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Brown's Chemistry Clinic
KS4 Science GCSE/IGCSE ==> Advanced AS/A2 Level Chemistry Revision-Information
Notes
Part
2 States of Matter
Revision Notes (advanced)
Part II More
advanced topics on the ideal gas
laws, calculations, kinetic particle model-theory
and ideal and non-ideal real gas behaviour
EMAIL query?comment
The
basic particle theory and properties of gases, liquids and solids, and state changes
and solutions are described on the Part I GCSE
gases-liquids-solids notes page
and advanced students should be familiar with ALL its contents
*
 Sub-index for Part II:
Section 4 Ideal gas behaviour and the gas laws:
Introduction-the kinetic
particle theory of an ideal gas
* 4a. Boyle's Law *
4b. Charles's-Gay Lussac's Law and the combined gas law equation
* 4c. The ideal gas equation PV=nRT * 4d. Dalton's Law of partial pressures
* 4e. Graham's Law of diffusion *
Section 5. Non-ideal real gas behaviour and Van der Waal's Equation:
5a. The deviations of a gases from ideal
behaviour and their causes * 5b.
The Van der Waals equation of state
* 5c Compressibility factors *
For other calculations see
the Calculations Index page
including Mole definition and
Avogadro Constant, molar gas volume
and reacting gas volume ratios.
The Maxwell
Boltzmann distribution of particle kinetic energies is discussed in the
KINETICS (rates of reaction) pages.
4. Ideal gas behaviour and
the gas laws
Introduction - the kinetic particle model of an ideal gas
-
The (advanced) kinetic
theory of gases is founded on the following six fundamental postulates:
-
Gases are composed of
minute discrete particles (usually molecules).
-
The particles are in
continuous chaotic motion moving in straight lines between very frequent collisions with
each other and the sides of the container (approximately 109/s).
-
The bombardment of the
container walls by the particles causes the phenomenon we call pressure
(i.e. force of impacts/unit area).
-
The collisions are
perfectly elastic i.e. no energy loss on collision due to friction.
-
At relatively low
pressures the average distance between particles is large compared to the
diameter of the particles and therefore the inter-molecular forces between
the particles is negligible.
-
The average kinetic
energy of the particles is directly proportional to the absolute temperature
on the Kelvin scale (K).
-
When a gas behaves
according to this model, the gas laws described in sections 4a to 4e are
obeyed.
-
However in real gases
things are not so simple and this non-ideal behaviour is
discussed in section 5.
 4a. Boyle's Law
-
Boyle's Law states that for given mass of gas
at a constant temperature, the product of the pressure multiplied by the
volume is a constant.
-
p x V = constant
-
Therefore, for initial
values of p1
and V1, which change to final values of p2 and V2, the following
equation applies ...
-
p1 x V1
= p2
x V2 (for fixed amount of gas at constant
temperature)
-
or
p2
= p1
x V1/V2 or V2
= p1 x V1/p2
-
The graph shows how the pressure and volume
vary according to Boyles Law at two different temperatures.
-
At lower temperatures the volume and pressure
values are lower (see next section).
-
You can use any volume or
pressure units you like as long as both p's and both V's have the
same units.
-
Using particle theory and
simple arithmetical values to explain Boyles Law.
-
If a gas is compressed to half its
original volume the concentration or density of the gas is doubled. Therefore
there will be twice as many collisions with the surface causing twice the
impact effect i.e. double the pressure.
-
If the volume of a gas is
increased by a factor of three, the concentration is reduced by the same
factor, so the chance of particle collision with the container walls is
similarly reduced, so the pressure decreases by a factor of three.
-
Examples of Boyle's Law calculations (constant temperature assumed)
-
Ex. Q4a.1
-
240cm3 of air
at a pressure of 100kPa in a bicycle pump is compressed to a volume of 150cm3.
-
What is the pressure of
the compressed air in the pump?
-
p1 x V1
= p2
x V2
, rearranging to scale up for the new
higher pressure
-
p2
= p1 x V1/V2 = 100 x 240/150 = 160 kPa
-
Ex. Q4a.2
-
10 m3 of butane
gas at 1.2 atm was required to be stored at 6 atm pressure. To what volume
must the gas be compressed to give the required storage pressure?
-
p1 x V1
= p2
x V2
, rearranging to scale down for the new
lower volume
-
V2 = p1 x V1/p2
= 1.2 x 10/6 = 2.0 m3
-
-
4b. Charles's/Gay-Lussac's Law and the combined
gas law equation
 
-
This law states that for a fixed mass of gas
...
-
(i) the volume of a gas is directly
proportional to the absolute temperature (K) at constant pressure
-
V = constant x T (left graph), or
-
V/T = constant,
or
-
V1/V2
= T1/T2 for conditions changing from 1 (initial)
to 2 (final),
-
or V1/T1
= V2/T2 for constant pressure
-
V2 = V1
x T2/T1
-
or T2 =
T1 x V2/V1
-
(ii) the pressure of a gas is directly
proportional to the absolute temperature (K) at constant volume,
-
p = constant x T (right graph),
or
-
p/T = constant,
or
-
p1/p2
= T1/T2 for conditions changing from 1 (initial)
to 2 (final),
-
or p1/T1
= p2/T2 for constant volume
-
p2 = p1
x T2/T1
-
or T2 =
T1 x p2/p1
-
In all calculations, the
absolute or Kelvin scale of temperature must be used for T (K =
oC + 273).
-
If all the laws described
in 4a and 4b are combined, you get the following general expression
-
p x V/T = a constant (for a
given mass of gas).
-
This can be expressed in
generalised form for calculations based on an
initial set of conditions1, changing to a new and final set of
conditions2 for a given mass of gas, giving the
combined pressure-volume-temperature gas calculation equation ...
-
| p1 x V1 |
|
p2 x V2 |
| ------- |
= |
-------- |
| T1 |
|
T2 |
- (p1 x V1)/T1
= (p2 x V2)/T2
- therefore the three
permutations for problem solving are ...
- p2 = (p1 x V1
x T2)/(V2 x T1)
- or V2 = (p1 x V1
x T2)/(p2 x T1)
- or T2 = (p2 x V2
x T1)/(V1 x T2)
-
Note:
-
If the temperature is constant you get Boyle's
Law.
-
If p or V is constant you
get Charles's/Gay-Lussac's Law.
-
You can use any volume or
pressure units you like as long as both p's or both V's have the
same units.
-
The graphs of p or V
versus temperature become invalid once the gas has condensed into a liquid BUT
when extrapolated back all the lines seem to originate from y = 0 (for p or
V), x = -273oC (for T).
-
This was part of the scientific evidence
that led to the belief that -273oC was the lowest possible
temperature, though there is no theoretical upper limit at all.
-
This led to the devising of a new thermodynamic absolute
temperature scale or Kelvin scale which starts at OK.
-
Examples of P-V-T calculations
-
Ex. Q4b.1
-
The pressure exerted by
a gas in sealed container is 100kPa at 17oC. It was found that
the container might leak if the internal pressure exceeds 120kPa. Assuming
constant volume, at what temperature in oC will the container
start to leak?
-
17oC + 273 =
290K
-
p1/T1
= p2/T2
-
rearranging to scale up to the higher
temperature
-
T2 = T1
x p2/p1
-
T2 = 290 x 120/100 = 348 K or 348 - 273
= 75oC
when the container might leak
-
Ex. Q4b.2
-
A cylinder of propane
gas at 20oC exerted a pressure of 8.5 atmospheres. When exposed
to sunlight it warmed up to 28oC. What pressure does the
container side now experience?
-
20oC = 273 +
20 = 293K, 28oC = 273 + 28 = 301K
-
p2 = p1
x T2/T1
-
p2 =
8.5 x 301/293 = 8.73 atm
-
Ex. Q4b.3
-
A student was
investigating the speed of reaction between limestone granules and different
concentrations of hydrochloric acid. However after doing a whole series of
experiments at different acid concentrations, there was no time to do the
last planned experiment. The volume of carbon dioxide collected after 5 minutes in a
100cm3 gas syringe was used to determine the rate of reaction.
All the experiments were done in one lesson at a temperature of 22oC
except for the last one. This was done in the next lesson, giving a carbon
dioxide volume of 47.0 cm3 after 5 minutes, but at a higher
temperature of 27oC (when in Kelvin call this T1,
and the other temperature T2).
-
To make the data
analysis fair, all the gas volumes should be ideally measured at the same
temperature, but a correction can be made for the last experiment.
-
(a) Calculate the volume
the of 47.0 cm3 of gas at 27oC, would occupy at 22oC.
-
V1/V2
= T1/T2
so V2 = V1
x T2/T1
-
V1
= 47.0 cm3, T1 = 273 + 27 = 300K, T2 = 273 +
22 = 295K
-
V2 =
47.0 x 295/300 = 46.2 cm3
-
(b) If the temperature
was ignored, what is the % error in the rate of reaction measurement?
-
Volume error = 47.0
- 46.2 = +0.8 cm3, therefore ....
-
% error = 0.8 x 100/47 = +1.7%
(so you would over calculate the reaction rate without this correction)
-
The % error in the
volume would be the same as calculated for the rate e.g. in cm3/min.
-
(c) Should the
calculated value for 22oC be used in the rate calculation
analysis? and are this still other sources of error?
-
The
theoretical-calculated gas volume for 22oC should be used for
calculating the rate, it will improve the accuracy a little, BUT there is
another problem!
-
If the reaction was
unfortunately carried out at a higher temperature (i.e. 27oC)
there is a second source of error. At a higher temperature the reaction is
faster, so you are bound to get a higher volume of gas formed in five
minutes. Therefore you will calculate a faster rate of reaction e.g. in cm3
gas/minute at 27oC, that would have occurred/been measured at 22oC
and so an unfair comparison with all the other results from the previous
lesson.
-
So, although you can
correct reasonably well the volume error due to an 'expanded' gas volume at
the higher temperature, the gas volume will still be too high because of
the faster rate of reaction at 27oC and there isn't much you
can do about that error except repeat the experiment at 22oC,
which is the best thing to do anyway!
-
Note that if the
temperature of a rates experiment was too low compared to all the other
experiments, the 'double error' would occur again, but this time the
measured gas volume and the calculated speed/rate of reaction would be lower
than expected.
-
(d) Would you need to do
any correction for the volume of acid added to the limestone? Explain your
decision.
-
No correction needed
for this at all. Although liquids expand/contract on heating/cooling, the volume
changes are far less compared to gas volume changes for the same temperature
change. This is because of the relatively strong intermolecular forces between liquid
molecules, which are almost absent in gases.
-
Ex. Q4b.4
-
25 cm3 of a gas
at 1.01 atm. at 25oC was compressed to 15 cm3 at 35oC.
-
Calculate the final
pressure of the gas.
-
p1 = 1.01 atm,
p2 = ?, V1 = 25 cm3, V2 = 15 cm3,
-
T1 = 25 + 273 = 298
K, T2 = 35 + 273 = 308 K
-
(p1 x V1)/T1
= (p2 x V2)/T2
-
p2 = (p1 x V1
x T2)/(V2 x T1)
-
p2 =
(1.01 x 25 x 308)/(15 x 298) = 1.74 atm
4c.
The
Ideal Gas Equation of State pV = nRT
-
A most 'compact molar' form of all the P-V-T
equations is known as the ideal gas equation and is the simplest possible
example of an 'equation of state' for gases (see also Van der Waals equation in
section 5b). The explanation of the use of the word 'ideal'
is explained in the 4. Introduction
-
The equation is
pV = nRT and requires a consistent
set of units, so see below for the two most common examples, and take care!
-
| pressure p |
volume V |
n = mass g/Mr |
Ideal gas constant R |
temperature T |
| Pa (pascal, 760mmHg = 1 atm =
101325 Pa) |
m3 (1m3 = 106 cm3 or m3 = cm3/106) |
mol |
8.314
J mol-1 K-1 |
K
(Kelvin =
oC + 273) |
| atm (atmospheres, 760
mmHg = 1 atm =
101325 Pa) |
litre or dm3 (litre = dm3
= 1000 cm3
and dm3 = cm3/1000) |
mol |
0.08206
atm dm3 mol-1 K-1 |
K
(Kelvin =
oC + 273) |
-
The first set are becoming the 'norm' since
they are the SI units, but the mass does not have to be in kg and can be in
the more 'practical unit' of g as long as Mr is in g mol-1.
-
Examples
of PV = nRT calculations
-
Ex.
Q4c.1
-
(a) Describe with the aid of a diagram a
simple gas syringe method for determining the molecular mass of a volatile
liquid.
-
-
A 100 cm3 gas syringe is mounted in
an oven (ideally thermostated) but a humble bulb will do and the temperature
is quite stable after an initial warming up period via internal convection.
Some of the liquid (whose Mr is toe determined), is sucked into a
fine 'hypodermic' syringe syringe (e.g. 0.2cm3) and the syringe
weighed. Quickly (to avoid evaporation losses), the liquid is injected into
the gas syringe via a self-sealing rubber septum cap and the syringe
re-weighed immediately. The difference in weighings gives the mass of liquid
injected. When the gas volume has settled to its maximum value the volume is
read (to the nearest 0.5cm3 if possible), together with the oven
temperature and barometric pressure (mercury barometer for best accuracy i.e.
in mmHg).
-
(b) In an experiment using the above
apparatus the following data were recorded and the molecular mass of the
voltile liquid calculated.
-
Mass of syringe + liquid = 10.6403 g
-
Mass of syringe after injection of liquid =
10.4227g
-
When volatilised the liquid gave 67.3 cm3
of gas.
-
The temperature of the oven = 81oC,
barometric pressure 752 mmHg.
-
Using the equation PV = nRT, calculate the
molecular mass of the liquid.
-
Mass of liquid injected = 10.6405 -
10.4227 = 0.2176 g
-
p = 101325 x 752/760 =
100258
Pa,
-
V = 67.3/106 =
6.73 x 10-5
m3, T = 273 + 81 = 354 K
-
PV = nRT, substituting for moles
n
gives PV = m/MrRT
-
and then rearranging gives ...
-
Mr = mRT/PV
-
= (0.2176 x 8.314
x 354)/(100258 x 6.73 x 10-5) = 94.9 (95 to 2sf)
-
(c) If the compound was formed from the
reaction of bromine and a hydrocarbon, suggest a possible molecular formula
for the compound.
-
(d) State very briefly, a method of
determining the molecular mass of ANY compound that can be vapourised intact.
-
Ex.
Q4c.2
-
Ex.
Q4c.3
-
For other gas
calculations see Mole definition and
Avogadro Constant, molar gas volume
and reacting gas volume ratios.
4d. Dalton's law of partial pressures
-
Dalton's Law of partial
pressures states that at constant temperature the total pressure exerted by a
mixture of gases in a definite volume is equal to the sum of the individual
pressures which each gas would exert if it alone occupied the same total
volume.
-
For a mixture of gases
1, 2, 3 ... ptot = p1 + p2 + p3
... where p1, p2 etc. represent the partial pressures.
-
The partial pressure ratio
is the same as the % by volume ratio and the same as the mole ratio of gases
in the mixture.
-
This means for a component
gas z:
-
Examples of partial pressure calculations
-
Ex. Q4d.1
-
In the
manufacture of ammonia a mixture of nitrogen : hydrogen in a 1 : 3 ratio is
passed over an iron/iron oxide catalyst at high temperature and high pressure.
-
N2(g) + 3H2(g)
 2NH3(g)
-
What are the partial pressures of nitrogen and hydrogen if the total pressure
of the gases is 200 atm prior to reaction?
-
The 1 : 3, N2
:H2 ratio means that nitrogen forms 1/4 of
the mixture, therefore
-
pN2 = 1/4 x
200 = 50 atm and
-
pH2 = ptot - pN2 =
150 atm (or from 3/4 x 200)
-
Ex. Q4d.2
-
Methanol can be
synthesised by combining carbon monoxide and hydrogen in a 1 : 2 ratio.
-
CO(g) + 2H2(g)
CH3OH(g)
-
In an experimental reactor
experiment, 300oC at a total pressure of 400kPa, the final
equilibrium gaseous mixture contained 10% carbon monoxide.
-
(a) Calculate the % of
hydrogen gas and % methanol vapour in the final mixture.
-
Whatever hydrogen is left,
its % must be double that of carbon monoxide since they were both mixed and
react in a 1 : 2 ratio, so there will 20% hydrogen in the equilibrium
mixture.
-
Therefore there will be
100 - 10 - 20 = 70% methanol in the final mixture.
-
(b) Calculate the partial
pressures of the three gases in the mixture.
-
(c) Calculate the value of
the equilibrium constant, Kp, under these reaction conditions
(use Pa pressure units).
-
-
4e. Graham's law of diffusion
-
Diffusion, or the
'self-spreading' of molecules, naturally arises out of their constant chaotic
movement in all directions, though on a time average basis, more molecules
will move in the direction of a region of lower concentration if such a
situation exists e.g. initially 'pouring bromine vapour into air' in gas jars
(see GCSE notes).
-
Molecules of differing
molecular mass diffuse at different rates.
-
It has been shown that,
assuming ideal gas behaviour and constant temperature, the relative rate of
diffusion of a gas through porous materials or a mixture of gases or a tiny hole
(effusion) is inversely
proportional to the square root of its density.
-
Since the density of an
ideal gas is proportional to its molecular mass, the relative rate of
diffusion of a gas is also inversely proportional to the square root of its
molecular mass*.
-
* PV = nRT, PV = m/MrRT,
Mr = mRT/PV, since d = m/V, then Mr is proportional to
density.
-
| r1 |
|
√d2 |
|
√M2 |
| --- |
= |
---- |
= |
---- |
| r2 |
|
√d1 |
|
√M1 |
- Which is the mathematical ratio
representation of Graham's law of diffusion for comparing two gases of
different molecular masses.
-
Graham's Law arises from
the fact that the average kinetic energy** of gas particles is a constant for
all gases at the same temperature.
-
**The formula for kinetic
energy is KE = 1/2mu2, where m = mass
of particle, u = velocity. This means the average mu2
is a constant for constant kinetic energy, so u is proportional to 1/√m and the m can be shown
via the Avogadro Constant to
be proportional to Mr, the molecular mass of the gas.
-
Examples of diffusion rate calculations
-
Ex. Q4e.1
-
-
Two cotton wool plugs are
separately soaked in concentrated aqueous ammonia and hydrochloric acid
solutions respectively and sealed in a long tube with rubber bungs.
-
Using a simple chemical
equation and Graham's Law of diffusion, account for (a) the appearance of a
'white smoke ring' and (b) the fact the smoke ring occurs about 2/3rds
along from the ammonia end of the tube.
-
(a) The aqueous
ammonia will give off ammonia fumes and the conc. hydrochloric acid gives off
hydrogen chloride fumes which will diffuse down the tube towards each other.
When the meet an acid base reaction gives fine crystals of the salt ammonium
chloride.
-
(b) Mr(NH3)
= 17, Mr(HCl) = 36.5
-
If r is the relative rate
of diffusion the following ratio applies,
-
| rNH3 |
|
√MrHCl
= √36.5 |
|
6.04 |
| ---- |
= |
------------- |
= |
---- |
| rHCl |
|
√MrNH3
= √17 |
|
4.12 |
-
and this shows that
ammonia will diffuse about 50% faster than hydrogen chloride so the smoke
ring will appear much nearer the HCl end of the tube.
-
Ex. Q4e.2
-
Zeolites are silicate
minerals that are porous at the molecular level and they are used as catalysts
and 'molecular sieves' in the petrochemical industry in processes such as
cracking and subsequent molecule separation.
-
(a) Calculate the relative
rates of diffusion of pentane CH3CH2CH2CH2CH3,
hexane CH3CH2CH2CH2CH2CH3
and 2-methylpentane (CH3)2CH2CH2CH2CH3
into a zeolite mineral.
-
Atomic masses: C = 12, H =
1
-
Hexane and 2-methylpentane
are structural isomers of C6H14 with the same molecular
mass.
-
| rC5H12 |
|
√MrC6H14
= √86 |
|
9.27 |
| ----- |
= |
------------- |
= |
----
= 1.09 |
| rC6H14 |
|
√MrC5H12
= √72 |
|
8.49 |
- Relative rate of diffusion is 1.09 : 1.00 for
pentane : hexane/2-methylpentane
-
(b) In practice
2-methylpentane does not diffuse into the zeolite as fast as hexane or maybe
not at all. Suggest a reason for this behaviour.
-
Ex. Q4e.3
-
Enriching uranium means to
increase the relative ratio of 235U/238U to produce
uranium metal suitable for use as fuel rods in nuclear reactors. It is the
235U isotope that is very fissile (readily undergoes fission) but
only occurs as a small % in uranium ores in which most uranium is the
non-fissile 238U. To produce 'enriched' uranium metal it is first
extrated by reduction from uranium oxide and then converted into gaseous
uranium(VI) fluoride (uranium hexafluoride). The 235UF6
is concentrated by a diffusion process in huge gas centrifuges before being
converted back to uranium metal. Atomic mass of F = 19
-
(a) Calculate the relative
rate of diffusion of the hexafluorides of the two uranium isotopes.
-
| r235UF6 |
|
√Mr238UF6
= √352 |
|
18.76 |
| ----- |
= |
------------- |
= |
-----
= 1.004 |
| r238UF6 |
|
√Mr235UF6
= √349 |
|
18.68 |
-
(b) Suggest why the
process must be repeated many times before enough enrichment has occurred.
5. Non-ideal real gas behaviour and Van der Waal's Equation
 5a. The deviations of
a gases from ideal
behaviour and their causes
-
Certain postulates in the
kinetic theory of gases (see 4.) are far from true in real gases, particularly at
higher pressures and at lower temperatures.
-
This can be clearly seen
in the diagram on the right.
-
If the gases conformed to
the ideal gas law equation PV=nRT, the product PV should be constant with increasing
pressure at constant temperature, clearly this is not the case.
-
It can also be seen
that the greatest deviation from ideal behaviour always tend to occur at
higher pressures (right diagram) and often at lower temperatures
(see the compressibility factor diagram in 5c) and
both postive and negative deviation occur.
-
Several points in the
theoretical kinetic particle model cannot be ignored in 'real gases'.
-
The actual
volume of the
molecules (Vmolecules) is significant at high pressures i.e. the free space for
random particle movement (Videal) is less than it appears from volume measurements.
-
Vreal = Videal
+ Vmolecules
-
At very high pressures
therefore, the value of PV becomes greater than the ideal value and
presumably outweighs the intermolecular force of attraction factor which would
tend to increase the closer the molecules are and decrease P (see forces
arguments next).
-
The deviation from ideal
gas behaviour due to the molecular volume factor will generally increase with
(i) the greater the pressure and (ii) the larger the volume of the molecule
(~increasing Mr).
-
Intermolecular
forces
always exist i.e. instantaneous dipole - induced dipole forces (Van der Waals
forces) between ANY molecules and at high pressures the molecules are closer
together and so attraction is more likely to occur. As a particle hits the
container side there is an imbalance of the intermolecular forces which act in
all directions in the bulk of the gas. Just as the particle is about to hit
the surface there will be a net greater attraction towards the bulk of the gas
as the molecule, so reducing its impact force i.e. reduces its 'ideal'
pressure (pideal)
by an amount (preduction).
-
preal = pideal
- preduction
-
At lower temperatures
when the KE of the molecules are at their lower values, the intermolecular
forces can have more of an effect in reducing P, so the PV value is less
than the ideal value. The effect becomes less as the temperature increases
(see graph in 5c.) and also as the pressure becomes
much higher when the molecule volume factor outweighs the intermolecular force
factor.
-
These intermolecular
forces will increase the bigger the molecule (~increasing number of electrons) and the more polar the molecule
where permanent dipole - permanent dipole forces can operate in addition to
the instantaneous dipole - induced dipole forces.
-
Also, the lower the
temperature, the kinetic energies are lower so its more likely that neighbouring molecules can
affect each other. The reduction in pideal also increases
with increasing pressure too, since the molecules will be on average closer
together.
-
There is direct
experimental evidence for the effects of intermolecular forces in gases from
adiabatic expansion or compression situations. Adiabatic means to effect a
change in a system fast enough to avoid heat transfer to or from the
surroundings. e.g.
-
(i) If a gas at high
pressure is suddenly released through a small nozzle it rapidly cools on
expansion into the lower pressure zone. The reason for the cooling is that
in order to expand the intermolecular forces must be overcome by energy
absorption, an endothermic process. The change is so rapid that the source
of heat energy can only come from the kinetic energy of the gas molecules
themselves, so the gas rapidly cools. This is observed when a carbon dioxide
fire extinguisher is used, just for a second bits of solid CO2
can be seen, which rapidly vaporise. However, it proves that the gas was
rapidly cooled from room temperature to -78oC!
-
(ii) When you rapidly
pump air into a bicycle tyre the gas warms up because the molecules are
forced closer together so the intermolecular forces can operate more
strongly, this, just like bond formation, is always an exothermic process.
-
Therefore generally
speaking for any gas the lower its pressure and the higher its temperature,
the more closely it will be 'ideal', i.e. closely obey the ideal gas equation PV=nRT
etc. Also the smaller the molecular mass or the weaker the intermolecular
forces, the gas will be closer to ideal behaviour.
-
However, for any gas at
a particular P and T, its all a question of what factor outweighs the others.
-
Note that both positive
and negative deviation from ideal gas behaviour can occur and there will be
situations where the different causes of non-ideal behaviour cancel each other out.
-
Check out the graphs
at the start of 5a. and 5c.
-
The measurement and
predictions of gas behaviour is very important in industrial processes and so
many mathematical developments have been devised to accurately describe the
real behaviour of gases. The Van der Waals equation
is one of the earliest and simplest equations to model real gas behaviour.
5b.
The Van der Waals equation of state
-
Equations such as the Van
der Waals equation for real non-ideal gases attempt to take into account the volume occupied by the
molecules and the intermolecular forces between them. The idea is to
incorporate 'corrective' terms to reproduce or model real gas P-V-T behaviour
with a modified equation of state.
-
The Van der Waals equation
for one mole of gas can be most simply stated in (i) as
-
(i) (p + a') (V - b') = RT
-
The term a' represents the
extra pressure the gas would exert if it behaved ideally. In real gases the
intermolecular forces are imbalanced at the point of impact on the container
wall, with a net attraction in the direction of the bulk of the gas. In the
bulk of the gas, each molecules is subjected to the same 'time averaged'
attractions in all directions, but heading for the container wall it is
considered to be 'dragged back a bit' by attraction with the bulk of the gas
surrounding it on all sides bar the surface of impact, which is therefore
reduced in force. (see also intermolecular forces
discussion in 5a.)
-
The term b' represents the
volume that the molecules occupy, so V-b' represents the actual volume of free
space the molecules can move in. (see also molecule
volume discussion in 5a.)
-
For n moles of gas the Van
der Waals equation is ...
-
(ii) [p + (an2/V2)] (V - nb) =
nRT
-
a and b are the Van der
Waal equation constants.
-
The factor n2/V2 is
related to the gas density, the more dense the gas (i.e. moles/volume), at
higher pressures, the more intense will be
the intermolecular attractive force field effects.
-
Dividing through by n,
using the (V - nb) term, gives the alternative version ...
-
(iii) [p + (an2/V2)] [(V/n) - b)]
= RT
-
(iv) p = [nRT/(V - nb)] - (an2/V2)
-
(v) p = [RT/(V/n - b)] - (an2/V2)
-
For 1 mole of gas the equation
simplifies to
-
A selection of a and b
Van der Waal's constants are given below.
-
| Data |
Van der Waals constants |
critical values of the gas |
| Gas |
a (Pa m6 mol-2) |
b (m3 mol-1) |
pressure pc (Pa) |
temp.
Tc (K) |
| air, av Mr(mix)
~ 29 |
0.1358 |
3.64 x 10-5 |
3.77 x 106 |
133 K |
| ammonia, Mr(NH3) = 17 |
0.4233 |
3.73 x 10-5 |
11.3 x 106 |
406 K |
| butane, Mr(C4H10) =
59 |
1.466 |
12.2 x 10-5 |
3.78 x 106 |
425 K |
| carbon dioxide, Mr(CO2)
= 44 |
0.3643 |
4.27 x 10-5 |
7.39 x 106 |
304 K |
|
dichlorodifluoromethane, freon CFC-11, Mr(CCl2F2)
= 121 |
1.078 |
9.98 x 10-5 |
4.12 x 106 |
385 K |
| helium, Mr(He) = 4 |
0.00341 |
2.34 x 10-5 |
0.23 x 106 |
5 K |
| hydrogen, Mr(H2) = 2 |
0.0247 |
2.65 x 10-5 |
1.29 x 106 |
33 K |
| nitrogen, Mr(N2) = 28 |
0.1361 |
3.85 x 10-5 |
3.39 x 106 |
126 K |
| water, Mr(H2O) = 18 |
0.5507 |
3.04 x 10-5 |
22.1 x 106 |
647 K |
-
The constant a
varies considerably from gas to gas because of the wide variety of
intermolecular forces e.g. very low for helium and non-polar hydrogen (2 e's
each, just instantaneous dipole-induced dipole forces), to much higher a
values for larger polar molecules like water or dichlorodifluoromethane (more
electrons and extra permanent dipole-permanent dipole intermolecular forces).
-
The constant b
varies less, and not unexpectedly, just tends to rise with increase in
molecule size.
-
Critical values of gas
behaviour.
 5c.
Compressibility factors
-
The compressibility
factor z, is defined as the ratio PV/nRT.
-
Since PV = nRT for an
ideal gas, then z = 1 for an ideal gas.
-
z varies with pressure or
temperature for any gas, see the PV versus P graph in
section 5a. which gives an indication of how z might vary with
pressure at a given temperature).
- Clearly from the graph on the right
for methane, z can be at least as high as 2, and, at least as low as 0.6,
showing considerable deviation from ideal gas behaviour, particularly at low
temperatures (influence of intermolecular forces stronger) and high
pressures (where the effect of both actual molecule volume and
intermolecular forces are important). See more detailed
discussion in 5a.
- As the pressure becomes lower and/or
temperature higher, the gas becomes more ideal in terms of its physical
behaviour and particularly 'ideal' as the pressure tends towards zero.
- Known values of z can be used to
calculate the real P-V values for a non-ideal gas.
- z = pV/nRT, pV = znRT, p = znRT/V
and V = znRT/p
ks4 science examinations gcse-igcse chemistry
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practice tests involving GCSE Science CHEMISTRY in the areas of REVISING
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academy/academies or home self-study. Hopefully it will encourage
interest and understanding of Chemistry, Earth Science and Radioactivity
in any country of the world, though the site is written entirely in English. The website is designed to help
and unofficially support students/teachers revise-learn/teach the chemistry for modular
or co-ordinated examination science
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360Science ,
Nuffield, Salters, Cambridge International (CIE), London International, WJEC, CCEA
exams etc.
Also, national award assessments-examinations for GCSE-IGCSE-KS4-O
level-BTEC-NVQ
applied, additional and chemistry national science courses. Also covers,
mainly via quizzes the UK National KS3 SATs Science-biology/chemistry/physics (SAT revision levels 3-5
or 5-7) and covers much of the revising, learning and teaching chemistry
examinations for the
national curriculum for secondary schools and colleges. The site does
not support the content of England, Wales or Northern Ireland primary
science KS1 or KS2. The notes should also provide some background theory
for a coursework assignment or project. BUT please note that
my on-line revision notes and quizzes are no substitute for good classroom
teaching-lecturing and thorough studying of your own notes and textbooks, practicing past papers
and a copy of the syllabus which are readily downloaded from the
examination board sites, but I hope here and there they will lend a
tutoring hand on some topic, unit, module etc. For final revision you
have to be intellectually honest about what you don't know or follow, YOU have to
take the stuff to pieces, analyse what you do/do not understand
and reconstruct it so it all makes sense in the end. There is no other
way, there are no magic secrets on how to revise and learn, its mainly
down to hard work and just good old fashioned study and employing teach-yourself
strategies without the need for extra tutors and tutoring lessons. I also think
there is too much hit and miss revision using past papers (which I do NOT
supply) and not enough
systematic revision. I also hope it will help teachers in planning
lessons and developing schemes of work for science-chemistry. There are no
lesson plans on the site but there are plenty of quizzes to incorporate into
classroom activities whether photocopied or on electronic whiteboard projector
for use as self-tuition-assessment purposes and a variety of teaching and
learning styles and the images may be used in Microsoft Word documents and powerpoint projections.
The site seems to be used by a large number of home study tutors, particularly
the revision notes. An individual tutor may print out the notes for
science-chemistry learning teaching-tuition purposes and for background material
for assignments and projects. I have no interest or time in producing WORD.doc or xxxx.pdf files
of the notes at the moment. Neither have I time to write up many practical
laboratory experiments ('lab'-'labs') at the moment, but the notes contain lots
of background information of chemical reactions in terms of
observations-balanced equations-reactants-products-theory etc. I also find it
difficult to recommend specific exam websites or syllabus textbooks, it depends exactly on
what you need, what you have time for, and there are so many of them to choose
from and I do not supply past examination papers for classes. The sites
resources include revision notes, quizzes and worksheets which provide support
for home study or tuition for homework and coursework help e.g. science
investigations for any of the key stage courses indicated, but I do not supply
lesson plans. Dr W P Brown gcse
10-11-2007 * ks4 science examinations gcse-igcse chemistry
revision *
ks4 science examinations-gcse-igcse chemistry revision * ks4 science
examinations-gcse-igcse chemistry revision * ks4 science examinations-gcse-igcse chemistry
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chemistry revision * ks4 science examinations-gcse-igcse chemistry
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GCE
advanced-A2-AS-subsidiary-IB level chemistry examinations * GCE advanced-A2--AS-subsidiary-IB
level chemistry examinations * GCE advanced-A2--AS-subsidiary-IB level chemistry
examinations SITE PURPOSE EDUCATION - online learning
or 'self-private-tuition' using revision notes, quizzes, practice
tests involving ADVANCED LEVEL CHEMISTRY in the areas of REVISING
only the CHEMISTRY at Doc Brown's Chemistry Clinic via HOMEPAGE in
secondary school/schools, 6th form college/colleges,
academy/academies or home self-study and may help with 1st year undergraduate
university chemistry courses. Hopefully it will encourage
interest and understanding of Chemistry, Earth Science and Radioactivity
in any country of the world, though the site is written entirely in English. The website is designed to help
and unofficially support students/teachers revise-learn/teach the chemistry for modular
or co-ordinated examination science
courses from UK QCA based AQA, OCR (Oxford and Cambridge),
Nuffield, Salters, Cambridge International (CIE), London International, WJEC, CCEA etc.
Also, national award assessments-examinations for BTEC-NVQ
applied, additional and chemistry science courses, Advanced Subsidiary
Level GCE-AS-A2-IB-KS5-BTEC-NVQ exams. National Chemistry assessment levels, International Baccalaureate,
K12 higher US grade level examinations for the
national curriculum for secondary schools and colleges. The notes should also provide some background theory
for a coursework assignment or project. BUT please note that
my on-line revision notes and quizzes are no substitute for good classroom
teaching-lecturing and thorough studying of your own notes and textbooks, practicing past papers
and a copy of the syllabus which are readily downloaded from the
examination board sites, but I hope here and there they will lend a
tutoring hand on some topic, unit, module etc. For final revision you
have to be intellectually honest about what you don't know or follow, YOU have to
take the stuff to pieces, analyse what you do/do not understand
and reconstruct it so it all makes sense in the end. There is no other
way, there are no magic secrets on how to revise and learn, its mainly
down to hard work and just good old fashioned study and employing teach-yourself
strategies without the need for extra tutors and tutoring lessons. I also think
there is too much hit and miss revision using past papers (which I do NOT
supply) and not enough
systematic revision. I also hope it will help teachers in planning
lessons and developing schemes of work for science-chemistry. There are no
lesson plans on the site but there are plenty of quizzes to incorporate into
classroom activities whether photocopied or on electronic whiteboard projector
for use as self-tuition-assessment purposes and a variety of teaching and
learning styles and the images may be used in Microsoft Word documents and powerpoint projections.
The site seems to be used by a large number of home study tutors, particularly
the revision notes. An individual tutor may print out the notes for
science-chemistry learning teaching-tuition purposes and for background material
for assignments and projects. I have no interest or time in producing WORD.doc or xxxx.pdf files
of the notes at the moment. Neither have I time to write up many practical
laboratory experiments ('lab'-'labs') at the moment, but the notes contain lots
of background information of chemical reactions in terms of
observations-balanced equations-reactants-products-theory etc. I also find it
difficult to recommend specific exam websites or syllabus textbooks, it depends exactly on
what you need, what you have time for, and there are so many of them to choose
from and I do not supply past examination papers for classes. The sites
resources include revision notes, quizzes and worksheets which provide support
for home study or tuition for homework and coursework help e.g. science
investigations for any of the key stage courses indicated, but I do not supply
lesson plans. Dr W P Brown GCE A AS A2
IB Advanced-Subsidiary Level Chemistry 10-10-2007 GCE
advanced-A2--AS-subsidiary-IB level chemistry examinations * GCE advanced-A2--AS-subsidiary-IB
level chemistry examinations * GCE advanced-A2--AS-subsidiary-IB level chemistry
examinations
Online free help resources for AQA,
Edexcel, OCR, CIE GCSE IGCSE BTEC Science, GCE, AS, A2 Advanced subsidiary
Chemistry A levels, IB Diploma and US K12 (K-12 grades) courses and examinations
and revising for the various syllabuses and specifications. Exploring the site
for lessons, plans, ideas for projects and coursework, professional development.
Through hard work the site has been built up over the course of many years with
no need of special pc software except FrontPage and Hot Potatoes (uvic) for quizzes and
worksheets. It is used in the classroom, home learning-tutoring-schooling and
guidance, private tuition, school retakes revision. Whether you are a
teacher/tutor teaching, a student studying, using the pages as self-study guides
for your science-chemistry studies etc. etc. I hope the site supports your endeavour.
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