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1
Quantitative
chemistry
es
Chemistry was a late developer as a physical science. Newton was working on
the laws of physics more than a century before the work of the French chemist
Antoine Lavoisier (1743–1794) brought chemistry into the modern age. Chemical
reactions involve changes in smell, colour and texture and these are difﬁcult to
quantify. Lavoisier appreciated the importance of attaching numbers to properties
and recognized the need for precise measurement. His use of the balance allowed
changes in mass to be used to analyse chemical reactions. There are practical
problems with this approach as powders scatter, liquids splash and gases disperse.
It is essential to keep track of all products and it is perhaps signiﬁcant that
Lavoisier was a tax collector by profession. A quantitative approach to the subject
helped chemistry to develop beyond the pseudoscience of alchemy.
Mole calculations are used to work out
the relative amounts of hydrogen and
oxygen needed to launch the space
shuttle.
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This chapter is central to the practice of chemistry as it builds a foundation for
most of the numerical work in the course. The two threads to this chapter, a
description of the states of the matter and its measurement, are both based on
a particulate model of matter. The unit of amount, the mole, and the universal
language of chemistry, chemical equations, are introduced.
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Quantitative chemistry
Assessment statements
1.1 The mole concept and Avogadro’s constant
1.1.1 Apply the mole concept to substances.
1.1.2 Determine the number of particles and the amount of substance
(in moles).
1.2 Formulas
1.2.1 Deﬁne the terms relative atomic mass (Ar) and relative molecular mass (Mr).
1.2.2 Calculate the mass of one mole of a species from its formula.
1.2.3 Solve problems involving the relationship between the amount of
substance in moles, mass and molar mass.
1.2.4 Distinguish between the terms empirical formula and molecular formula.
1.2.5 Determine the empirical formula from the percentage composition or
from other experimental data.
1.2.6 Determine the molecular formula when given both the empirical
formula and experimental data.
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1.3 Chemical equations
1.3.1 Deduce chemical equations when all reactants and products are given.
1.3.2 Identify the mole ratio of any two species in a chemical equation.
1.3.3 Apply the state symbols (s), (l), (g) and (aq).
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1.4 Mass and gaseous volume relationships in chemical reactions
1.4.1 Calculate theoretical yields from chemical equations.
1.4.2 Determine the limiting reactant and the reactant in excess when
quantities of reacting substances are given.
1.4.3 Solve problems involving theoretical, experimental and percentage yield.
1.4.4 Apply Avogadro’s law to calculate reacting volumes of gases.
1.4.5 Apply the concept of molar volume at standard temperature and
pressure in calculations.
1.4.6 Solve problems involving the relationship between temperature,
pressure and volume for a ﬁxed mass of an ideal gas.
1.4.7 Solve problems using the ideal gas equation, PV  nRT .
1.4.8 Analyse graphs relating to the ideal gas equation.
1.5 Solutions
1.5.1 Distinguish between the terms solute, solvent, solution and
concentration (g dm–3 and mol dm–3).
1.5.2 Solve problems involving concentration, amount of solute and volume
of solution.
1.1
The mole concept and Avogadro’s
constant
Measurement and units
Scientists have developed the SI
system – from the French Système
International, to allow the scientiﬁc
community to communicate
eﬀectively both across disciplines
and across borders.
2
Scientists search for order in their observations of the world. Measurement is a
vital tool in this search. It makes our observations more objective and helps us ﬁnd
relationships between different properties. The standardization of measurement
of mass and length began thousands of years ago when kings and emperors used
units of length based on the length of their arms or feet. Because modern science
is an international endeavour, a more reliable system of standards, the Système
International is needed.
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A lot of experimental chemistry relies on the accurate measurement and recording
of the physical quantities of mass, time, temperature and volume. The SI units for
these are given below.
Property
Unit
Symbol for unit
mass
kilogram
kg
time
second
s
temperature
kelvin
K
volume
cubic metre
m3
pressure
pascal
Base SI units of some physical quantities
used in chemistry; volume is a derived
unit as it depends on length.
Pa or N m2
1 m3
Figure 1.1 A cube of 1 m3. This has
a volume of 100 cm  100 cm 
100 cm  1000 000 cm3. A typical gas
syringe has a volume of 100 cm3 so ten
thousand of these would be needed to
measure a gas with this volume.
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1m
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These units are however not always convenient for the quantities typically used in
the laboratory. Volumes of liquids and gases, for example, are measured in cubic
centimetres.
pl
1 cm
1m
mass
time
Unit
gram
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Property
m
1m
Symbol for unit
g
minute
min
degree celsius
°C
volume
cubic centimetre
cm3
pressure
atmosphere
atm
temperature
Other units used in chemistry.
Amounts of substance
Chemists need to measure quantities of substances for many purposes.
Pharmaceutical companies need to check that a tablet contains the correct amount
of the drug. Food manufacturers check levels of purity. In the laboratory, reactants
need to be mixed in the correct ratios to prepare the desired product. We measure
mass and volume routinely in the lab but they are not direct measures of amount.
Equal quantities of apples and oranges do not have equal masses or equal volumes
but equal numbers. The chemist adopts the same approach. As all matter is made
up from small particles (see Chapter 2), we measure amount by counting particles.
If the substance is an element we usually count atoms, if it is a compound we
count molecules or ions.
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A chemical species may be an
atom, a molecule or an ion.
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Quantitative chemistry
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A standard unit of amount can be deﬁned in terms of a sample amount of any
substance. Shoes and socks are counted in pairs, eggs in dozens and atoms in
moles. A mole is the amount of a substance which contains the same number of
chemical species as there are atoms in exactly 12 grams of the isotope carbon-12
(see Chapter 2). The mole is a SI unit with the symbol mol. The word derives from
the Latin for heap or pile.
A mole is the amount of a
substance which contains the
same number of chemical
species as there are atoms in
exactly 12 g of the isotope
carbon-12.
As the average relative mass of a carbon atom is actually greater than 12 owing to
the presence of heavier isotopes (see Chapter 2), one mole of the element carbon
has a mass of 12.01 g. The mass of one mole of atoms of an element is simply the
relative atomic mass (see page 41 in Chapter 2) expressed in grams. One mole of
hydrogen atoms has a mass of 1.01 g, one mole of helium 4.00 g and so on.
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Some elements exist as molecules not as individual atoms. The composition of a
molecule is given by its molecular formula. Hydrogen gas, for example, is made
from diatomic molecules and so has the molecular formula H2.Water molecules
are made from two hydrogen atoms and one oxygen atom and have the molecular
formula H2O. The relative molecular mass (Mr ) is calculated by adding the
relative atomic masses of the atoms making up the molecule.
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• The mass of one mole of a
species is called the molar
mass. It is the relative mass
expressed in g and has units
of g mol1.
• The molar mass of an
element which exists as
atoms is the relative atomic
mass expressed in g.
• The relative molecular mass
(Mr) is deﬁned as the sum of
the relative atomic masses of
the atoms in the molecular
formula. The molar mass (M)
of a compound is the relative
molecular mass expressed
in g.
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One mole of carbon-12.
Worked example
Calculate the relative molecular mass of ethanol C2H5OH.
Solution
The compound is made from three elements, carbon, hydrogen and oxygen.
Find the number of atoms of each element and their relative atomic masses from
the Periodic Table.
relative atomic mass
C
H
O
12.01
1.01
16.00
2
516
1
number of atoms in one
molecule of compound
Calculate the relative molecular mass of the molecule.
Relative molecular mass  (2  12.01)  (6  1.01)  16.00  46.08
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The molar mass of ethanol is 46.08 g mol1. The molar mass of a compound is
calculated in the same way as that of the elements. It is the relative molecular mass
in grams.
It is incorrect to use the term relative molecular mass for ionic compounds as they
are made from ions not molecules. The term relative formula mass is used. It is
calculated in the same way.
Once the molar mass is calculated and the mass is measured, the number of moles
can be determined.
Worked example
Calculate the amount (in mol) in 4.00 g of sodium hydroxide, NaOH.
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M  40.00 g mol1
4.00  0.100 mol
m  _____
n  __
M 40.00
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It is important to be precise when calculating amounts. One mole of hydrogen
atoms has a molar mass of 1.01 g mol1 but one mole of hydrogen molecules, H2,
has a mass of 2  1.01 2.02 g mol1.
Counting particles
Examiner’s hint: Pay attention
to decimal places. When adding or
subtracting, the number of decimal
ﬁgures in the result should be the same
as the least precise value given in the
data.
Number of moles (n)
mass (m)
 ______________
molar mass (M)
Solution
The relative atomic masses are Na: 22.99, O: 16.00 and H: 1.01.
The relative formula mass  22.99  16.00  1.01  40.00
The relative formula mass of
an ionic compound is the sum
of the relative atomic masses
of the atoms in the formula.
The molar mass of an ionic
compound is the relative
formula mass expressed in g.
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As the mass of an individual atom can be measured using a mass spectrometer
(see Chapter 2), the mole is a counting unit.
Examiner’s hint: Pay attention to
signiﬁcant ﬁgures. When multiplying
or dividing, the number of signiﬁcant
ﬁgures in the result should be the same
as the least precise value in the data.
Examiner’s hint: Use the accepted
shorthand to solve problems more
quickly in exams, for example n for
moles, m for mass and M for molar mass.
Avogadro’s constant (L) has the
value 6.02  1023 mol1. It
has units as it is the number of
particles per mole.
From mass spectrometer measurements: mass of 1 atom of 12C  1.99252  1023 g
12
Number of atoms in one mole (12 g of 12C)  ______________
1.99252  1023
 602 000 000 000 000 000 000 000.
This is a big number. It is called Avogadro’s number (L) and is more compactly
written in scientiﬁc notation as 6.02  1023. It is the number of atoms in one mole
of an element and the number of molecules in one mole of a covalent compound.
Examiner’s hint: Although
Avogadro’s constant is given in the IB
Data booklet you need to know its value
for Paper 1.
• If 6.02  1023 pennies were distributed to everyone currently alive they could all
spend two million pounds every hour, day and night of their lives.
• 6.02  1023 grains of sand would cover a city the size of Los Angeles to a height of
600 m.
• 6.02  1023 soft drink cans would cover the surface of the earth to a height of over
300 km.
This is the number of carbon atoms in a tablespoon of soot!
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Imagine you emptied a glass of labelled water molecules into the sea and then
allowed suﬃcient time for the molecules to disperse throughout the oceans. What is
the probability that you could catch one of the original molecules if you placed the
same glass into the sea? (see the Answer section). The magnitude of Avogadro’s constant
is beyond the scale of our everyday experience. This is one reason why ‘moles’ is a
challenging subject and why natural sciences don’t always come naturally!
Number of particles (N)
 number of moles (n) 
Avogadro’s constant (L)
N  nL
Although we could never count to L, even with the most powerful computer, we
can prepare samples with this number of atoms. The atoms are counted in the
same way as coins are counted in a bank; we use a balance:
3.01  1023 atoms of C  _12 mol  0.5  12.01 g  6.005 g.
3.01  1023 carbon atoms are ‘counted out’ when we prepare a sample of 6.005 g.
Worked example
 2.99 mol
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Solution
Use the shorthand notation: N  nL
N
n  __
L
1.80  1024
 __________
6.02  1023
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Calculate the amount of water, H2O, that contains 1.80  1024 molecules.
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Note the answer should be given to 3 signiﬁcant ﬁgures – the same precision as the
data given in the question. If the amount given was 1.8  1024 the correct answer
would be 3.0.
Worked example
Calculate how many hydrogen atoms are present in 3.0 moles of ethanol, C2H5OH.
Solution
In 1 molecule of ethanol there are 6 H atoms.
In 1 mole of ethanol molecules there are 6 moles of H atoms.
In 3 moles of ethanol there are 18 moles of H atoms.
Number of H atoms  18L  6.01  1023  18  1.08  1025
Exercises
1 Calculate how many hydrogen atoms are present in 0.040 moles of C2H6.
2 Calculate the molar mass of magnesium nitrate, Mg(NO3)2.
3 Calculate how many hydrogen atoms are contained in 2.3 g of C2H5OH (Mr  46).
4 The relative molecular mass of a compound is 98.0. Calculate the number of molecules in a 4.90 g
sample of the substance.
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1.2
Formulas
Finding chemical formulas in the laboratory
When magnesium is burned
in air its mass increases as it
is combining with oxygen.
The mass changes can be
investigated experimentally.
crucible
magnesium ribbon
pipe clay
triangle
heat
tripod
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Figure 1.2 This apparatus
reduces the chance of the product
escaping when magnesium is
heated.
Mass/g (0.001)
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Item
empty crucible
25.000
crucible with magnesium before heating
25.050
crucible with solid after heating
Magnesium burns in air to produce a
white residue of magnesium oxide.
Examiner’s hint: The uncertainties
in all the measurements should be
included in all data tables. This is
discussed in Chapter 11.
25.084
Mass/g (0.002)
Moles
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Element
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The masses of the magnesium and oxygen are then calculated.
25.050  25.000  0.050
oxygen
25.084  25.050  0.034
m
magnesium
0.050  0.0021
 ______
24.34
0.034  0.0021
 ______
16.00
The precision of the calculated value
is limited by the precision of the
mass measurements to 2 signiﬁcant
ﬁgures.
The empirical formula gives the
ratio of the atoms of diﬀerent
elements in a compound.
It is the molecular formula
expressed as its simplest ratio.
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The ratio of magnesium : oxygen atoms  0.0021:0.0021  1:1.
This is expressed as an empirical formula: MgO.
Worked example
A 2.765 g sample of a lead oxide was heated in a stream of hydrogen gas and
completely converted to elemental lead with a mass of 2.401 g. What is the
empirical formula of the oxide?
Solution
The mass loss is caused by a loss of oxygen.
Set out the calculation in a table.
Pb
O
mass/g
2.401
2.765  2.401  0.364
moles
2.401  0.01159
 _______
207.19
0.01159  1
 ________
0.01159
0.364  0.0228
 ______
16.00
0.0228  1.97  2
 ________
0.01159
simplest ratio
Empirical formula: PbO2
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Worked example
A hydrocarbon contains 85.7% by mass of carbon. Deduce the empirical formula.
Solution
A hydrocarbon is a compound of carbon and hydrogen only (see Chapter 10).
When data is given in percentages consider a 100 g sample.
C
Carry out your own carbon
hydrogen analysis.
Now go to www.heinemann.co.uk/
hotlinks, insert the express code
4259P and click on this activity.
H
mass/g
85.7
100  85.7  14.3
moles
85.7  7.14
 ______
12.01
14.3  14.16
 _____
1.01
simplest ratio
7.14  1
 _____
7.14
14.16  1.98  2
 ______
7.14
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Empirical formula: CH2
Exercises
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5 An oxide of sulfur contains 60% by mass of oxygen. Deduce the empirical formula.
6 Pure nickel was discovered in1751. It was named from the German word ‘kupfernickel’,
meaning ‘devil’s copper’. A compound of nickel was analysed and shown to have the following
composition by mass: Ni 37.9%, S 20.7 %, O 41.4 %. Deduce the empirical formula.
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The molecular formula shows
the actual number of atoms
of each element present in a
molecule.
pl
Molecular formula
m
The empirical formula does not give the actual number of atoms in the molecule.
The hydrocarbon in the previous worked example had an empirical formula of
CH2 but no stable molecule with this formula exists. The molecular formula,
which is a multiple of the empirical formula, can only be determined once
the relative molecular mass is known. This can either be measured by a mass
spectrometer (Chapter 2.2, page 40) or calculated from the ideal gas equation
(Chapter 1.4, page 24).
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State symbols indicate the state
of a substance:
(s) is for solid, (l) is for liquid, (g)
is for gas and (aq) is for
aqueous – dissolved in water.
The molecular formulas of some
compounds. The state symbols identify
the state at room temperature and
atmospheric pressure.
Substance
Formula
Substance
hydrogen
H2(g)
carbon dioxide
CO2(g)
oxygen
O2(g)
ammonia
NH3(g)
nitrogen
N2(g)
methane
CH4(g)
water
H2O(l)
glucose
C6H12O6(s)
Worked example
What is the empirical formula of glucose?
Solution
From the table above, the molecular formula  C6H12O6
Express this as the simplest ratio: CH2O.
8
Formula
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Worked example
The compound with the empirical formula of CH2 is analysed by a mass
spectrometer and its relative molecular mass found to be 42.09. Deduce its
molecular formula.
Solution
Empirical formula  CH2
Examiner’s hint: Practise empirical
formula calculations. All steps in the
calculation must be shown. ‘Keep going’
as errors are carried forward so that a
correct method in a later part of the
question is rewarded even if you have
made earlier mistakes. One common
problem is the use of too few signiﬁcant
ﬁgures in intermediate answers.
Molecular formula  CnH2n (where n is an integer)
Mr  42.09  (12.01n)  (2n  1.01)  14.03n
42.09  3
n  _____
14.03
Molecular formula: C3H6
Exercises
I only
II only
Both I and II
Neither I nor II
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B
C
D
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7 Which formula can be determined by only using the percent mass composition data of an
unknown compound?
I Molecular formula
II Empirical (simplest) formula
8 CFCs are compounds of carbon, chlorine and ﬂuorine which catalyse the depletion of the ozone
layer. The composition of one CFC is shown below.
chlorine
17.8%
1.5%
52.6%
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States of matter
28.1%
m
The value of its Mr is 135.
Determine the molecular formula of the CFC.
ﬂuorine
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hydrogen
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carbon
If you were hit with 180 g of solid water (ice) you could be seriously injured,
but you would be only annoyed if it was 180 g of liquid water. 180 g of gaseous
water (steam) could also be harmful. These three samples are all made from same
particles – 10 moles of water molecules. The difference in physical properties is
explained by kinetic theory. The basic
ideas are:
• All matter consists of particles (atoms or
molecules) in motion.
• As the temperature increases, the
movement of the particles increases.
The three states can be characterized in
terms of the arrangement and movement
of the particles and the forces between
them.
Most substances can exist in all three
states. The state at a given temperature and
pressure is determined by the strength of
the interparticle forces.
Solid: the particles are
closely packed in fixed
positions. The
interparticle forces
restrict the movement
to vibration about a
fixed position. Solids
have a fixed shape.
Figure 1.3 Comparison of the three
states of matter.
Liquid: the particles are
still relatively close
together. The interparticle
forces are sufficiently weak
to allow the particles to
change places with each
other, but their movement
is constrained to a fixed
volume. Liquids can change
shape but not volume.
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Gas: the interparticle
forces between the
particles are negligible;
they are zero for an ideal
gas (see Section 1.4). The
particles move freely
occupying all the space
available to them. Gases
have no fixed shape or
volume.
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Changes of state
The movement or kinetic energy of the particles depends on the temperature.
When the temperature increases enough for the particles to have sufﬁcient energy
to overcome the interparticle forces, a change of state occurs.
The heating curve below shows how the temperature changes as ice is heated from
40°C to steam at 140°C.
Figure 1.4 Heating curve for water.
The phase change (l) →(g) needs more
energy than (s)→(l) as all the interparticle bonds are broken during this
process.
60
40 solid
20 and
liquid
0
�20
�40
solid
liquid
melting
heat
To understand these changes it is more helpful to use the absolute or Kelvin scale
of temperature.
Consider a sample of ice at 40°C  233 K. The water molecules vibrate at this
temperature about their ﬁxed positions.
• As the ice is heated the vibrational energy of its particles increases and so the
temperature increases.
• At the melting point of 273 K the vibrations are sufﬁciently energetic for the
molecules to move away from their ﬁxed positions and liquid water starts to
form. The added energy is needed to break the bonds between the molecules
– the intermolecular bonds. There is no increase in kinetic energy so there is
no increase in temperature.
• As the water is heated, the
particles move faster and so
the temperature increases.
• Some molecules will have
sufﬁcient energy to break away
from the surface of the liquid
so some water evaporates.
• At the boiling point of water
there is sufﬁcient energy to
break all the intermolecular
bonds. The added energy is
used for this process, not to
increase the kinetic energy, and
so the temperature remains
constant.
• As steam is heated the
average kinetic energy of the
molecules increases and so the
temperature increases.
pl
Ice cubes melting over a period of 4
hours. The water is absorbing heat from
the surroundings to break some of the
intermolecular bonds.
boiling
es
temperature/°C
melting
point
liquid and gas
80
Sa
The absolute temperature of a
substance is proportional to the
average kinetic energy of its
particles.
10
100
m
Temperature diﬀerences
measured on either the Celsius
or Kelvin scale are the same.
Absolute zero  0 K. This is
the temperature of minimum
kinetic energy.
boiling
point
e
The kelvin is the SI unit of
temperature.
Temperature in kelvin 
temperature in °C  273
gas
120
pa
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The Celsius scale of temperature is
deﬁned relative to the boiling and
freezing point of water. The original
scale, developed by the Swedish
astronomer Anders Celsius, made
the boiling point of water zero and
the freezing point 100. This may
now seem absurd but the modern
scale is just as arbitrary.
140
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Worked example
In which sample do molecules have the greatest average kinetic energy?
A He at 100 K
B H2 at 200 K
C O2 at 300 K
D H2O at 400 K
Solution
Answer  D The sample at the highest temperature has the greatest kinetic energy.
The kinetic energy of a particle
depends on its mass (m) and
speed (v). All gases have the
same kinetic energy at the
same temperature, so particles
with smaller mass move at
faster speeds. Kinetic energy
 _12 mv2.
Some substances change directly from a solid to gas at atmospheric pressure. This
change is called sublimation.
Figure 1.5 The diﬀerent phase
changes.
melting
freezing
es
liquid
Sublimation is the conversion of a solid
directly to a vapour state. Dry ice, CO2(s),
sublimes when it is mixed with water
producing thick clouds of fog.
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solid
pa
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lim
sub
g/
ilin ng
bo ati
r
po
eva
ng
nsi
de
con
ati
on
rev
sub er
lim se
ati
on
gas
The coldest place in nature is in
the depths of outer space at a
temperature of 3 K. The 2001 Nobel
Prize in Physics was awarded to
a team who cooled a sample of
helium atoms down to only a
few billionths (0.000 000 001) of a
degree above absolute zero. Under
these conditions helium atoms
crawl along at a speed of only
about 3 mm s1!
Exercise
9 When a small quantity of perfume is released into the air, it can be detected several metres away
in a short time. Use the kinetic theory to explain why this happens.
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We can smell perfumes because they
evaporate at body temperature. In a
mixture of molecules the most volatile
evaporate ﬁrst and the least volatile
last. The skill of the perfumer is to use
the laws of chemistry to make sure that
chemicals are released steadily in the
same proportions.
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es
The Celsius scale gives an artiﬁcial
description of temperature and the
Kelvin scale a natural description.
Do the units we use help or hinder
our understanding of the natural
world?
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Worked example
A ﬂask contains water and steam at boiling point. Distinguish between the two
states on a molecular level by referring to the average speed of the molecules and
the relative intermolecular distances.
Solution
As the two phases are at the same temperature they have the same average kinetic
energy and are moving at the same speed. The separation between the particles in
a gas is signiﬁcantly larger than that in a liquid.
Exercise
10 Which of the following occur when a solid sublimes?
I The molecules increase in size.
II The distances between the molecules increase.
A
B
C
D
12
I only
II only
Both I and II
Neither I nor II
UNCORRECTED PROOF COPY