Part 3 More
advanced topics on the ideal gas
including molecular mass determination, partial pressures and Graham's law of
diffusion, kinetic particle model–theory
and ideal and non–ideal real gas behaviour (Sections
4c to 4e)
chemistry revision notes: basic school chemistry science GCSE chemistry, IGCSE chemistry, O level
& ~US grades 8, 9 and 10 school science courses or equivalent for ~14-16 year old
science students for national examinations in chemistry plus more advanced
The ideal gas equation PV = nRT, ideal
gas theory, how to determine the relative molecular mass Mr of a volatile liquid,
Dalton's Law of partial pressures, ideal gas behaviour and non–ideal gas
behaviour, Graham's Law of diffusion, Van der Waals equation of state,
compressibility factors, critical pressure, critical temperature. This
page is well above GCSE level but useful for A Level chemistry or A Level
GCSE Chemistry Revision
This is a BIG
website, you need to take time to explore it [SEARCH
mobile phone or ipad etc. in 'landscape' mode
Sub-index for this page
Ideal Gas Equation of State pV = nRT AND determining molecular mass of a
examples of the PV = nRT ideal gas equation calculations
Dalton's law of partial pressures and calculations
Graham's Law of Diffusion and calculations
Non–ideal real gas behaviour, Van der Waals
equation, compressibility factors, critical point
[5.(e) is not usually dealt with these
days with UK pre-university advanced level chemistry courses]
The basic particle theory and properties of gases,
liquids and solids, state changes & solutions are described on
GCSE/IGCSE notes on particle models of gases–liquids–solids,
describing and explaining their properties and advanced students should be familiar with ALL its contents
before studying this page ....
AND P-V-T gas law calculations
are on a separate page
Sub–index for more advanced sections:
Introduction–the kinetic particle
theory of an ideal gas * Kelvin scale of
temperature * Boyle's Law *
Charles's–Gay Lussac's Law and the combined gas law
equation * The ideal gas equation PV=nRT *
Dalton's Law of partial pressures * Graham's Law of diffusion
* The deviations of a gases from ideal
behaviour and their causes * The Van der Waals equation of state
* Compressibility factors * The
Critical Point – The Critical Temperature and Critical Pressure
For other calculations see
the Calculations Index page
Mole definition and
AND molar gas volume
and reacting gas volume ratios,
studying these helps to follow this page too.
Boltzmann distribution of particle kinetic energies is discussed in the
Ideal Gas Equation of State pV = nRT AND determining molecular mass of a
The 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 5.(e)). The explanation of the use of the word 'ideal'
is explained in the
Section 4. (a) Introduction
The equation combines both Boyle's law
and Charles law plus moles of gas involved.
The equation, known as the
ideal gas equation, is
p = pressure in
pascals (unit Pa)
V = volume in
cubic metres (m3)
n = moles of gas
(mol = mass in g / molecular mass of gas Mr)
ideal gas constant = 8.314 joules per kelvin per mol (J K-1
T = temperature
in kelvin (K)
these units for a correct calculation using pV = nRT
Make sure can do all the
V = nRT /p,
T = pV/nR, p = nRT/V, n =
pV/RT and R = pV/nT
The equation is pV = nRT and requires a consistent
set of units, so see below for a comparison of the two most common examples, and take care!,
and SI units are pretty standard now and
my calculation examples primarily use SI
units (but I have left in a few examples in 'old' non-SI units).
||n = mass g/Mr
||Ideal gas constant R
and its units
760mmHg = 1 atm =
Pa = kPa x 1000
Pa = MPa x 106
1m3 = 106 cm3
so m3 = cm3/106
or dm3/1000 = m3
mol = mass (g)/Mr
J mol–1 K–1
oC + 273)
mmHg = 1 atm
| litre or dm3
1 litre = 1 dm3
= 103 cm3
dm3 = cm3/1000
mol = mass (g)/Mr
atm dm3 mol–1 K–1
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.
of PV = nRT calculations (all calculations assume ideal gas
(a) Describe with the aid of a diagram a
simple gas syringe method for determining the molecular mass of a volatile
An emptied 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 (e.g. 0.2cm3) and the syringe
Quickly (to avoid evaporation losses), the liquid is injected into
the gas syringe via a self–sealing rubber septum cap and the syringe
The difference in weighings gives the mass of liquid
When the gas volume has settled to its maximum value the volume is
read (to the nearest 0.5cm3 if possible),
Then note the oven
temperature and barometric pressure (mercury barometer for best accuracy i.e.
(b) In an experiment using the above
apparatus the following data were recorded and the molecular mass of the
volatile liquid calculated.
Mass of syringe + liquid = 10.6403 g
Mass of syringe after injection of liquid =
When volatilised the liquid gave 67.3 cm3
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 =
Pa, (converting pressure from mmHg to Pa)
V = 67.3/106 =
6.73 x 10–5
m3, T = 273 + 81 = 354 K
PV = nRT, substituting for moles
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.
5.(b) More examples of
the PV = nRT ideal gas equation calculations
TOP OF PAGE
5.(c) 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
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
Examples of partial pressure calculations
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)
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)
Methanol can be
synthesised by combining carbon monoxide and hydrogen in a 1 : 2 ratio.
CO(g) + 2H2(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
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).
TOP OF PAGE
5.(d) Graham's Law of Diffusion
Diffusion, or the
'self–spreading' of molecules, naturally arises out of their constant chaotic
movement of particles in all directions, though on a time average basis, more molecules
will move in the direction of a region of lower concentration down a diffusion
gradient 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.
The smaller the molecular mass, the
greater the average speed of the molecules at constant temperature.
The greater the average speed of the
particles the greater their rate of diffusion.
See notes on the
Maxwell–Boltzmann distribution of molecular velocities.
This conceptually explains Graham's Law
of diffusion, explained below.
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
* PV = nRT, PV = m/MrRT,
Mr = mRT/PV, since d = m/V, then Mr is proportional to
- 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.
You have to think of the molecules
'hitting' the space of the pore or tiny hole and passing through the. The
greater the speed the more chance the particle has of passing through this
Examples of diffusion rate calculations
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
= 17, Mr(HCl) = 36.5
If r is the relative rate
of diffusion the following ratio applies,
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.
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,
and 2–methylpentane (CH3)2CH2CH2CH2CH3
into a zeolite mineral.
Atomic masses: C = 12, H =
Hexane and 2–methylpentane
are structural isomers of C6H14 with the same molecular
- 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.
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
extracted 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
rates of diffusion of the hexafluorides of the two uranium isotopes.
(b) Suggest why the
process must be repeated many times before enough enrichment has occurred.
TOP OF PAGE
5.(e) Non–ideal real gas behaviour and Van der Waals Equation
The deviations of
a gases from ideal
behaviour and their causes
Certain postulates in the
kinetic theory of gases (see
section 4.(a)) are far from true in real gases, particularly at
higher pressures and a 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 and
both positive and negative deviation occur.
Several points in the
theoretical kinetic particle model cannot be ignored in 'real gases'.
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
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
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
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'
by an amount (preduction).
preal = pideal
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
(graph above-right) and also as the pressure becomes
much higher when the molecule volume factor outweighs the intermolecular force
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
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
(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.
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 5.(e)
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.
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
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
For n moles of gas the Van
der Waals equation is ...
(ii) [p + (an2/V2)] (V
– nb) =
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)
(iv) p = [nRT/(V – nb)] – (an2/V2)
(v) p = [RT/(V/n – b)] – (an2/V2)
For 1 mole of gas the equation
A selection of a and b
Van der Waal's constants are given below.
Van der Waals constants
critical values of the gas
a (Pa m6 mol–2)
b (m3 mol–1)
pressure pc (Pa)
|air, av Mr(mix)
||3.64 x 10–5
||3.77 x 106
|ammonia, Mr(NH3) = 17
||3.73 x 10–5
||11.3 x 106
|butane, Mr(C4H10) =
||12.2 x 10–5
||3.78 x 106
|carbon dioxide, Mr(CO2)
||4.27 x 10–5
||7.39 x 106
dichlorodifluoromethane, Freon CFC–11, Mr(CCl2F2)
||9.98 x 10–5
||4.12 x 106
|helium, Mr(He) = 4
||2.34 x 10–5
||0.23 x 106
|hydrogen, Mr(H2) = 2
||2.65 x 10–5
||1.29 x 106
|nitrogen, Mr(N2) = 28
||3.85 x 10–5
||3.39 x 106
|water, Mr(H2O) = 18
||3.04 x 10–5
||22.1 x 106
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
Critical values of gas
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
start of section 5.(e). 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
discussion at start of section 5.(e).
- 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
The Critical Point –
The Critical Temperature and Critical Pressure
Question! If you increase
the pressure of a gas it can change into a liquid. But, increasing the pressure,
also increases the temperature, so shouldn't the gas remain a gas?
Gases can be converted
to liquids by compressing the gas at a suitable temperature and this is done
commercially at as lower temperature as possible e.g. liquefaction of air to
fractionally distil off nitrogen and oxygen or liquefying petroleum gas.
Gases become more
difficult to liquefy as the temperature increases because the kinetic
energies of the particles that make up the gas also increase and the
intermolecular forces have less influence i.e. more easily overcome.
When you increase the pressure of a gas
you force the molecules closer together and if the extra intermolecular
force is strong enough liquefaction occurs. Remember the force of electrical
attraction is proportional to the numerical +ve charge multiplied by the
–ve charge divided by the distance squared.
However when you
compress a gas it can heat up. This is because heat is generated by the
increased intermolecular interaction (remember bond formation is also
exothermic) but here its just weak molecule association due to the
intermolecular attractive forces.
BUT liquefaction =
condensation and is an exothermic process, so heat must be removed to effect the
state change of gas ==> liquid. If the temperature is low enough and the
heat is dispersed liquefaction can still happen.
If it is too hot it
would stay as a gas. So liquefaction conditions are all about temperature,
pressure and heat transfer i.e. the ambient conditions.
However, above a certain
temperature called the critical temperature (Tc)
you cannot get a liquid with a 'surface', what you get is an extremely dense
gas that is close to being a liquid but not quite!
The critical temperature
of a substance is the temperature at and above which vapour of the substance
cannot be liquefied, no matter how much pressure is applied.
The critical pressure
(Pc) of a substance is the minimum pressure required to
liquefy a gas at its critical temperature i.e. the critical pressure is the
vapour pressure at the critical temperature.
critical point denotes the conditions above which distinct liquid and gas
phases do not exist and a meniscus no longer exists!
The point at the critical temperature and critical
pressure is called the critical point of the substance.
OTHER USEFUL PAGES
notes on gas law calculations, kinetic
model theory of an IDEAL GAS & non–ideal gases
See also for gas calculations
Moles and the molar volume of a gas, Avogadro's Law
Reacting gas volume
ratios, Avogadro's Law
& Gay–Lussac's Law Calculations
All other calculation pages
What is relative atomic mass?,
relative isotopic mass and calculating relative atomic mass
formula/molecular mass of a compound or element molecule
Law of Conservation of Mass and simple reacting mass calculations
Composition by percentage mass of elements
in a compound
Empirical formula and formula mass of a compound from reacting masses
(easy start, not using moles)
Reacting mass ratio calculations of reactants and products
moles) and brief mention of actual percent % yield and theoretical yield,
and formula mass determination
Introducing moles: The connection between moles, mass and formula mass
– the basis of reacting mole ratio calculations
(relating reacting masses and formula
moles to calculate empirical formula and deduce molecular formula of a compound/molecule
(starting with reacting masses or % composition)
Moles and the molar volume of a gas, Avogadro's Law
Reacting gas volume
ratios, Avogadro's Law
and Gay–Lussac's Law (ratio of gaseous
Molarity, volumes and solution
concentrations (and diagrams of apparatus)
do volumetric titration calculations e.g. acid–alkali titrations
(and diagrams of apparatus)
Electrolysis products calculations (negative cathode and positive anode products)
e.g. % purity, % percentage & theoretical yield, volumetric titration
apparatus, dilution of solutions
(and diagrams of apparatus), water of crystallisation, quantity of reactants
required, atom economy
Energy transfers in physical/chemical changes,
Gas calculations involving PVT relationships,
Boyle's and Charles Laws (this page)
Radioactivity & half–life calculations including
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