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3.1 An Introduction to
Entropy
Entropy
and the direction of physical change or chemical change
Why do chemical
reactions occur? Why does magnesium react with hydrochloric acid? Why
doesn't hydrogen react with magnesium chloride solution? Why does ice
melt at 273 K (0oC)?
These are questions not concerned
with the rates of change (kinetics), even though this may have a bearing
on their apparent spontaneity, but the above questions have everything
to do with the energy changes which are possible i.e. what changes are
feasible and why? and what is the 'energetic' driving force of these
changes?
3.1a Events
that happen, are the ones most likely to happen i.e. the most probable
outcome!
This sounds easy? sounds
logical? obvious? BUT is it
always obvious? Hmm! I'm afraid not.
For most exothermic reactions the
very negative enthalpy change is usually indicative that the
reaction will occur if initiated in some way
i.e. just mixing chemicals e.g.
mixing an acid and alkali,
H+(aq)
+ OH–(aq) ==> H2O(l)
ΔHθneut,298K
= –57.1 kJ mol–1
or applying a means of ignition to a
combustible mixture e.g. burning methane,
C3H8(g) + 5O2(g) ==> 3CO2(g) +
4H2O(l) ΔHθcomb,298K(propane)
= –2219 kJ mol–1
and frankly, you would be a bit surprised if these reactions didn't
happen.
At one time it was thought
that only exothermic reactions where spontaneous and the more heat
released to the surroundings the more likely the change would take
place BUT although in the minority, there are plenty of spontaneous
endothermic changes in which heat is absorbed from the surroundings
and a temperature fall can be recorded with a thermometer e.g.
(i) The dissolving of the
salt ammonium nitrate in water
NH4NO3(s)
+ aq ==> NH4+(aq) + NO3–(aq)
ΔH = +25.8 kJ mol–1
(ii) The reaction between
hydrated cobalt(II) chloride and sulfur dichloride oxide (thionyl
chloride) which should be done in a fume cupboard due to its
spectacular gas producing performance BUT not due to its
exothermicity!
CoCl2.6H2O(s)
+ 6SOCl2(l) ==> CoCl2(s) + 6SO2(g)
+ 12HCl(g) ΔH = +? kJ mol–1
ΔHθreaction,298 =
∑ΔHθf,298(products)
– ∑ΔHθf,298(reactants)
ΔHθreaction,298 =
{ΔHθf(CoCl2(s))
+ 6xΔHθf(SO2(g))
+ 12xΔHθf(HCl(g))}
– {ΔHθf(CoCl2.6H2O(s)) +
6xΔHθf(SOCl2(l))}
ΔHθreaction
= {–326 –6x297 –12x92.3} – {–2130 –6x206} = +150 kJ mol–1
This rapid reaction is
very spontaneous and very cooling i.e. it is clearly and
endothermic reaction! The huge volumes of extra gas
molecules lead to a very large increase in entropy (a very +ΔS)
which will explained in detail below – but remember this
example.
(iii)
However, contrast these two spontaneous
reactions initiated at room temperature with the decomposition of
limestone, whose is
ΔH is +179 kJ mol–1,
but only becomes feasible–spontaneous at over 900oC!
The thermal
decomposition of calcium carbonate.
CaCO3(s)
==> CaO(s) + CO2(g) ΔH =
+179 kJmol–1
Unlike examples (i) and
(ii), (iii) is not feasible at room temperature BUT will go
spontaneously and commercially feasible at temperatures above 1000oC where the
reaction is just as endothermic as at room temperature!
Clearly the feasibility of
process or change cannot solely depend on the enthalpy change and in
order to explain further you need to consider the concept of
ENTROPY and some of its ramifications.
Concept: The entropy of a system is a measure
of the number of ways a system can be arranged. This can been spatial
arrangement–distribution of the particles or the distribution of all
the quanta of energy internal to an atom, ion, molecule etc.
In Parts 1 Thermochemistry
and Part 2. on the Born–Haber cycle we relied exclusively on the 1st
Law of thermodynamics i.e. energy is conserved. In this section Part
3. we need get into the 2nd second law of
thermodynamics states that the entropy of a system not at
equilibrium will increase to give the maximum entropy state i.e. it
will change to the most probable state. Its a bit abstract and
takes a bit of getting used to and beware of simple explanations
relying too heavily on relative order and disorder – though this
idea has its place in coming to terms with the concept of entropy.
SO BEWARE – increase in entropy
is often described as increasing chaos–disorder
– but always try express your answers in terms of ... 'the more ways of
arranging the outcome' – but this will only become clearer as you read
further on. However in many ways it is conceptually useful at first
to think of the greater the number of ways of ... the greater the
entropy of the system and expressed in words as more
disordered, spread out, mixed up or the more chaotic etc. etc.
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3.1b The more ways an event can happen, the more
probable is that event and the higher the entropy of the system
If there are more ways to arrange a
'system' the more likely is that 'system' situation will be attained.
You may be talking about the number of possible ways the particles
in a system can be arranged or the number of ways the quanta of
energy can be distributed i.e. the internal energy of the particles
in a system (atoms, ions, molecules) which may be in the form of
occupied rotational, vibrational or electronic quantum levels –
remember the possible levels are governed by quantum rules. Each
possible way of arranging a system is often referred to as a
microstate. A system will adopt the 'arrangement' that offers
the maximum statistical probability of microstates.
You don't need to know
this, but entropy is defined in a mathematical way as
S = klnW in which S = entropy, k = Boltzmann constant and lnW = the
logarithm of the number of microstates possible for a
system.
This equation is carved
on the tombstone of Ludwig Boltzmann in recognition of his
incredible intellectual contribution to the development of
quantum physics and statistical thermodynamics. He committed
suicide on Sept 5th 1906 after suffering bouts of depression and
vilification from other minority physicists such as Ernst Mach
who refused to accept the theories proposed by Boltzmann and
others.
3.1c The entropy of a system
not at equilibrium will always try to
increase, determining the direction of change
 This is another way of
expressing the 2nd law of thermodynamics. The use of the word
equilibrium should be taken in its widest sense i.e. a system in
equilibrium shows no net change on the macroscale.
e.g. gases will always diffuse into each other
(e.g. bromine–air),
the random motion of the molecules ensures that the most likely
situation occurs i.e. multiple random distribution situations are possible
and the outcome will be the most mixed up system possible. There
is zero probability that they will unmix! In other words the outcome
here is the situation that leads to the maximum possible ways of
spatially arranging the particles.
Miscible liquid layers will mix
in the same way as gases do, for exactly the same reason. However,
unlike in gases, intermolecular forces can intervene to restrict the
outcome. For example, on mixing the immiscible liquids oil and water,
the two layers reform after shaking. Why? The strong hydrogen
bonding between water molecules is stronger than any possible
oil–water interaction.
immiscible liquid layers
 VERSUS miscible liquids
3.1d The entropy
of substances increases gas >
liquid >> solid Why?
From left to right there is
an increasing effect of inter–particle forces restricting the ways
the particles or their kinetic energies can be distributed. In a gas
the molecules are free to move randomly so there are endless
permutations of spatial arrangement. In a liquid there is random
movement but it is restricted by the intermolecular forces and
clumps of molecules exist albeit for a tiny fraction of time before
bombardment breaks them up – BUT clumps of molecules amount to tiny
transient pockets of lower entropy. In a solid the
particles can hardly move around so the spatial position possibilities
are very limited.
Note: The above arguments
only discuss spatial positioning and freedom of movement, but the distribution of energy
e.g. kinetic is another contribution to the entropy of a
substance and this factor is discussed in the next paragraph
3.1e. However in terms of the order/disorder solids are clearly
highly ordered (short range and long range), liquids have a
little short range order and gases are totally randomised at any
given instant in time with virtually no inter–particle order at
all. So in terms of possible 'arrangements' i.e. entropy, the
general expected entropy order is not surprisingly is
gases > liquids > solids. This is a general trend for a
range of substances and a specific trend for every individual
substance. The entropy of a substance increases with temperature
(see next section 3.1e and associated graph) and the most
dramatic increases occur when there is a change in state (s) ==>
(l) and (l) ==> (g).
See section Thermodynamics
section 3.2 for some examples of
entropy values
The production of a gas in reactions is quite
a driving force for 'unfavourable' reactions e.g. thermal decompositions
like limestone ==> lime OR the cracking of alkanes are both very
endothermic because the net increase in gas molecules considerably increases
the entropy of a system driving the reaction in
the decomposition direction so at higher temperatures many endothermic reactions
become feasible. More on feasibility later.
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3.1e Heat
Capacity, energy distribution and entropy
The term specific
heat capacity indicates quantitatively the energy required
to increase a specified unit of mass (g or kg) of a substance by
a specific unit of temperature (K or oC). The
specific heat capacity at constant pressure is denoted by Cp.
For water it is 4.2 J g–1 K–1 (or 4200 J
kg–1 K–1), and the former is more
convenient to use in laboratory calorimeter experiment
calculations.
What
happens when you heat a substance? Why does the entropy of a
substance increase with increase in temperature? See diagram on
the left. Tm = melting point and Tb =
boiling point. At each change of state there is a dramatic rise
in the entropy S of a substance and in between a steady increase
in the entropy of a substance.
The temperature of a
substance depends on the kinetic energies of the particles –
that is the motion of the particles or its component atoms.
The more energetic the particles in terms of KE the higher the
temperature and the 'hotter' it feels! So, how does this
relate to the concept of entropy?
Notice that the entropy
graph has its origin at 0,0 (S = 0 when T = 0). Theoretically at
0K all vibrations or rotations etc. have stopped and if a
perfect crystal was formed its entropy is zero for this sole
possible 'frozen' microstate.
The 3rd Law of
Thermodynamics states that at absolute zero all perfect crystals
have zero entropy.
S = klnW = 0
if W = 1 possible microstate
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Molecules possess three forms
of kinetic energy:
Translation – moving from
one place to another as in a gas or liquid (virtually no translation in
solids).
Rotation – the molecule or
a grouping in the molecule spinning around.
Vibration – all atoms
constituting bonds vibrate in some way either by stretching–compression
along the bond axis or bending–relaxing movement.
All of these forms of energy
are quantised, that is, only specific KE energies of translation,
rotation and vibration are allowed and the more energy absorbed in
raising the temperature of a substance the more of the higher quantum
levels of KE are accessible i.e. more ways in which the energy can be
distributed i.e. an entropy increase. Note that we are now talking about
the distribution of energy and not just the possible spatial
arrangements and freedom to move around.
The order of these three types of
quantum level is vibrational > rotation > translational
The translational quantum levels
are so close together that virtually any measurable KE or velocity is
likely to be observed – but it does seem strange to think of this motion
in a quantised way. Although not required at advanced level, it is
possible to show by direct experiment that freely moving electrons,
atoms or molecules behave as a wave as well as particle – but the
quantum world is 'wacky one' so don't expect the reality of our macro
physical world to match with the quantum world. However, the
quantisation of vibrational and rotational levels shouldn't present a
conceptual problem – just think that the vibrations and rotations can
only occur at specific frequencies determined by the quantum
number rules – which you don't have to worry about. Also, remember
microwave absorption is due to rotational quantum level changes and
infra–red absorption is due to vibrational level changes.
As you heat a substance more
and more of these 'kinetic energy' quantum levels can be accessed as the
temperature of the substance increases, therefore more ways to
distribute the energy, therefore a higher entropy state is obtained.
3.1f
Electronic energy levels and entropy
If you continue to heat a
substance eventually the kinetic energy of collisions–vibrations etc. is
sufficient to cause electronic energy quantum level changes.
Molecules can be raised to an 'excited' state when an electron is raised
to a higher quantum level, and, if raised sufficiently highly, bond
breaking will occur. This does not necessarily increase the temperature
of substance because electronic quantum level changes do not affect the
motion of the molecules i.e. the kinetic energy of the molecules.
However, what it does mean is that yet more 'energy levels' are
available in which to distribute the energy absorbed i.e. yet again this
will lead to a higher entropy level being attained.
So the complete order of the four
types of quantum level is electronic > vibrational > rotation >
translation and, depending on the physical state and temperature of
a substance, there is continuous interchange of energy between the
possible energy levels e.g. via particle collision – remember energy
cannot be created or destroyed, but can be changed in 'form' or
'distribution'.
Incidentally, excited
molecules can be produced by e.g. uv light, but if no bond fission
occurs, the molecules 'relax' by changing the absorbed electronic
quantum level energy into some form of kinetic energy – vibrational,
rotational and translational, so in the end the temperature of the
substance is increased. Another example is holding your hand in
sunlight – it becomes hotter because the molecules of your skin
absorb IR and higher vibrational levels are accessed – in the end
the molecules relax and the absorbed energy ends up as translational
kinetic energy – but either way, the temperature of your skin
increases as does its entropy!
There is a further conceptual
quantum level complication because these types of quantum levels,
although specific for a given bond or molecule etc. are not independent
of each other. Each electronic level has its own set of associated
vibrational levels, each vibrational level has its own set of associated
rotational levels and each rotational level has its own set of
translational levels. BUT don't worry about this! the point is that when
a system–substance absorbs energy, the energy will be distributed in the
maximum possible number of ways i.e. to attain the highest possible
entropy state.
3.1g A
summing up so far before we get into examples and then calculations!
We have considered the concept of the entropy
state of a substance in terms of :–
A substance/system not at
'equilibrium' (i.e. a 'no net change state') will try to attain the
highest entropy state by physical or chemical change. We can envisage
the entropy of a substance in terms of the spatial distribution of the
particles in the substance and the distribution of the energy of
individual particles between the available translational, rotational,
vibrational and electronic energy levels of each particle.
Therefore, entropy is a measure
of the number of ways particles can be spatially distributed AND the
number of ways the quanta of energy can be distributed i.e. the way the
energy is 'arranged' in the system.
Energetics-Thermochemistry-Thermodynamics Notes INDEX
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QUICK INDEX for
Energetics:
GCSE Notes on the basics of chemical energy changes
– important to study and know before tackling any of the three Advanced Level
Chemistry pages Parts 1–3 here
* Part 1a–b
ΔH Enthalpy Changes
1.1 Advanced Introduction to enthalpy changes
of reaction,
formation, combustion etc. : 1.2a & 1.2b(i)–(iii)
Thermochemistry – Hess's Law and Enthalpy
Calculations – reaction, combustion, formation etc. : 1.2b(iv)
Enthalpy of reaction from bond enthalpy
calculations : 1.3a–b
Experimental methods
for determining enthalpy changes and treatment of results and
calculations :
1.4
Some enthalpy data patterns : 1.4a
The combustion of linear alkanes and linear
aliphatic alcohols
:
1.4b Some patterns in Bond
Enthalpies and Bond Length : 1.4c
Enthalpies of
Neutralisation : 1.4d Enthalpies of
Hydrogenation of unsaturated hydrocarbons and evidence of aromatic
ring structure in benzene
:
Extra Q page
A set of practice enthalpy
calculations with worked out answers **
Part 2 ΔH Enthalpies of
ion hydration, solution, atomisation, lattice energy, electron affinity
and the Born–Haber cycle : 2.1a–c What happens when a
salt dissolves in water and why? :
2.1d–e Enthalpy
cycles involving a salt dissolving : 2.2a–c
The
Born–Haber Cycle *** Part 3
ΔS Entropy and ΔG Free Energy Changes
: 3.1a–g Introduction to Entropy
: 3.2
Examples of
entropy values and comments * 3.3a ΔS, Entropy
and change of state : 3.3b ΔS, Entropy changes and the
feasibility of a chemical change : 3.4a–d
More on ΔG,
free energy changes, feasibility and
applications : 3.5
Calculating Equilibrium
Constants from ΔG the free energy change : 3.6
Kinetic stability versus thermodynamic
feasibility - can a chemical reaction happen? and will it happen?
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