Expansion of Gasses

Gas is a state of matter with no fixed shape and no fixed volume. It takes the volume of its container. There are two ways by which the dynamics of gasses can be described. These are macroscopic and microscopic description. Macroscopic description is description based on the averge properties of the particles comprising the gas. Microscopic description is a statistical description of the properties of the particles comprising the gas.

Macroscopic Description

The state variables used for the macroscopic description of gasses are temperature ( T ) , pressure ( P ) and volume ( V ). Volume is the volume of the container of the gas. Pressure is the average force per unit area exerted on the container by the particles of the gas. Temperature is the average of the kinetic energies of the particles. These three variables are related by a law called the combined gas law.

The Combined Gas Law

The combined gas law states that the volume of a gas is directly proportional to its temperature ( in °K) and inversely proprtional to its pressure. Mathematically, this means the state of the gas might change but the ratio between the product of volume and pressure, and temperature is a constant. If the state of a gas changes from the state (V1 , T1 , P1 ) to the state ( V2 ,T2 ,P2 ), then the following equation holds.

V1 P1 ⁄ T1 = V2 P2 ⁄ T2

The temperature must be in °K when using this equation. °C and °F can't be used. There are two special cases of this law called Boyle's law and Charles Law.

Boyle's law states that if temperature is kept constant, then the volume of a gas is inversely proportional to its pressure. That is, the product of volume and pressure of a gas remains constant.

P1 V1 = P2 V2

Charles' law states that if pressure of a gas is kept constant, then the volume and the temperature of the gas are directly proportional. That is, the ratio between volume and temperature is a constant.

V1 ⁄ T1 = V2 ⁄ T2

The freezing temperature of water ( 0 °C) and the atmospheric pressure at sea level ( 101300 Pa ) are commonly referred as standard temperature and standard pressure respectively. The abbreviation STP stands for standard temperature and pressure.

Brief Review of Chemistry

The mass of atoms is measured by a unit of mass called atomic mass unit, abbreviated as u. Atomic mass unit is defined to be (1 ⁄ 12 )th of the mass of a carbon atom. The mass of the atom of an element measured in atomic mass unit is called atomic mass ( M ) of the element. For example the mass of a carbon atom ( MC ) is 12 u and the mass of an oxygen atom ( MO ) is 16 u. Molecular mass of a compound is defined to be the sum of the atomic masses of the atoms in the chemical formula of the compound. For example the molecular mass of carbon di-oxide whose chemical formula is CO2 is the sum of the atomic mass of a carbon atom and twice the atomic mass of oxygen: MCO2 = MC +2MO = ( 12 + 2 * 16 ) u = 44 u. The gram molecular weight ( Mg ) of a compound is defined to be its molecular mass expressed in grams ( That is Mg = ( M ⁄ u )g ) . For example the gram molecular weight of carbon di-oxide is 44 g because its molecular mass is 44 u. One gram molecular weight of any substance contains 6.02e23 molecules. The number 6.02e23 is called Avogadro's number and denoted as NA.

NA = 6.02e23

One gram molecular weight of a substance is also called one mole of the substance. Thus the mass of one mole of a substance is equal to gram molecular weight of the substance and there are Avogadro number of moleccules in one mole of a substance. The number of moles, n, in a sample of mass m (in grams) may be obtained by deviding the mass of the sample in grams by the molecular weight of the sample, Mg.

n = m ⁄ Mg

Also, the number of moles in a substance can be obtained as a ratio between the number of molecules in the sample, N, and Avogadro number.

n = N ⁄ NA

The Ideal Gas Equation

The combined gas law was stated for a fixed amount of gas. For a situation where the amount of gas also might change, the proportionality between volume and number of moles should also be included. The combined gas law should be restated to say the volume of a gas is directly proportional to temperature and number of moles, and inversely proportional to pressure. Mathematically, this means the ratio between the product of volume and pressure to the product of number of moles and temperature is a constant ( PV ⁄ nT = constant ). This constant is a universal constant called universal gas constant, denoted by R.

PV = nRT

This is the equation known as the ideal gas equation. The value of R is 8.3 J ⁄ °K ⁄ mole.

R = 8.3 J ⁄ °K ⁄ mole

The ideal gas equation is applicable to real gasses only approximately. It applies exactly only for the ideal case where the interaction energy between the moleccules can be neglected completely and the energy of the molecules can be assumed to be purely kinetic energy. In using this equation, SI units ( That is m 3 for volume, Pa for pressure and °k for temperature ) shuold be used. But Liter (Abbreviated as L) is a very common unit of volume. If Liter is used for volume, then kilo Pascal ( kPa ) should be used for pressure. The unit for temperature must be in °K. °C and °F can not be used.

The ideal gas equation can also be expressed in terms of the number of molecules instead of the number of moles. Replacing n in the ideal equation by N ⁄ NA, the equation PV = NRT ⁄ NA can be obtained. The ratio R ⁄ NA is a constant called Boltzman's constant and is denoted by κβ.

PV = Nκβ T

The value of Boltzman's constant is 1.38e-23 J ⁄ °K ⁄ mole.

κβ = 1.38e-23 J ⁄ °K ⁄ mole

Energy of Ideal Gas Molecules

An ideal gas is a gas where the interaction energy between the molecules is neglected and the energy is purely kinetic energy. This energy is directly proprtional to temperature ( in °K ) and depends on the number of degrees of freedom of the molecules comprising the gas. Number of degrees of freedom is equal to the number of different ways by which the particles can acquire energy. The random motion of ideal gas molecules can be decomposed into motion along the x-axis, y-axis and z-axis. That is, the number of degrees of freedom of the particles is three. The energy per particle per degree of freedom is equal to κβ T ⁄ 2. Therefore, since a particle of an ideal gas has three degrees of freedom, the energy of one particle of an ideal gas is three times the energy per degree of freedom.

E = 3κβ T ⁄ 2

E is the energy of one molecule of an ideal gas at a temperaure T ( in °K ). The total energy ( ET ) of the sample is obtained by multiplying the energy of one particle by the number of molecules ( N ) in the sample.

ET = 3Nκβ T ⁄ 2

The total energy also can be expressed in terms of the number of moles in the sample. Replacing κβ by R ⁄ NA and noting that the ratio N ⁄ NA is equal to the number of moles n, the following alternative expression for the total energy can be obtained.

ET = 3nRT ⁄ 2

RMS speed of Ideal gas molecules

The molecules of a gas are moving randomly with all kinds of speeds. The root mean square speed abbreviated as RMS speed is defined to be the square root of the average of the squares of the speeds of all the particles. The energy of one particle of an ideal gas may also be obtained as the kinetic energy of a particle whose speed is the RMS speed. Thus, if the mass of one molecule is denoted as mo, the equation mo vRMS 2 ⁄ 2 = 3κβ T ⁄ 2 holds ; and it follows that

vRMS = √ (3κβ T ⁄ mo )

m0 ( in grams ) can be obtained by dividing the gram molecular weight by avogadro number, since there are avogadro number of particles in one gram molecular weight. It needs to be in kg though. mo in kg may be given as follows.

mo = Mkg ⁄ ( 1000NA )

Where Mkg is molecular mass expressed in kg ( That is Mkg = ( M ⁄ u )kg ). Replacing κβ by R ⁄ NA, the following alternative expression for the RMS speed can be obtained.

vRMS = √ ( 3000RT ⁄ Mkg )