11 Ideal Gases

Ideal gases
Ideal gas
 1 all molecules behave as identical, hard, perfectly elastic spheres.

 2 the volume of the molecules is negligible compared with the
volume of the containing vessel.
 3 there are no forces of attraction or repulsion between molecules
 4 there are many molecules, all moving randomly.
The gas laws
 Equation of state of the gas:

 The volume V of the gas is inversely proportional to its pressure p,
provided that the temperature is held constant.
 Boyle's law
V  1/ p
pV  cons tan t
p1v1  p2 v2
Charles’ law :
For a given mass of gas maintained at constant pressure, the
volume V of the gas is directly proportional to its thermodynamic
temperature T.
V T
V1 V2

T1 T2
 Gay-Lussac’s law, or law of pressures:
 For a given mass of gas maintained at constant volume, the
pressure p of the gas is directly proportional to its thermodynamic
temperature T.
p T
p1 p2

T1 T2
 Relation between pressure p, and volume V and temperature T for
a fixed mass of gas.
pV  T
p1V1 p2V2

T1 T2
pV  mT
pV  AmT
 The mole is the amount of substance which contains as many
elementary entities as there are atoms in 0.012kg of carbon-12.
 The Avogadro constant NA is the number of atoms in 0.012 kg
of carbon-12. That is, it is the number of atoms in a mole of
atoms. It has the value 6.02×1023 mol-1.
N
n
NA
 The relative atomic mass Ar is the ratio of the mass of an atom to
one-twelfth of the mass of an atom of carbon-12.
 The relative molecular mass Mr is the ratio of the mass of a
molecule to one-twelfth of the mass of an atom of carbon-12.
 The atomic mass unit u is one-twelfth of the mass of an atom of
carbon-12, and has the value 1.66×10-27 kg.
 Molar mass=relative atomic(molecular) mass in value.
 Unit: g/mol
m
M
n
Increase the number of molecules
Increase the pressure
Decrease the volume ;
Increase the pressure
Pressure increases with temperature
Pressure is caused by collisions
Gas particles have random motions. Each time a particle

collides with the walls of its container there is a force
exerted on the wall. The force per unit area on the wall is
equal to the pressure in the gas.
pV  nT
pV  nRT
pVm  RT
pV  NkT
R  kN A
 R is called the molar gas constant (R=8.3 J K-1 mol-1).
 Vm is the volume occupied by one mole of the gas.
 N is the number of molecules in the gas and k is a constant called
the Boltzmann constant.(k=1.38×10-23 J K-1)
 An ideal gas is one which obeys the equation of state
 pV=nRT
 At all pressures, volumes and temperatures.
 Eg
 Find the volume occupied by 1 mole of air at standard temperature
and pressure (273 K and 1.01×105 Pa), taking R as 8.3J K-1 mol-1
for air.
 Eg
 Find the number of molecules per cubic metre of air at standard
temperature and pressure.
 Eg
 A syringe contains 25×10-6 m3 of helium gas at a temperature of
20℃ and a pressure of 5.0×104 Pa. The temperature is increased to
400℃ and the pressure on the syringe is increased to 2.4×105 Pa.
Find the new volume of gas in the syringe.
A microscopic model of a gas
 Kinetic theory of an ideal gas

 Brownian motion
The molecules in the gas
have enough kinetic
energy, and therefore the
intermolecular force is
small, and the typical
distance between
neighboring molecules is
very large. Gas has no
definite shape or volume,
but the space occupied is
confined.
 A solid has rigidity: the atoms are held in more or less positions
by much stronger interatomic forces than exist in liquids or gases.
 The atoms are arranged quite close together in a solid, whereas in
a gas there is a great del of empty space.
 Atoms in a solid can still move, but their motion is restricted to
vibration about their equilibrium positions.
Some basic assumptions of the
kinetic theory of gases
All gases are made up of very small atoms or molecules.

All molecules move randomly.
The volumes of molecules are negligible as compare to the
volume of container.
Except during collisions, the interactions among molecules are
negligible.
The molecular are spherical in shape, and elastic in nature.
As compare to the time spent traveling between collisions, the
duration of a collision is negligible.
The molecules move with constant velocity between collisions,
and gravity has no effect on the motion.
 a gas molecule in a cubic container
x  component of momentum changes by
px  2mcx
the rate at which this molecule transfers momentum to the wall is
px
 2mcx  2 L / cx   mcx 2 / L
t
p  F / A  Nmcx 2 / L3  Nmcx 2 / V
1 
pV  Nm c 2 
3  2 1 2  2
 pV  N  m c   N Ek  NkT
1 2  3 2  3
Ek  m c
2 
3
Ek  kT , c 2  3kT / m
2
c 2  3kT / m
1
pV  Nm c 2
3
3
Ek  kT ,
2
The average translational kinetic energy of a
molecule is proportional to T.
 Root-mean-square speed or r.m.s. speed(crms)
c 2
 3kT / m
 The faster the molecule moves, the higher the temperature is, and
the higher the temperature is, the faster the molecule moves.
 Eg
 The speeds of seven molecules in a gas are numerically equal to
2,4,6,8,10,12 and 14 units. Find the numerical values of (a)the
mean speed; (b) the mean speed squared; (c) the mean square
speed; (d) the r.m.s speed.
 Eg
 Find the total kinetic energy of the molecules in one mole of an
ideal gas at standard temperature (273 K).
 Eg
 Find the root-mean-square speed of the molecules in nitrogen gas
at 27℃. The mass of a nitrogen molecule is 4.6×10-26 kg.

11 Ideal Gases

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Ideal gases

 1 all molecules behave as identical, hard, perfectly elastic spheres.

 Equation of state of the gas:

Gas particles have random motions. Each time a particle

 Kinetic theory of an ideal gas

All gases are made up of very small atoms or molecules.

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