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 H_{2}O 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, H_{2}; oxygen, O_{2}).

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

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

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(H_{2}O) = 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 **N _{A}**, whose value is

**N _{A}** = 6.023 x 10

The number of molecules (**N**) of a substance is the number of moles (**n**) it contains, multiplied by **N _{A}**.

Example

1kg of water contains

**n** = 1000/18.016 = 55.51 moles, and

**N = n N _{A}** = 55.51 x (6.023 x 10

= 334.34 x 10^{23} 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 = 10^{3} Pa

Normal atmospheric pressure = 101 kPa

= 1.01 x 10^{5} 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 = P_{1} +P_{2} + …

where P_{1} and P_{2} 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

P_{o} is the pressure of the gas at 0^{o}C, P_{T} at T^{o}C, 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 P_{1} is the gas pressure when its volume is V_{1} and P_{2} is its pressure when its volume is V_{2}, then Boyle’s law states that

P_{1}V_{1} = P_{2}V_{2} 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**, V_{1} is the volume of the sample at the absolute temperature T_{1} and V_{2} at its temperature T_{2}.

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 0^{o}C (273.15K) and pressure of 1atm (1.013x10^{5}Pa) 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 = Nk_{B}T

where k_{B} = R/N_{A} = 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 k_{B}T

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 3x10^{23} hydrogen molecules in a gas of volume 12.42 litres is 6.21x10^{-21}J.

Find the temperature and pressure of the gas.