7.7.1  Gases

 

The equation pV=nRT and its application to the determination of relative molecular masses. (Correction of gas volumes to STP not required.) (Derivation of equation not required.)

Properties of gases

1.               A gas occupies the whole volume of a containing vessel.

2.              Gases are easily compressed compared to solids and liquids.

3.              Gases are completely miscible in any proportions provided they do not react chemically.

4.              Gases have low densities.

 

Gas Laws

 

·         Boyle’s Law

The pressure of a fixed mass of gas is inversely proportional to the volume at constant temperature.

                                                              

         P                                        p                                  pV

 


                                           OR                                        OR

 

                          

p α  1/V                           V                                  1/V                                           p

p = c1/V

or pV = c1

 

·                Charles’ Law

          The volume of a fixed mass of gas is directly proportional to the absolute temperature at constant pressure.

 

 


                        V

 

 

 


  -273                                       T oC

          V α T                             

          V = c2T

 

·                Avogadro’s Law

Equal volumes of gases under the same conditions of temperature and pressure contain the same number of molecules.

If I mole of gas is considered then it occupies 22.4 dm3 at STP (1 atm and 0 oC) or 24 dm3 at 1 atm and 20 oC.

1 mole of any substance contains the Avogadro’s number of particles = 6.02 x 1023.

 

·         Ideal Gas Equation

          Combining Boyle’s Law and Charles’ Law gives the Ideal Gas Equation

                                    PV = constant

                                      T

For 1 mole of gas

                                    PV = R

                                      T

          where R is the molar gas constant.

   If p is in Nm-2 or Pa (1 atm. = 105 Nm-2), V is in m3 and T is in K then R = 8.31 J mol-1 K-1.

          For n moles of gas:

                        PV = nRT

 

Determining Molecular Masses

          For n moles of gas pV = nRT.

          but n = mass of gas in g

                     Mr of gas in g

          ∴ pV = m RT

                     Mr

          Rearranging gives 

 

                                    Mr = mRT

                                             pV

 

Exercise 1

1        Calculate the relative molecular mass of a gas G if 0.574 g occupies 548 cm3 at 22oC and a pressure of 740 mm Hg.

          (R = 8.31 J mol-1 K-1  : 760 mm Hg = 1 atm = 105 Nm-2)

 

2        The relative molecular mass of a liquid substance was determined by vaporising 0.30 cm3 of the liquid to give 80.0 cm3 of vapour at a pressure of one atmosphere and at 25 oC. If the density of the liquid is 1.50 g cm-3, what is the relative molecular mass?  70

 

 

 

 

Note that m/V = density d of the gas

   Mr = mRT

             pV

∴ Mr = dRT

             p

 

 

 

Kinetic Molecular Theory

It is assumed that A gas consists of widely spaced particles (i.e. the particles are separated from each other by distances which are large in comparison to the size of the particles).

1.                    The particles are in continual rapid motion in straight lines in all directions. They collide with each other and the walls of the container (this causes the gas pressure).

2.                   The particles occupy a negligible volume compared to the volume of the container.

3.                   Collisions are perfectly elastic (i.e. the attractive forces between the particles are negligible.

4.                   The average kinetic energy of the particles is proportional to the absolute temperature. Increase in temperature causes the average kinetic energy to rise.

 

From these assumptions can be derived the fundamental equation.

pV =   ¯ x  mnc2

       

 

Where p= pressure

            V= volume

            m = mass of one molecule

            n = number of molecules

            c = root mean square velocity (a measure of the average speed of the particles)

 

The equation above can be used to prove some important gas laws.

Boyle’s Law

The pressure of a gas is inversely proportional to the volume at constant temperature.

p α 1/V

 or pV = constant (at constant temperature)

 

The kinetic energy of a gas is proportional to the temperature (K). For a fixed mass of

gas at constant temperature

K.E. = ½ mnc2 is constant

pV = 1/3 mnc2 = 2/3 x ½ mnc2 = 2/3 x K.E.

∴  pV = constant

 

Charles’ Law

The volume of a gas is directly proportional to its absolute temperature at constant

pressure.

V α T

 

pV =1/3  mnc2

V= 1/3P x mnc2 = 2/3P x ½ mnc2 = 2/3P x K.E.

At constant pressure

V = constant x K.E.

But the kinetic energy is proportional to the temperature T.

∴ V α T