Gas Laws

The main distinguishing property of gases is their uncanny ability to be compressed into smaller and smaller spaces. Gases are also the least complex state of matter. Don't get it wrong, just because they are the simplest doesn't mean that they are not one of the most interesting and useful states of matter.

Properties of Gases
Boyle's Law
Charles's Law
STP and Gas Collected over Water
Combined Law
Ideal Gas Law

Properties of Gases

Gases are easily expandable and compressible unlike solids and liquids. Gases have a measurement of pressure. Pressure is defined as force exerted per unit area of surface. It can be measured in several units such as kilopascals (kPa), atmospheres (atm), and millimeters of Mercury (mmHg). Gas has a low density because its molecules are spread apart over a large volume. A gas will fill whatever container that it is in. An example of this is a bottle of ammonia being opened in a room and the smell traveling throughout the room.

The Kinetic Molecular Theory is the basis of the many properties of gases. The five postulates to the Kinetic Theory are as follows:

Gases are composed of molecules whose size is negligible compared to the average distance between them.
Molecules move randomly in straight lines in all directions and at various speeds.
The forces of attraction or repulsion between two molecules in a gas are very weak or negligible, except when they collide.
When molecules collide with one another, the collisions are elastic; no kinetic energy is lost.
The average kinetic energy of a molecule is proportional to the absolute temperature.

diffusion animation

Effusion and Diffusion are the two ways that gases mix with other gases. Diffusion is a process in which a gas enters a container with another gas and the two mix to form a uniform mixture. Effusion occurs when a gas moves through a small hole in its current container into another container. An example of diffusion is the ammonia mentioned earlier where the ammonia moves into the room with the air. An example of effusion is if a coke bottle had a small hole in it just small enough for the gas inside to escape.

Since one of the properties of a gas is compressiblity, a gas at a certain volume can be compressed by adding pressure. The mass of the gas will remain unchanged. Since the mass remains the same and the volume decreases, the density of the gas is greater.

This can be observed by using the density equation D=m/V.

If the mass of the gas is .50 grams and the volume of the gas is one liter then the density of the gas is .50 grams/liter. However, if the gas is compressed to only take up one half a liter then the density will change to 1 gram/liter.

The picture illustrates the compressing of a gas.

Compressing a Gas

Many of the properties of gases can be measured in different ways. Conversion from one unit of pressure to another is very important. To achieve this there has to be a conversion factor to move from one unit to another. Here is a list of equivalent amounts of pressure:

1 atm
760 mmHg
76 cmHg
101.3 kPa
760 torr
29.92 inches Hg

If you are in atms and want to convert to mmHg then # of atms * (760 mmHg / 1 atm) = # of mmHg because the atms cancel. This dimensional analysis method of canceling the units will work for changing from any unit to another.

Boyle's Law

Boyle's Law states the volume of a definite quantity of dry gas is inversely proportional to the pressure, provided the temperature remains constant.

Mathematically Boyle's law can be expressed as P₁V₁ = P₂V₂

V₁ is the original volume
V₂ is the new volume
P₁ is original pressure
P₂ is the new pressure

Suppose you have a gas with 45.0 ml of volume and has a pressure of 760.mm. If the pressure is increased to 800mm and the temperature remains constant then according to Boyle's Law the new volume is 42.8 ml.

(760mm)(45.0ml) = (800mm)(V₂)
V₂=42.8ml

Charles's Law

Charles's Law can be stated as the volume occupied by any sample of gas at a constant pressure is directly proportional to the absolute temperature.

V / T =constant

V is the volume
T is the absolute temperature (measured in Kelvin)

Charles's Law can be rearranged into two other useful equations.

V₁ / T₁ = V₂ / T₂

V₁ is the initial volume
T₁ is the initial temperature
V₂ is the final volume
T₂ is the final temperature

V₂ = V₁ (T₂ / T₁)

V₂ is the final volume
T₂ is the final temperature
V₁ is the initial volume
T₁ is the initial temperature

Important: Charles's Law only works when the pressure is constant.

Note: Charles's Law is fairly accurate but gases tend to deviate from it at very high and low pressures.

STP

STP is Standard Temperature and Pressure. STP is O^o Celcius and 1 atmosphere of pressure. Gases properties can be compared using STP as a reference.

To obtain the pressure of gas collected over water the partial pressure of the water must be taken into consideration. The reason for this is as the gas bubbles through the water the gas picks up water vapor. The amount of water vapor the gas picks up only depends on the temperature. To calculate the pressure of the gas the partial pressure of the water must be subtracted from the pressure in the container. The partial pressure of the water can be obtained from the table below.

Temperature (^oC)	Pressure (mmHg)
0	4.6
5	6.5
10	9.2
11	9.8
12	10.5
13	11.2
14	12.0
15	12.8
16	13.6
17	14.5
18	15.5
19	16.5
20	17.5
21	18.7
22	19.8
23	21.1
24	18.7
25	23.8
26	25.2
27	26.7
28	28.3
29	30.0
30	31.8
35	42.2
40	55.3
45	71.9
50	92.5
55	118.0
60	149.4
65	187.5
70	233.7
75	289.1
80	355.1
85	433.6
90	525.8
95	633.9
100	760.0
105	906.1

For example if a gas is collected over water at 22^oC and 1 atm of total pressure, the pressure of the gas would be calculated as follows:

1 atm = 760 mmHg therefore 760 mm - 19.8 mm = 740.2 mmHg would be the pressure of the gas.

Combined Law

The combined gas law is a combination of Boyle's Law and Charles's Law; hence its name the combined gas law. In the combined gas law, the volume of gas is directly proportional to the absolute temperature and inversely proportional to the pressure.

This can be written as PV / T = constant. Since for a given amount of gas there is a constant then we can write P₁V₁ / T₁ = P₂V₂ / T₂.

P₁ is the initial pressure
V₁ is the initial volume
T₁ is the initial temperature (in Kelvin)
P₂ is the final pressure
V₂ is the final volume
T₂ is the final temperature (in Kelvin)

This equation is useful if you have the current volume, temperature, and pressure of a gas, and if you have two of the three final values of the gas.

For example if you have 4.0 liters of gas at STP, and you want to know the volume of the gas at 2.0 atm of pressure and 30^o C, the equation can be setup as follows:

(1.0)(4.0) / 273 = (2.0)(V₂) / 303
(V₂)(2)(273) = (1)(4)(303)
V₂ = 2.2
Therefore the new volume is 2.2 liters.

Ideal Gas Law

The ideal gas law is a combination of all the gas laws. The ideal gas law can be expressed as PV = nRT.

P is the pressure in atm
V is the volume in liters
n is the number of moles
R is a constant
T is the temperature in Kelvin

The constant R is calculated from a theroretical gas called the ideal gas. The most commonly used form of R is .0821 L * atm / (K * mol). This R will allow the units to cancel so the equation will work out.

To find the volume of 2.00 moles gas that is at 1.00 atm of pressure and 235 Kelvin, use the ideal gas law equation.

(1.00 atm)(V) = (2.00 mol)(.0821 L * atm / (K * mol))(235 kelvin)

V = (38.587 L * atm) / (1.00 atm)

V = 38.6 L