Gasses

Basics

Molecules

Gasses consist of molecules. These molecules move around and bounce to each other. Molecule or atom mass is conveniently expressed in terms of some standard mass m0. In accordance with an international convention (Unified Scale of Atomic Weights, 1960) this reference mass (in gram) is defined with respect to the mass mC (in gram) of one atom of a particular Carbon isotope 12C:

m 0  =   m C 12

(1)

The atomic weight μ of an atom or molecule with mass m is defined as:

μ  =   m m 0

(2)

A convenient macroscopic number N_a of molecules (or atoms) to work with is the number of molecules with molecular mass 1 that make up 1 gram of a gas:

N a  =   1 m 0   =  6.02251 10 23

(3)

N_a is called Avogrado's Number. One mole of a gas is defined as quanttity of the gas consisting of N_a molecules. N_a molecules of a gas having a molecular weight μ weigh μ grams.

Let us have a look at vessel containing 1 liter (dm³) of Nitrogen (N₂) gas at atmospheric pressure. Nitrogen is the main constituent of air. The atomic weight of one Nitrogen atom is 14. The atomic weight of one molecule (consisting of two atoms) is 28. The mass of one liter (1 dm³) of Nitrogen is experimentally found to be 1.15 gram. Hence the number of moles and the number of molecules N in our liter Nitrogen is:

N mole  =   1.15 28   =  0.041 mole

So one liter of gas contains a extreme large number of molecules. If we could take one million molecules out of our vessel every second, it would take 7.8 billion years to empty the vessel!

The radius r of an molecule is about 10^-10 meter. The volume of one molecule is about:

v  =   4 3  π r 3  =   4 3  π  ( 10  - 10 )  3  =  4 10  - 30 m 3

(5)

The molcules in our vessel occupy the volume:

So only a small fraction consists of matter (0.1 ml). Most space (999.9 ml) is void. The matter is distributed over the space in an extremely large number of molecules, which are extremely small.

We could look at the average distance a molecule travels before it collides into another molecule. Consider the molecules to be a spheres with radius a. A molecule A travels along a line. A second molecule B is considered to be stationary. If the distance b of (the center of) molecule B to the line is larger than 2a, molecule A will miss molecule B. If the distance c of (the center of) molecule C to the line is smaller than 2a, molecule A collides with C. Now consider molecule A carrying a disk with radius 2a. If the molecule travels a distance L, the disk defines cylinder with volume:

Now we define the distance L_mfp such that in the volume defined by the disk and L_mfp contains only one other molecule. Given n is the average number of molecules per volume, this means:

This means for L_mfp:

L mfp  =   1 4 π a 2 n   =  3 10  - 7

(9)

The distance L_mfp is called mean free path and is the average distance a molecule A travels between to subsequent collisions.

The ideal gas

In an ideal gas mutual interaction (like attractive or repulsive forces) between molecules is neglegible. Stated more scientific: the mean potential energy of mutual interaction between the molecules is small compared to their mean kinetic energy. For

Pressure

When an amount (number of molecules) of gas is confined in a closed volume, the molecules excert a force on the walls of the volume. This force is due to collisions of the molecules with the walls. One could show that the average force eache gas molecule excerts on a wall of a volume at a given temperature is constant and independent of the mass of the molecules. 'Pressure' is the average force exerted on a unit surface by the molecules. The more molecules in the volume, the higher the density and hence pressure will be.

Daltons Law - partial pressure

Daltons law states: "The pressure a gas mixture excerts is the sum of the pressures of each of the individual gasses that make up the mixture, each gas acting as if it was alone occupying the entire volume". This means each of the gasses can be regarded independently of the others. The pressure each individual gas excerts is called 'partial pressure'. Atmospheric air at sea level contains mainly 21% oxigen (molecules) and 79% Nitrogen (molecules). The partial pressure of the oxigen is 21/100 * 1 bar = 0.21 bar. Similar the Nitrogen partial pressure is 0.79 bar.

Boyle's law

Boyle's law states: P*V is constant at a given temperature. P is the pressure in of a gas confined in a volume V. For example if a diver takes a balloon filled with air at roughly 1 bar from the surface to a depth of 10 meters (the pressure will be roughly 2 bar) the volume of the balloon will reduced to about half of the original volume.

Charles' law

Charles' law states: P*V/T is constant. T is the abolute temperature (K). For example a diver has a cylinder filled to P1=200 bar at 20 °C (T1=293 K). He leaves the cylinder in a car, which is placed in the sun. In the car the temperature rises to 70 °C (T2=343 K). We assume the volume V of the cylinder remains the same. Since :

P l V T 1   =   P 2 V T 2

(10)

the new pressure P2 in the cylinder will be:

P 2  =   P 1 V T 2 T 1 V   =   200 343 293   =  234 bar

(10)

If the cylinder is guarantied to 230 bar, the diver might have a problem.

Henry's law - dissolving of gasses

Gasses can become dissolved in water. Henry's law states: "At a given temperature the amount of dissolved gas is proportional to the partial pressure of that gas". When a liquid is exposed to a gas, gas will start to dissolve. Gas in solution still excerts pressure (gas tension). In equilibrium this gas tension equals the partial pressure of the gas to which the liquid is exposed. Similarly, when the gas pressure drops, dissolved gas will emerge from the liquid until a new equilibrium is established. This is exactly what happens when you open a bottle of soda to quickly. The pressure in the bottle drops. Dissolved Carbondioxide emerges from the gas and even forms bubbles.

Air

Atmospheric air around us mainly consists of Nitrogen and Oxigen:

• 78.084% Nitrogen N₂
• 20.946% Oxigen O₂
• 0.934% Argon Ar
• 0.033% Carbon Dioxide CO₂
• 0.003% Trace elements