This content explains non-ideal gas behavior by introducing the compressibility factor and the van der Waals equation, which account for molecular volume and intermolecular forces that deviate from ideal gas assumptions.
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the topic of this video is non-ideal gas behavior
behavior
the learning objectives are on the
screen so go ahead and pause the video
to write those down to jump
into non-ideal gas behavior i think
it's useful to make a prediction about
ideal gas behavior
and that starts by looking at the
ideal gas law pv equals nr
t and we can think about what
is something that should always be true
of an ideal gas from this equation
if we divide both sides by pv
for example what we'll get is that the
left hand side
should um become one right and
so then we have this expression that one
is equal to
nrt over pv and we can also
for this particular case we can just
make the assumption that
for the case for n equals one mole
okay then one equals
r t over p v
so that means that for any gas that
behaves in an ideal manner
the ratio of the ideal gas constant
multiplied by temperature divided by the
pressure and volume of the gas should always
always
equal one now
um of course not all all gases
will behave in an ideal manner and many
in many cases they will deviate from
ideal behavior
we'll talk about why that is in a second
so rather than always calling this one
we can actually
use this expression rt over pv to define
something else so we can actually
call it a a variable capital z which
we'll call the compressibility factor
and that will allow to be equal to
pressure times the
over rt so again
this term right here capital z
compressibility factor
and v sub m is the molar
volume okay now what ends up
being the case like if we can actually
we can experimentally measure
what this ratio is for different gases
as a function of different
conditions so this is the plot that i
want to show here
as a function of pressure in atmospheres
and what we see here is that for an
ideal gas
that should be one across the board
but you see that for all of these gases
there's uh the compressibility factor
changes quite a bit
uh at different pressure values
so what does this mean it kind of we can
kind of see here
that at sort of low pressure values
things tend to be close to ideal
behavior of about one
but then they start deviating as the pressure
pressure increases
so um one way that we can sort of
talk about why that deviation might occur
occur
is that ideal gas molecules
we assume don't interact with one
another and we also assume that they
didn't have a volume
but in reality they do have a volume and
they do interact with one another
in other than perfectly elastic collisions
collisions
so if we do not hold the volume constant
but we do hold the pressure constant
if there if we do allow for
interactions to occur sort of attract slight
slight
attractive interactions between
molecules what that means is that there
less energetic collisions
and also there could be you could
envision um fewer collisions because
molecules will
interact with another other with each
other for a little bit longer
per collision resulting in fewer
collisions with other molecules or
the walls of the container per second
and that would cause the pressure to decrease
decrease
in order to compensate for that pressure
decrease the volume would
decrease we can also think about
in the case where the volume is held
constant what would happen
to the pressure
and so if the the pressure
or if the volume is held constant and we
do allow for these interactions then
again in a similar manner when this gas
particles interact with one another and
are slightly attracted to one another
uh it will decrease the collision
frequency per second with other
molecules of the walls of the container
um and the pressure will decrease here
if the volume is held constant they'll
just be a net pressure decrease
relative to the ideal case so there are multiple
multiple
mathematical expressions for talking
about non-ideal gas behavior
the one that we're going to talk about
now is called the
van der waals equation
and what we're going to do is we're
going to take our ideal gas constant pv
equals nrt and we're actually going to
expand out the p
and the v components so the p
we're going to add on a correction
factor so
it's not just pressure anymore but we're
going to add on to it a correction factor
factor
that is composed of a n squared a
is uh one of the van der waals constants
and i'll
talk about a uh in a second so it's it's
plus a
n squared over v squared this term right here
here
is a pressure correction factor okay so
it's a
correction factor v
also is going to have a correction
factor um
added into its term so it's going to be
v minus
and then b is another van der waals
constant so this
nb term is another correction factor
this time it's a volume correction factor
factor
and then this equals n r t
so what i've just written is the van der waals
waals equation
equation
and the van der waals constant a
corresponds to the
forces between
gas molecules
okay and that's one of the reasons why
it is
a part of the correction factor to pressure
pressure
because it's talking about the the
magnitude of the attractive forces
that then can influence pressure in a
fixed volume container for example
and then b is going to be it's a van der
waals constant that corresponds to the size
size
of gas molecules or their volume
size of gas sorry i just missed that
molecules
it corresponds to it okay so uh and
that's why it's part of the volume
uh correction factor so what does this mean
mean
for um if we think through what are the
conditions in which
a non-ideal gas will behave in an ideal manner
manner
it would be conditions in which those
can essentially be made as close to zero
as possible
so pretty much in cases where we have
small amounts of gas so if we have low
end values and we also have um
a large container a large volume
so high volume if we have a high volume
right then this v squared term
that v squared if it's a large volume
becomes a very large number
so and also if n is a very small
number a small number squared is an even
smaller number so if a very small number
over a very large number this term can
essentially start to go to
zero under the conditions of small
amount of gas
in a large volume and also
if n is really small over here that also
makes the correction factor
over here uh increasingly go to zero and
if those two
go to zero or become zero then we just
have the ideal gas log and pv equals nrt
so conditions under which we have a
small amount of gas
and a large volume this is pretty much uh
uh
unanimous with low pressure situations
so non-ideal gases tend to behave
more like ideal gases under low pressure circumstances
now that we've discussed the van der
waals equation and what the van der
waals constants
are and how they can be used to make
correction factors here
is a table of van der waals constants
for some common gases and what we see
here is that
a has units of liter squared atmosphere
per mole squared
and the van der waals constant b has
units of liter per mole um and
what we can sort of see here and what
you would predict is that
smaller values of a and b
would indicate less of a correction to
the ideal gas constant or the ideal gas law
law
so the only noble gas in this entire list
list
helium has this is no coincidence
it has the smallest a value
and it also has the smallest
b value or a and b constants
for the van der waals equation so
for a quick example of the calculating
the pressure of a gas
using the ideal gas constant for the
versus the van der waals equation we're
going to look at
water in the gaseous state so we have a
5 liter flask
containing 3 moles of of water at 373 kelvin
kelvin
so the question is what is the pressure
calculated under an ideal circumstance
versus the van der waals circumstance
uh we'll just use pv equals nrt
and so p is equal to n r
t over v and so
i'll plug in the the the variables that
way to get an answer and then to save time
time
i'm just going to set it up and then
actually write all of the values but i
just want to show you the the expression
to solve for pressure using the van der
waals equation
equation is actually going to look like
this when it's rearranged
p equals nrt over
v minus nb so this is
the denominator minus n squared
times a over v squared
um so go ahead and look back in the at
the table for the
the van der waals constants for water
and do this on your own pause the video now
now
solve for the i the pressure solved for
the ideal case versus the van der waals
case and then
um return to the video and we'll compare
answers okay
so in the ideal case what i calculated
was 18.4 atmospheres
and the units cancel out like we've seen
before we have
kelvin kelvin liters liters
mole and mole cancel out leaving us with
units of atmospheres the
van der waals cases a bit more uh a few
more units to keep track of
on the top of the first part we have
kelvin cancel out with kelvin
we have moles cancel out with moles and
down in the denominator we have moles
cancel out with moles
and then it's a liter minus liter unit
um so so the bottom is liters so liters
will actually end up canceling out on
the top and the bottom
leaving us with atmospheres and then
over on the right
we have um liters squared on the bottom
cancel out with liter squared down here
moles squared cancel out with moles
squared here leaving us with atmospheres
so we have
atmospheres and atmospheres so we can
actually subtract those two
values and what we get is 16.7 atmospheres
atmospheres
so in the van der waals case we do
predict a lower
pressure for water at this under these conditions
conditions
and the reason there is because we would
predict then that
this deviation from ideal behavior is
because of attractive forces between
water molecules in the gas phase
indicative of that is the um
is is actually the van der waals
constant a
which has to deal with the correction
factor um
for pressure and that is actually the
one of the highest
um in the table um not the highest but
one of the highest for
this so we would uh uh it would make
sense then that there
might be we would predict more
attractive forces between
um gas molecules of water than for
example something like
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