00:00:00
in this session we'll be looking at how
00:00:02
I ins move through solutions and how
00:00:04
this affects their ability to transport
00:00:06
charge through that solution so when we
00:00:09
think about ion transport there's a
00:00:10
number of factors which affect their
00:00:12
ability to move through the solution
00:00:14
ions move through an electric field and
00:00:16
any electric field has magnitude and
00:00:18
direct all of these exerted coulombic
00:00:19
force on our ion but as it's moving
00:00:21
through the solution it experiences
00:00:23
resistance it experiences movement
00:00:24
against officer ions and the experience
00:00:26
is solvent drag we covered this in terms
00:00:28
of an ionic atmosphere and salvation'
00:00:30
shells and these are the models that
00:00:33
we've explored already without the
00:00:35
electric field however ions will not
00:00:37
migrate they won't move through solution
00:00:39
and while it seems like we're stating
00:00:40
the obvious it is important to remember
00:00:42
that the electric field is absolutely
00:00:44
vital for electrochemistry so
00:00:45
understanding how it works how it
00:00:47
behaves is a necessary component of our
00:00:50
study the electric field itself is
00:00:53
measured in volts per meter so a
00:00:54
potential difference over a distance so
00:00:57
if we think about what's going on
00:00:59
between two uniform plates the electric
00:01:02
field we assume is uniform between these
00:01:04
plate electrodes by convention it
00:01:06
travels from positive to negative so our
00:01:09
field goes from the positive electrode
00:01:10
to the negative electrode that's simply
00:01:13
a convention this field exerts a force
00:01:16
on the ions in solution which we can
00:01:18
fairly easily quantify where the force
00:01:20
is the charged number of the ion
00:01:22
multiplied by the charge on an electron
00:01:24
and the electric field itself this is
00:01:26
the force which is driving our migration
00:01:28
forward it's the thing that's pushing
00:01:29
our ions through the solution now one
00:01:33
thing I'd like you to make sure you're
00:01:34
happy with is that the units are indeed
00:01:36
congruent here so make sure you
00:01:37
understand the units of force the units
00:01:39
of charge and the units of electric
00:01:41
field and make sure that you understand
00:01:43
that these units are indeed congruent
00:01:45
the shape of the electrode is also
00:01:47
important in defining the electric field
00:01:49
so if we think about paddle electrode so
00:01:51
there's plate electrodes we've been kind
00:01:53
of looking at if we look at them from
00:01:54
above we imagine them as two plates our
00:01:56
positive and negative but if we think
00:01:59
about rod electrodes sticking in to the
00:02:01
solution they would have a circular
00:02:03
cross-section between these two diagrams
00:02:04
we're going to have a fixed potential
00:02:06
difference and as we explore this we're
00:02:08
looking at the electric field between
00:02:09
them so quick convention a longer field
00:02:12
line represents
00:02:13
a weaker field we justified this as the
00:02:15
electric field along a field line is
00:02:17
equal to the change in voltage across
00:02:19
that line divided by the length of it or
00:02:20
if we're looking in solution we look at
00:02:23
the potential this Delta Phi divided by
00:02:26
the length over which it acts okay so
00:02:29
these are the kind of things that we're
00:02:30
looking at with our electric field if we
00:02:31
imagine the electric field in our plate
00:02:34
electrodes the electric field has
00:02:37
straight lines between the plates so
00:02:39
there's a uniform field between these
00:02:41
plates but outside we start to get
00:02:43
curving of these field lines so we get
00:02:45
this variable field longer line weaker
00:02:48
field because of this field equation
00:02:50
that we have here if we think of rod
00:02:52
electrodes however they're not uniform
00:02:54
anymore those lines curved through the
00:02:57
solution which means no matter how we
00:02:59
were act with a with a rod electrode we
00:03:01
have a variable field so this is
00:03:04
imagining the shape of the electric
00:03:05
field and understanding how these things
00:03:06
will potentially behave we now want to
00:03:10
think about the drag forces on our ions
00:03:12
now remember the drag forces impede that
00:03:14
ionic flow in this case the diagrams
00:03:16
representing the effect of solvation
00:03:18
shells as well as the ionic atmosphere
00:03:20
it's affected by a number of factors so
00:03:23
we've got the things attached the iron
00:03:25
but we also need to think about the
00:03:26
solvent viscosity this carries a simple
00:03:28
eater here we think of the radius of the
00:03:31
migrating species whether that's the
00:03:32
radius of the iron plus the solvation
00:03:34
shell this is all summed up in a term
00:03:37
called the hydrodynamic radius it
00:03:40
includes the solvation shells and
00:03:41
includes any averaging effects that goes
00:03:43
on there there's also the speed of
00:03:46
motion the faster an ion moves the more
00:03:48
drag you would expect it to experience
00:03:49
and all of these factors are summed up
00:03:52
in Stokes law which relates the force on
00:03:55
our iron to the viscosity of solvent and
00:03:57
the hydrodynamic radius we'll come back
00:04:01
to this later on it's important to
00:04:02
recognize that the drag force on our
00:04:04
iron is a combination of all of these
00:04:06
factors together okay so let's think
00:04:09
about the drift speed of our ions what
00:04:11
is the maximum speed that our ions
00:04:13
travel so this is the constant speed
00:04:15
reached when all the forces balanced so
00:04:17
when the electrical force pushing the
00:04:18
iron through solution balances the drag
00:04:20
force well we've got our definitions
00:04:22
we've defined the drag force we've
00:04:24
defined the electrical force and all we
00:04:26
need to do is just
00:04:27
simply equate them this allows us to
00:04:29
find the ionic drift speed s as a
00:04:32
function of our electric field solvent
00:04:33
factor and the hydrodynamic radius
00:04:35
another way of looking at this is to
00:04:37
consider ionic mobility so if we define
00:04:40
another term ionic mobility of U this is
00:04:42
simply a measure of how the iron
00:04:44
responds to the electric field this u
00:04:46
term is simply everything in this
00:04:48
equation except for the electric field
00:04:50
so it simply says how the speed varies
00:04:53
as we vary the electric field mobility
00:04:56
is a microscopic property so remember we
00:05:00
talked about conductivity before
00:05:01
conductivity looks at the entire solvent
00:05:03
it's a bulk property of the solvent but
00:05:06
the ionic mobility just looks at the
00:05:07
individual ions and we have many
00:05:09
different factors which affect the
00:05:11
mobility of our solvent so we looked at
00:05:13
a Onix size so we'd think that would
00:05:16
affect our ionic radii if we look at our
00:05:18
Group one ions lithium is absolutely the
00:05:22
smallest iron but the ionic size isn't
00:05:24
the main important main factor we need
00:05:26
to look at the hydration shells so we
00:05:29
spoke before about how lithium even
00:05:31
though it's the smallest ion has the
00:05:33
largest primary hydration shell so we
00:05:37
have these two factors we have
00:05:38
conductivity which is a bulk property we
00:05:40
have mobility which is a microscopic
00:05:42
property so can we link the two together
00:05:44
it would seem logical that we should do
00:05:46
this because ultimately the bulk
00:05:48
property that we observe the
00:05:49
conductivity must be a factor of the
00:05:52
mobility and it turns out that yes we
00:05:54
absolutely can relate these the
00:05:57
derivation of this is outside the scope
00:05:59
of this course however when we work it
00:06:01
through we find that there is a
00:06:02
connection between our limiting molar
00:06:04
conductivity that we spoke about the
00:06:06
last few videos and the mobility that
00:06:09
you factor of our ion and there's a very
00:06:13
simple relationship between them if we
00:06:15
simply sum up the contribution from the
00:06:17
positive ions and the negative ions we
00:06:19
can derive the limiting molar
00:06:22
conductivity
00:06:24
the next factor we wish to look at is
00:06:26
transport numbers as said before we have
00:06:28
looked extensively at bulk properties in
00:06:30
terms of conductivity but we need to
00:06:33
think about what's going on we need to
00:06:35
think about how the charge moves around
00:06:36
in our solution so how do we look at the
00:06:39
movement of the individual ions when all
00:06:41
we can measure is the external current I
00:06:44
so what we need to do is need to
00:06:45
consider both sides of this particular
00:06:48
cell we need to factor in the external
00:06:51
circuit going through the ammeter and we
00:06:53
need to consider the internal circuit
00:06:55
going through the solvent because the
00:06:57
total current will be the overall sum of
00:06:59
all components so inside the solution we
00:07:02
have an ions carrying our negative
00:07:05
charge across we have our cations
00:07:08
carrying and positive charge to the
00:07:10
other electrode and this results in a
00:07:13
flow of electrons around the external
00:07:15
circuit the transport number simply
00:07:18
represents the share of the total
00:07:20
measured current carried by any
00:07:22
individual ion so it's a fraction so the
00:07:25
transport number simply indicates how
00:07:27
much of the current is carried by the
00:07:28
cation and how much is carried by the
00:07:30
anion ions rarely carry an equal share
00:07:33
of the current it's tempting to believe
00:07:34
that since we have an equal number of
00:07:36
positive charges as negative charges
00:07:38
that each would carry an equal share but
00:07:41
it depends on the mobility if something
00:07:42
is more able to move through the
00:07:44
solution if it's able to move more
00:07:46
freely through the solution it can carry
00:07:48
its charge faster through the solution
00:07:50
and can deposit it quicker at the
00:07:52
electrode so we need to consider this
00:07:54
factor as well it's a tricky thing to
00:07:57
visualize but let's start building up
00:07:59
how charge is transported through a
00:08:01
solution if we look at our standard
00:08:04
solution we have our two electrodes here
00:08:05
we're going to do a few things to this
00:08:07
system what firstly we're going to
00:08:09
divide it into sections so we have an
00:08:11
area close to the anode we call the
00:08:13
anode area so this is the area space
00:08:16
which is mainly affected by the anode
00:08:17
likewise we have an area of space which
00:08:20
is affected by the cathode and in the
00:08:22
middle we have a bulk area which is not
00:08:24
affected by either the anode or the
00:08:26
cathode the next thing we need to say is
00:08:28
that the overall charge in each section
00:08:30
must be zero so whenever we have a
00:08:33
solution in equilibrium the charge is
00:08:35
moving around we must have
00:08:37
an equal charge distribution so in the
00:08:39
anode area we have an equal number of
00:08:41
positive and negative charges in the
00:08:43
bulk area we also have an equal quantity
00:08:45
of each and likewise in the cathode area
00:08:47
we have an equal quantity of each so
00:08:49
this sets up our model to start
00:08:50
consideration so let's move into what
00:08:54
happens when we allow a current to flow
00:08:56
when we connect those we're going to say
00:08:58
that the cation is twice as mobile as
00:09:01
the anion okay that's the first thing
00:09:04
we're going to say if we have a very
00:09:05
small nimble cation but a very big bulky
00:09:08
anion then we would expect it to move
00:09:11
quicker and it would resolve charge
00:09:12
imbalance quicker so let's think about
00:09:15
what happens the cations they pick up
00:09:18
let's start with these cations pick up
00:09:20
three electrons from the cathode and are
00:09:22
removed from solution so these three
00:09:24
pick up another electron and are removed
00:09:26
from solution likewise to keep the
00:09:29
external current equal the electrons
00:09:32
flow through the circuit and
00:09:33
redistribute themselves which means the
00:09:35
anions have to deposit three electrons
00:09:38
at the anode these are also removed from
00:09:41
solution so what we've ended up with is
00:09:43
a charge imbalance so we need to rectify
00:09:47
this charge imbalance in each area the
00:09:48
bulk area is still charged balanced but
00:09:51
the anode area is not and neither is the
00:09:53
cathode area so it would seem sensible
00:09:54
that the rectification of charge comes
00:09:56
from the bulk so how does this happen
00:09:59
well we've said that the cations are
00:10:03
considerably more mobile than the anions
00:10:05
so let's say they flow twice as fast so
00:10:07
we would expect two cations to flow in
00:10:10
here to make a three plus and we would
00:10:13
get one anion flowing out of this area
00:10:15
into the bulk giving us an overall zero
00:10:18
charge in the cathode area we would
00:10:20
expect the same thing to happen in the
00:10:22
anode area and this gives us an equal
00:10:26
distribution of charge in each area
00:10:28
there are several ways of accomplishing
00:10:30
this we can have we can have all three
00:10:32
charges being balanced by one type of
00:10:34
ion or we can have it going the other
00:10:36
way as well in this example we have
00:10:39
simply said that our anion is less
00:10:40
mobile so only one anion moves for a
00:10:44
pair of cations so what we can say here
00:10:47
then is that the
00:10:49
cation is carrying twice as much current
00:10:51
as the anion but notice what's happening
00:10:54
we get a depletion of charge happening
00:10:58
in the anode area and an accretion of
00:11:00
charge eventually in the cathode area
00:11:02
this will become more significant as we
00:11:05
go through the course experiments such
00:11:07
as this allows to determine ionic
00:11:09
transport numbers if we look at a table
00:11:11
of data it's immediately quite dry to
00:11:13
look at but let's just quickly sort out
00:11:16
a few things
00:11:16
this solution normality is simply a way
00:11:19
of reflecting the basicity of the ions
00:11:21
and the only thing we need to worry
00:11:22
about is it's related to the
00:11:23
concentration that's related to
00:11:25
concentration it's related to activity
00:11:27
so let's look at what happens as we go
00:11:29
down a particular group so let's look at
00:11:31
the barium chloride what we see here is
00:11:34
we have a gradual decreasing in
00:11:37
transport number so that means the
00:11:39
barium is carrying less and less of the
00:11:42
charge as its concentration increases
00:11:44
that goes along what we were saying
00:11:46
about ionic atmospheres and the
00:11:48
solvation shells so as we increase the
00:11:51
number of solvation shells the more
00:11:52
resistance there is to that flow so
00:11:53
barium is less able to carry the charge
00:11:56
so we see that decrease if we now look
00:11:59
across the next three columns looking at
00:12:01
the group one cations they follow the
00:12:04
mobility we mentioned earlier lithium
00:12:06
chloride has a bigger solvation shell so
00:12:09
is able to move through less experience
00:12:11
more drag so its mobility is decreased
00:12:13
which means it's less able to carry that
00:12:15
charge if we look at sodium it's
00:12:18
slightly better able and potassium it's
00:12:20
about 50/50 okay so we can see all of
00:12:23
these values here are with the exception
00:12:26
of lithium around about 1/2 for a lot of
00:12:29
them so there's if we've got transport
00:12:31
number about 1/2 that means there's an
00:12:33
equal share of current being carried by
00:12:35
each of the anions and cations however
00:12:37
let's turn our attention now to hydrogen
00:12:39
chloride or HCl in this case we're
00:12:43
seeing a proton transport number which
00:12:45
is exceptionally high so everything else
00:12:47
is about 1/2 plus or minus a little bit
00:12:50
but the proton transport number is huge
00:12:52
it's carrying over 80 percent of the
00:12:54
current why is it the protons are able
00:12:57
to carry this and it comes down to the
00:13:00
phenomenon known as the growth house
00:13:01
mechanism
00:13:02
we covered this very briefly in first
00:13:04
year but let's look at it again remember
00:13:07
protons appear to be unusually mobile
00:13:09
they are moving much quicker through the
00:13:11
solution than we would otherwise predict
00:13:13
and carrying a massive share of the
00:13:14
current so let's think about what is
00:13:18
happening we're doing these measurements
00:13:19
in water so what is water well h2o if we
00:13:23
populate our system with more and more
00:13:24
water molecules and then think about
00:13:26
what happens as we introduce hydrogen
00:13:28
ion well remember this is inherently
00:13:30
unstable it will associate with a water
00:13:32
molecule to create the hydronium ion
00:13:35
this can then undergo a proton exchange
00:13:37
with a nearby water molecule which can
00:13:40
then go undergo another exchange and
00:13:43
another one and as you can see the
00:13:45
hydroxo Nia mine appears to be moving
00:13:47
through solution which will then deposit
00:13:49
a proton as an electrode so it's not
00:13:53
that an individual proton is
00:13:55
particularly mobile it's just we have
00:13:57
this grotus exchange we have this
00:13:59
continuous proton transfer throughout
00:14:01
the solvent and that's what's driving
00:14:03
our proton transport number makes it
00:14:05
much higher than we would otherwise
00:14:06
expect whenever we consider what happens
00:14:10
at an electrode anion comes in it drops
00:14:14
off its electrons or picks electrons up
00:14:15
in forms and neutral species it then
00:14:17
needs to move away from the electrode in
00:14:20
order to let other ions at the electrode
00:14:22
this process is diffusion and we need to
00:14:26
think about how diffusion behaves as
00:14:27
well to understand our electric
00:14:28
processes diffusion is fundamentally a
00:14:31
transport across a concentration
00:14:32
gradient so if we have a system where we
00:14:36
have a very concentrated analyte at one
00:14:38
end and free solution at the other end
00:14:41
this is going to diffuse across that
00:14:43
concentration gradient it's going to
00:14:45
distribute itself until we have an even
00:14:47
concentration diffusion applies to all
00:14:50
species but we need to particularly
00:14:51
focus on neutral species for the reason
00:14:53
that we said these things need to
00:14:54
diffuse away from the electrode surface
00:14:56
I'm going to call you back to your
00:14:58
first-year thermodynamic diffusion
00:15:00
argument go and reference that go and
00:15:03
have a look at that make sure that
00:15:04
you're happy with that content but what
00:15:06
we're going to look at here is we're
00:15:08
going to look at a concentration
00:15:09
gradient we're going to look at how the
00:15:11
concentration varies over a region of
00:15:13
space
00:15:14
so if we have a concentration gradient
00:15:15
if there's a difference in the
00:15:17
concentration we would expect to have
00:15:19
diffusion so at what rate does that
00:15:22
particular species flow so what is its
00:15:25
flux so that's what we're looking for
00:15:27
we're looking at how it flows through a
00:15:29
given space we proposed a particular
00:15:31
model such as this where we have the
00:15:34
rate of trap of flow so the flux j is
00:15:37
equal to some constant multiplied by the
00:15:39
concentration gradient so DC by DX the
00:15:42
rate of change of concentration across a
00:15:44
given area of space so if we look at
00:15:47
this we think ok well our concentration
00:15:49
is high on the left low on the right so
00:15:52
that means our concentration gradient is
00:15:54
pointing to the left but if we think
00:15:57
about the direction of motion the
00:15:58
direction of motion is going to go to
00:15:59
the right so this means we have a
00:16:01
negative value here this is exactly the
00:16:03
same as a rate law that you've looked at
00:16:05
in kinetics it's the same mathematics
00:16:07
it's the same expression so you're
00:16:09
comfortable with this so what would
00:16:11
happen over time if we suddenly let this
00:16:13
system equilibria well we're going to
00:16:16
get a flow from left to right across
00:16:18
this little region of space and the
00:16:21
boundary between it becomes more and
00:16:23
more diffuse until we get to a point
00:16:28
where there is no difference between the
00:16:31
two so the rate of change of
00:16:33
concentration with distance becomes zero
00:16:35
and we have no more flow happening this
00:16:38
relationship is known as ficks first law
00:16:39
of diffusion and it simply states that
00:16:42
the flux is proportional to the change
00:16:43
in concentration over distance the D
00:16:46
term here is the diffusion constant and
00:16:49
we can determine it fairly easily from
00:16:52
this particular relation here there's no
00:16:54
need to derive it but this is known as
00:16:56
the Stokes Einstein relation notice that
00:16:59
there's no charge present in this
00:17:00
equation at all and the diffusion is
00:17:02
purely related to viscosity so this ETA
00:17:06
term here now we've seen this already
00:17:08
today we've looked at this and we've
00:17:10
considered it in terms of ionic drag
00:17:13
forces this k-beauty you should
00:17:16
recognize as the thermal energy in the
00:17:18
system so the thermal energy per
00:17:20
molecule in that system and remember
00:17:22
that a is our hydrodynamic radius as
00:17:24
well chemical potential also plays an
00:17:27
important role
00:17:27
in diffusion concentration is
00:17:29
fundamentally linked to chemical
00:17:31
potential which means we can use our
00:17:33
chemical potential understanding from
00:17:34
previously this allows us to determine a
00:17:37
thermodynamic force for diffusion so
00:17:40
remember that the chemical potential is
00:17:42
simply the partial Gibbs energy of a
00:17:45
particular component and we're looking
00:17:47
at a the activity in this case so
00:17:50
remember it's a potential so if we do a
00:17:52
derivative of this this will give us a
00:17:54
thermodynamic force as such so we simply
00:17:57
do a first derivative of this with
00:17:59
respect to X so that's across that
00:18:02
concentration gradient there if we're
00:18:04
using an ideal solution we can replace
00:18:06
the activity a with our concentration C
00:18:08
and we end up with this expression here
00:18:12
we get a very straightforward diffusion
00:18:14
force on our species one thing I'd like
00:18:18
you to consider is why does the standard
00:18:19
chemical potential disappear in the
00:18:21
derivative so we have a great deal going
00:18:24
on we have diffusion and migration
00:18:26
processes happening we have diffusion
00:18:28
due to concentration gradients we have
00:18:31
migration due to electric fields but we
00:18:33
need to consider both whenever we're
00:18:35
looking at our system ions migrate to
00:18:38
the electrode they give up electrons or
00:18:40
they receive electrons depending whether
00:18:42
an ions or cations and become uncharged
00:18:44
species and those uncharged species must
00:18:46
diffuse away so we have three forces
00:18:48
going on here we have the diffusion as
00:18:51
things move across their concentration
00:18:53
gradients and we have the electrical
00:18:55
force which is which is drawing our ions
00:18:57
to the electrode and fundamentally each
00:18:59
of them is experiencing a drag force but
00:19:02
only the electrical charge species are
00:19:04
subject to that electrical driving force
00:19:06
ok we've covered a lot of concepts let's
00:19:08
start putting them all together we see
00:19:11
many of the same terms appearing so in
00:19:13
our diffusion we say that we've got a
00:19:16
solvent factor and the thermal factor
00:19:20
but we also have our ionic mobility
00:19:22
which again has a solvent factor but now
00:19:24
we're looking at the charge on our ions
00:19:26
we've got a simple equality here and we
00:19:28
can rearrange these and substitute
00:19:30
values in to express our ionic mobility
00:19:33
in terms of diffusion so this allows us
00:19:36
to relate them via the Einstein relation
00:19:39
so for a
00:19:41
and molecule we can relate our diffusion
00:19:44
coefficient to the charges but if we
00:19:48
want to look at it in terms of molar
00:19:49
terms we can relate it to our molar
00:19:53
thermal energy and the faraday constant
00:19:55
okay so that gives us our ionic mobility
00:19:57
in terms of the diffusion constant this
00:20:01
can also be related to molar
00:20:02
conductivity so remember what we did
00:20:04
last time looking at molar conductivity
00:20:05
and we have a link between our mobility
00:20:09
and connectivity from the start of this
00:20:10
session and this allows just to simply
00:20:13
do a quick substitution to express our
00:20:16
molar conductivity in terms of the
00:20:18
diffusion coefficient in summary we need
00:20:21
to look at ionic mobility ionic mobility
00:20:24
is key to how ions move through solution
00:20:25
if they are less mobile they have a
00:20:27
lower contribution to the ionic current
00:20:29
and fundamentally the conductivity as
00:20:31
well
00:20:32
ions rarely carry an equal share so
00:20:35
generally they are carrying an unequal
00:20:37
share of that current with more mobile
00:20:39
ions were carrying more of the current
00:20:41
through that solution and remember as
00:20:44
well at diffusion and migration of
00:20:46
complementary processes they are
00:20:47
intrinsically linked and we have to
00:20:49
consider both in our electrochemical
00:20:51
processes
