Gas Laws

Elements are the fundamental substances of which all matter is composed.

Two or more elements can combine to form a compound.

Example

Each molecule of water contains two hydrogen atoms and one oxygen atom, as its symbol H2O indicates.

The particles of an element are called atoms, and those of a compounds are called molecules.

Example

Elemental gases may consist of atoms (helium, He; argon, Ar) or of molecules (hydrogen, H2; oxygen, O2).

The masses of atoms and molecules are expressed in atomic units (u), where

1 atomic mass unit = 1 u = 1.660 x 10-27kg

The mass m of a molecule is the sum of the masses of the atoms of which it is composed.

Example

Mass of water molecule

m(H2O) = 2m(H) + m(O)

= 2 x 1.008 + 1 x 16.00 = 18.016 u

A relatively small sample of matter used in both industry and laboratory contains a large number of molecules.

In SI units the amount of sample is measured in moles.

A mole (or Gram-mole) of any substance is that amount of it whose mass is equal to its molecular mass expressed in grams instead of atomic mass units.

Example

A mole of water has a mass of 18.016g since a water molecule has a mass of 18.016 u.

The number of molecules in a mole of any substance is Avogadro’s number NA, whose value is

NA = 6.023 x 1023 molecules/mole

The number of molecules (N) of a substance is the number of moles (n) it contains, multiplied by NA.

Example

1kg of water contains

n = 1000/18.016 = 55.51 moles, and

N = n NA = 55.51 x (6.023 x 1023 )

= 334.34 x 1023 molecules.

Pressure of a gas

Consider a cylindrical container with uniformly distributed gas in it.

The gas exerts a force F at any point of the container.

Average force per unit area measures the pressure, P, of a gas in this container.

SI unit of pressure is 1 Pascal (1 Pa)

1 Pa = 1Nm-2

1 kPa = 103 Pa

Normal atmospheric pressure = 101 kPa

= 1.01 x 105 Nm-2 = 1 atmosphere (1 atm)

= 1.01 bars (in meteorology) = 760 torr or

760 mm Hg (in medicine and physiology).

For a mixture of gases the total pressure equals to the sum of all component

P = P1 +P2 + …

where P1 and P2 are the pressure of gas 1 and gas 2 respectively.

Thermal expansion of gases

For a gas, even a very small variation in pressure changes the volume by a significant amount.

For a constant volume, the pressure of a gas increases with temperature according to relation

where

Po is the pressure of the gas at 0oC, PT at ToC, and

for all gases.

The three gas laws

1. At constant temperature, the volume of a sample of gas is inversely proportional to the pressure applied to the gas.

The greater the pressure, the smaller the volume.

This relationship is known as Boyle’s law.

If P1 is the gas pressure when its volume is V1 and P2 is its pressure when its volume is V2, then Boyle’s law states that

P1V1 = P2V2 T = constant

Gas undergoes isothermal process.

2. The relationship between the temperature and volume of a gas sample at constant pressure can be expressed as

In this formula, which is called Charle’s law, V1 is the volume of the sample at the absolute temperature T1 and V2 at its temperature T2.

The temperatures are expressed in an absolute scale in Kelvins.

Gas undergoes isobaric process.

3. Gay-Lussac’s law says, that at a constant volume, ratio of pressure and an absolute temperature of gas is constant.

Then gas undergoes isochoric process.

Ideal Gas Equation

Combination of these three laws leads to the ideal gas law which relates the pressure, volume and temperature of a gas.

An ideal gas is one which obeys this law under all circumstances.

The ideal gas law is obeyed fairly well by all gases through a wide range of pressures and temperatures. The temperatures are expressed in an absolute scale.

For convenience, a temperature of 0oC (273.15K) and pressure of 1atm (1.013x105Pa) are taken as the standard temperature and pressure (STP).

Experimentally it is found that 1 mole of any gas at STP occupies a volume of 22.4 litres (22.4 L).

Ex. At STP 2.5 mol of a gas will occupy the volume:

V = (2.5mol) x (22.4L/mol) = 56 L

The complete ideal gas law for n moles is usually written in the form

PV = nRT

where R is the universal gas constant and has the value R = 8.31 J/(mol K) calculated using STP.

For N molecules

PV = NkBT

where kB = R/NA = 1.38x10-23 JK-1 is called a Boltzmann’s constant.

Temperature and Molecular Energy

Simple model of a gas

- large number of moving in random fashion indentical spheres (gas molecules)

- only elastic collision with other molecules or with the container (translational kinetic energy)

- average distance between molecules >>diameter of molecule (no interaction between collisions)

- the pressure of the gas results only from the collisions between the molecules and with the container walls

It can be proved, that pressure, P, is related to the average kinetic energy of gas molecules, (KE)ave

N is the number of molecules

V is the volume of gas

Since

P V = N kBT

the average internal, translational kinetic energy of gas molecules is directly related to the absolute temperature of the gas

Example

The average kinetic energy of 3x1023 hydrogen molecules in a gas of volume 12.42 litres is 6.21x10-21J.

Find the temperature and pressure of the gas.