Gas laws

Laws of Boyle, Gay-Lussac, Avogadro and Dalton, and the universal gas law

Basic principles

 

A gas is one of the three aggregation states of a substance, solid, fluid and gaseous. A gas is compressible and often a mixture. It occupies all available space uniformly and completely, has a small specific mass, diffuses and mixes rapidly, and is mono- (He), di- (N2, O2, CO), tri- (CO2, O3), or poly-atomic (NH3, methane etc). In a so-called ideal gas, the particles (being molecules or atoms) have no size and do not influence each other since they show no mutual attraction (no cohesion).

Gas particles move at random with velocities of many hundreds m/s (Table 1). The mean particle-particle distance is in the nm-range. Gas volume is empty for ca. 99.9% and therefore the gas particles compared to liquids infrequently collide with each other. At 300 K and 1 bar the mean free path of H2 is ca. 66 nm, more than 10 times than in the liquid phase. Collisions with constant temperature (isotherm) are pure elastic, also with a wall at the same temperature.

The gaslaws are those of Boyle (and Mariotte), Gay-Lussac, the universal gas one, Avogadro and Dalton.

 

Table 1

    Particle

diameter of particle

nm

velocity v

(273 K)

m/s

Molecular

Mass m

Km/h

cp/cV ratio

or

V*

 

He

0.027

1304

4690

4.003

1.66

H2

 

1838

6620

2.016

1.41

O2

0.034

461

1740

32.00

1.4

N2

0.037

493

1860

28.016

1.4

CO2

0.040

393

1480

44.011

1.29

H2O

0.027

 

 

18.016

1.33

(vapor)

 

* cp/cV = cp/(cp-R) (see Boyle’s law, The universal gas law and Adiabatic compressions or expansions).

 

Medical applications

 

These laws are fundamental for anesthesiology, the (patho)physiology of pulmonology, the medicine of diving, hyperbaric medicine (HBO, application of pure O2 as breathing gas under pressure), aviation (all recreation types) and aerospace medicine, mountaineering. The law of Dalton is basic for the physiology of respiration.

 

More info

 

These laws can only be applied when there is heat exchange with the environment. Then the process is called isothermic or diabatic. In practice this seldom holds. For instance when high-pressure helium tank is filled in the factory it raises in temperature (adiabatic compression). This compression is so fast that the produced heat has not the time to be transferred to the environment. For non-diabatic (non-isothermic compressions or expansions see Adiabatic compressions or expansions). The ideal gas laws assume an isothermic process and, self-evident with an invariable amount of gas mass.

 

 

The law of Boyle (and Mariotte)

 

In the derivation of Boyle's law it is assumed that the gas particles make elastic collisions with the wall surrounding the gas. The collisions exert a force on the wall, which is proportional to the number of particles per unit of volume, their velocity and their mass. Pressure (p) is defined as force (F) per area (A). By doubling the density of the gas (this is doubling the number of particles in a given volume), the number of collisions doubles, and hence the exerted force and so the pressure. Hence, the equation:

p1·V1 = p2·V2 = constant       or

p1/p2 = V2/V1, or p = constant·V-1,       (1)

is obtained. This law holds well for moderate gas densities (pressures < 300 bar, with regular temperatures) and under isothermic conditions, i.e. the process is diabatic. For higher pressures the condition that the total volume of the particles can be neglected and that the particles do not influence each other does not hold any longer. The Law of Boyle is refined with the Waals corrections. The Van der Waals corrections make sense for mass calculations of commercials gases in high-pressure tanks applied in medicine, especially the expensive helium. For the van der Waals corrections and further refinements see the physical textbooks.

 

 

The law of Gay-Lussac

 

The ratio of pressure and absolute temperature is constant provided that volume of the gas is constant:

p1/ p2 = T1/T2 = constant.        (2)

It has been proved that the squared velocity <v2> is proportional with T and reciprocal to the gas mass:

<v2> = 3RT/(NAm),

with <v2> the mean of the squared velocity of the particles, R the molar gas constant (= 8315 J/kmol·K), NAm the gas mass with NA the Avogadro’s number and m the particle mass.

Conceptually, the correctness of the law can be understood by realising that ½∙m∙v2 is kinetic energy of a particle. So, for a certain type of particle an increase in T gives an increase of v2 and consequently of p. When p and n are constant it holds that: V1/T1 = V2/T2 = constant, the law of Charles.

 

 

The universal gaslaw (of Boyle and Gay-Lussac)

 

This is a combination of the law of Boyle and the law of Gay-Lussac:

pV = n∙R∙T,                            (3)                                          

with n the number of kmoles of the gas and with R the molar gas constant (= 8315 J/kmol·K).

It holds that p∙V = ⅓∙NA∙m∙<v2> = constant and that <v2> = 3RT/(NAm). After substitution of <v2> and applying this for n moles the law follows.

 

 

The laws of Avogadro

 

V1/n1 = V2/n2 = constant       (4)

Since equal volumes of ideal gasses at equal pressure and equal temperature comprise an equal number of particles, the law follows directly.

 

The law of Dalton

 

The pressure of a mixtures of gasses is the sum of the pressure of the individual gasses (defined as the partial pressures) since the kinetic energy (½∙m∙<v2>) of all types of particles, irrespective their type, is the same: m<v2> = 3RT/NA = constant (see law of Gay-Lussac). So:

             ptotal = p1 + p2 + p3 .... (5)