into variables and formulas.  As stated above there is a direct correlation between volume and temperature. So the following formulas would be true:

Using the second relationship listed, it is possible to set two of them equal to each other to find the difference in a state change, either in volume or in temperature. The following is the crucial formula:

Lets take some data and see how the formula works, again will take some data where we can predict the results before doing the algebra.

• V1: 150.00 ml
• T1: 202.30 K
• T2: 101.15 K
• Find V2

V2 = 75 mL

This again a very simple problem, but it serves to prove the formula, and you can test it again and again to check the results.  This Law will always work except for negative temperature values (which shouldn't happen if you are using Kelvins), and zero values (which again should not happen if you are using Kelvins or remember that nothing can be at absolute zero).  Again I make the disclaimer that these proofs and equations were made by better scientists with larger amounts available to them.  It is not imperative that you understand the proofs or leaps of logic used here, just be content that they work for most all values, except for those discussed.

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## Gay-Lussac's Law

### The Concept

This law relates the temperature to the pressure of a gas.  Recalling the kinetic theory of matter, the theory that is behind the ideal gas, as temperature increases the gas molecules move around more.  At a fixed volume of gas, this means the gas molecules would hit the container more and more often, creating a higher pressure.  Again this is a direct relationship, where the temperature increase, the pressure increases also.

### Mathematical Proof

Given the words above I will again bring in some experimental data and change words into formulas.  Given that it is a direct relationship, the following formulas will hold true:

Also by some logic, as used in the other mathematical sections, you can set two states of a gas against each other to have a useful formula for finding pressure as the temperature of a gas changes or vice versa.  The following is the formula:

Lets again take some data with a result we can fairly well predict.

• P1: 101.325 kPa
• T1: 293.15 K
• T2: 308.15 K
• Find P2

The result is fairly predictable: P2 = 106.509 kPa

This result may have been less predictable than the others presented, but I decided to make the problem a little more complicated.  Again, as you can see the formulas do work.  I cannot lay claim to developing them myself, because, as I said before, better scientists and better data and I'm just a programmer.  If you're in doubt, you can crunch the numbers through several times to see that is holds fairly well true if you remember the significant digits.

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## The Combine Gas Law

### The Concept

Obviously a gas does not undergo only one property change over time.  As you have probably guessed by this point, all the separate laws above can actually be combined in a very sensible relationship using very little algebraic thought.  In the combined gas law, all three properties of pressure,  volume, and temperature are combined to show their property.  Also this law is now the common ancestor for all the other gas laws, that is all other gas laws are direct derivatives of the combined gas law.

### Mathematical Proof

As stated above, the combined gas law is a combination of pressure, volume, and temperature based on their various relationships with one another.  Boyle's Law stated that P · V = k, Charles' Law stated V = k · T, and Gay-Lussac's Law stated that P = k · T.  Using these simple relationships and a little algebraic know-how you arrive at the mighty formula:

And again by setting two copies of the formulas together we come up with a highly useful formula that we can use to do all kind of work with gases.  That formula is:

I am not going to go through any experimental data with this one, since all its component parts have been proven, and its formula was arrived by simple algebraic methods.  But as you can see from the formula, it is not difficult to see how the other formulas are arrived at if a constant is put into one of the variables.

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## The Existence of Absolute Zero: Mathematical Proof

This proving of Absolute Zero requires a little lee-way for playing with numbers.  If you're really not into the mathematics of it, you can rest assured that absolute is approximately -237.15 °C and that Kelvins are of the same magnitude as Celsius degrees, that is for one degree change in Kelvins, there is one Celsius degree change.  Here we go.

Lets take a big amount of data listed below and do a bit of arithmetic manipulation:

 Temperature (°C) Volume (mL) 273 1092 100 746 10 566 1 548 0 546 -1 544 -73 400

After taking the data and averaging a set of slopes and noting the X-intercept as 546 °C, the equation for the above line is y = 2 · X = 546.  Moving along the line, most of the values mach fairly well.  After a little more manipulation, the X value (temperature) at a zero volume (see Ideal Gas Law Theory) the resulting temperature is -273 °C.  This is not exactly the true value Celsius value of absolute value, neither is it the value most commonly used, but it is incredibly close.  The data listed above is from an experiment by Jacques Charles, and the data that led to his noting of the 1 over 273 ratio between temperature change and volume change, and the development of his law.  Today's Celsius value for absolute zero is slightly different, given that Charles' data was probably flawed by imprecise instruments.  The more common value used today is -273.15 °C, and even that is not quite accurate, the real value is some number with a large amount of decimal places, but it suffices.  Even -273 °C suffices but leads to greater inaccuracies over large spreads of data.  Even though this data may be slightly flawed, it shows very well an approximation of absolute zero.

Using limits, one can do a bit of calculation using Charles' Law to approximate the Celsius value of absolute zero, but I really won't go into that, since that is a subject for and advanced physics or chemistry class, and the above explanation suffices for this program.

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