all right let's do some chemistry
all right equilibrium
and i want QUIET
today okay?
no chit-chat, no nothing!
no nothing
that's a double negative
forget that
no anything
so we've talked about
physical equilibrium
okay going from
gas to liquid or liquid to solid
and this time we're going to be talking about
equilibrium
of chemicals okay
this thing's been posted, however it
looks a lot different
now when it did when it was posted so
i'll get the new posting up
...sometime early this afternoon
so we've talked about this
that at equilibrium
things are going as much in one
direction as they are in another
direction
do you--if it's
physical it's ice and water going back
and forth from solid to liquid in your
drink
...if it's, you've got
a bell jar
liquid water completely, closed system
you've got water molecules jumping out of
the water, liquid molecules jumping out, you got gas
molecules jumping back in
can't see a change
but there's a lot going on. All right,
Nv2Ov4
is colorless
NOv2
is that yucky brown haze
that you can see on a
nice sunny day early in
the morning in Irvine as we look
toward Los Angeles
how many have ever looked north, or whichever way LA is
and seeing kind of this layer--it doesn't look
like it's a big tall layer
it's kind of this
rusty looking brown
what's this?
and this ends up
changing
this gets photolized and all
but this is a reaction that can occur,
okay
so this is colorless
and this is yucky
so this folks, is got some of this in here
okay there's some NO--NOv2, sorry, in this
and depending on how much there is of this versus this
will determine
how colorful it is, okay if there is 
hardly any of this
then it's a very light, faint color
... if however there's a lot of it
then you get this
so we're going to be using
the Nv2Ov4
NOv2...relationship
a lot here
okay. So
this
first one, this is concentration
versus time
okay
and if we look at it, so I'm going to ask
this guy in the front
who better not send me an email
asking for more points, so this one right
here--if this is concentration, this time
which
gas
or both gases do we start off
with?
which one has no concentration at the
beginning?
so which one has concentration at the beginning?
NOv2
so what we've got is we've got a bowl
that we put pure NOv2 in it--at the
beginning
and then at a certain time
I'm not sure how long, okay?
we don't know anything about how long it
takes
okay
equilibrium is set up here this could be
a second
this could be a year
okay we have no knowledge of how fast
this is happening here, all I see is that we start
off with pure NO2
as it drops in concentration, the Nv2Ov4 increases
okay
look at the next one
in this case
we started with
N2Ov4 and no NOv2
boom boom
so we're at equilibrium here, this is equilibrium here, and here we start with both.
okay
both, but guess what, in this reaction
we didn't have enough NOv2
all right, we didn't have enough NOv2, we had too much NO--sorry, Nv2Ov4
we we had too much of this
so we lose a little bit of this, we make
this
and we reach equilibrium
now these are different temperatures
okay so that's why they don't all
end up at the same spot
but this is how it looks, this is
equilibrium, you can start with pure one
you start with both
...you can put in as much as you want
and let it equilibriate
okay
and this is the equilibrium process
now let's look at this one this one right here
look at the equation
okay on the left side the product
of the reactant side
what is the coefficient?
one
what is the coefficient for the product
side
okay so what does that mean?
what does that mean to you?
look at this right here
so we started with
certain concentrations of both
and let it go
to equilibrium
okay
so
in this case we had to make
some more NO-or Nv2Ov4 right? Look--
we started of the time
the concentration increases
in Nv2Ov4, everybody understand that?
so as it's increasing
what's happening here?
okay what's the rate of decrease?
for this versus that?
do they look like they're the same?
no, why? Because we have two of these
okay
it takes two of these
NOv2s to make one of those
so that's why, if we start with NO2
pure NOv2 and no of this, this slope is twice as steep as this one
same thing here, look at this nice gradual
decrease here, this fast
increase here
same thing here, this is dropping twice as fast because, folks
it takes two
folks, this is critical that you understand how important that coefficient is
if you don't understand that
chapter ten is going to be a nightmare
for you
so you have this in your notes
this is the general formula
write it down
I don't care if it's in your notes, write it on your notes
it'll help you remember it
it is
products over reactants
what a concept
so it's not products minus reactants, it's products
over reactants
so little a big A plus
little b, big B
is in equilibrium with
little c big C plus little d
big D
case of c, the concentration
okay and this is in
moles per liter
is equal to then
the concentration of c to the
the exponent is the coefficient
so it's c to the C d to the D a to the A, b to the B
and that's a constant, at a certain temperature, folks, it's always the same
okay
equilibrium concept is the same and was--I'll show you that in a minute
so let me ask you a question
if all the coefficients are one
and the concentration of...
a and b
is one molar each
and the concentrations of c and d are two molars each, what is
case of c
four over one, yes four
because all you did was just say okay, this is a two
two here
two there, one there, one there
case of c is equal to four
how many think that's simple? Good. Good answer
this is just
not that important but basically
the dimensions of case of c are
really
the change in the number of moles
okay, and the number of moles, folks, is the coefficients
okay c plus d minus a plus d, products minus reactants
so in that
scenario I gave you just a second ago where these were each one, one, one, and one, what is delta N?
what is delta N?
this is getting tough, I know
you got to go one plus one
minus one plus one
what is delta N? (zero) Yeah
okay so in that case
this is dimensionless, okay
but let's say that the C's had been--that had been a two and this was a
three
what's delta N now?
two three
three, yes, this is going to be five minus
two is three
folks, you're going to have to understand that
delta N, it gets critical in
about half an hour
okay so
in this case right here, this is how it's
setup
if case of c is much much greater
than one
what does that mean to you?
what does a case of c greater
than one
mean to you?
and I don't mean like it means eighty to you
 
it's a big number, okay, what does it mean
in terms of this?
it means either that you've got these
are bigger than this
these have to be bigger
this whole thing times--it's got to be
bigger than this, that means the product side
is much bigger than the reactant side
what happens if KvC of c is much, much less
than one
then this is bigger than this right?
but what that means is that things are going to lie on one side of these arrows versus
the other
if you have a big KvC
the majority of it is products
if you have a very small KvC, then it never makes it over here
just hangs around is as the reactants
so you'd have products or reactants, that's what this is telling us
once again
if KvC
is much bigger than one
lies to the right side, okay
looks like this
if it's much smaller, it looks like this
okay and in fact, folks, these can be almost zero
okay, you can have a reaction that goes so far, that there's hardly anything left
or
a reaction that
really doesn't even happen
that the equilibrium is so small that
you have ninety-nine point nine nine
percent left of this stuff okay
so think of it in terms of
multiple-choice questions
i could phrase this very very dif--in many different ways
and you'd have to understand the concept
the concept of KvC being
big case in c being small
are we okay so far? Now
there's something called a reaction
quotient
always before, folks
look at that, KVC is equal to
cc dd aa bb at equilibrium. At
equilibrium
okay
so its
we can only
figure out what KvC is
when all of the compounds
are at equilibrium
this one however, is the reaction
quotient
look at this
same setup, ccc ddd--blah blah blah
but initial
okay
so initial concentrations, that means
if I say you start with one mole of this
and one mole of this and one mole of this and one mole of this
what is the QvC?
what is the answer?
what's the answer?
if I say you started with one mole of this and one mole of this and you add one mole of this
and one mole of this
what is the QvC? One
okay, so QvC would be one
all right, so this is only going to tell you
this is going to help you decide which
way the reaction's going to go
that's the only thing it's good for
okay so if QvC
is much greater than KvC, then it
goes from the right side to the left side
okay--if this is bigger than this
it's got to come back this direction
so if QvC is equal to KvC
equilibrium
okay it's at equilibrium, its right there
okay
if QvC is smaller than KvC, like there
it needs to make more products so it's
going to go more to the right
equilibrium
and if the QvC is bigger than KvC, you
got to go back this direction so
that this drops down
so that's like a... multiple choice kind
of a question
I'm not big on QvC but what is
nice, folks
is
you can plug in
those initial concentrations and then
you have an idea as to which direction
things are going to go, and sometimes, as I
tell you
it's nice to know
have an idea
of the range of what your answers
should be
so that's all q does for you
it just tells you which way
are things are going to go, from the initial
concentrations
this is just a cartoon provided to me by
your book
so this is a reaction quotient
we've already done that--once again, these, folks
q-these are the initial concentrations
okay, a, b on the--in the denominator-c,d in
the numerator and you've got those up
 
coefficients there
so, if this is how we start
look at this, we started with only A and B
only A and B in this trial
could you do that on the exam if I gave
you this
and I said what
is the starting concentration of C and D
could you tell me zero?
does that make sense to you? Okay
these started at  zero--it almost doesn't look like a zero, but I'm going to say it's a zero
and these had something
so, if you had none of this
zero of this
okay
doesn't matter how big this is
what is your cue?
so q
at zero is probably smaller than KvC, even if KvC is small
so if you start with only
stuff on the left side
then the reaction has to go this
direction
but the QvC is going to tell you that
so this is what it is, and from right about
here on
is where we're at equilibrium, right about there
and once again, from--at equilibrium
Q is equal to k and that T there means at a specific temperature
okay, we've already done this
wonderful multiple choice question,
folks
perfect multiple choice question
this is in your notes
okay so, concentration
so far, has been molar
moles per liter
okay that's what those--those brackets are always moles per liter
okay
we're going to switch in a minute to some
gases but at this point it's all molar
all right, folks
last night's class couldn't even do this--I
was depressed
so
okay
if this is at equilibrium
at equilibrium
what is
what is K in this case?
what is k?
one half, she says
how did she get--how many said one half?
why didn't you say that?
you bunch of cowards
okay, so how many A B,what's the--the
concentration here
might as well be moles per liter
how--what is--how many A B's are there?
two!
so that goes with the denominator
sorry that goes in the nu--the
numerator
okay that's the product side
and then over that, then how many if we're going to see these green ones are the As
what is the concentration of A?
two--what's the concentration of B?
two, so it's going to be two over two times
two is
four which is two
or one-half right? We've got two over four is a half
that then has a
KvC of one half
how about this one?
think three over two?
and how about this one?
you're wrong on this one!
three overr one times one
and in this case, you got one over three
times three so this is one ninth
you see how it's done?
very simple! Folks, it doesn't get any easier than this
it's going to get hard in just a minute though, okay
I hope it doesn't, but it did last night
di-- is there everybody could do this?
everybody, I hope, if you can't
go to the notes, look at it
go on Google
okay
there's another cartoon
at times zero
I want you to look at this and
I'm going to call on somebody in twenty
seconds
to give me an answer
it's a little hint here folks
here's the initial time
at time ten seconds
we've had
some red balls turn into white balls
okay
at ten seconds from ten seconds on, it doesn't change anymore
well you look at this one
are there still three or so? Oh there's more
so you go, okay--we're not at equilibrium here
how about here, one, two, three, four, five,
balls that are white
how about here
one, two, three, four, five, six
one, two, three, four, five, six, seven, one, two
three, four, five, six
so here we have six, seven, six seven--you know, whatever
so what point would you say we're at 
equilibrium? (student response) Yes
thirty to forty seconds, you're right
because look, at thirty
we had six white balls
we don't have half balls, okay
probably if we had these in halves
then this one would have maybe--maybe
this one wouldn't be
six--whatever but
you go from six to seven to six to
seven to six, so we're at equilibrium
right in here
okay.  So at about thirty
or forty seconds, we're at equilibrium
what is the KvC?
what's case KvC?
and he's saying six over fourteen
I think I agree
i think there's twenty balls
okay
and if we're going here at thirty, then we've got six over fourteen
that's the KvC
you go here, it's seven over thirteen, doesn't matter, it's the same answer
okay
so KvC then
is equal to six over fourteen or seven
over thirteen
do we understand that?
thank you very much
we got that, folks?
this is how it can look. It can be the equilibrium
 
it could be the equilibrium
and then all of a sudden, boom. You dump something into the pot
okay, you dump it into the pot and look at what happens
this jumps up as a step function
this responds, okay, this decreases until
at some point we're at equilibrium, okay somewhere
at about in here, we're at equilibrium
so you can change things
you can add something, or you can take it
away
look at this one
so here we have H2
we have N2
and we're making ammonia, all right
so here, we start with no ammonia
and then we end up with products here
okay
...these are molar concentrations, now
what's the difference between this and
this one, we start with ammonia
and no
H2 or N2, it does just the opposite, okay
this is equilibrium right about here
look at how fast certain things
change here, folks
which one
which has got steepest slope?
this one, why do you think that's the
case?
because it's N2 plus three H2s
goes to NH3, okay
so, you're using more H2s
and therefore, the slope is steeper here
or steeper there
and this is just another one showing that
that..
this is partial pressure now
okay look at that, partial pressure
we can use partial pressure
or we can use
concentration in molar
okay those are the two things
one is KvC
one is KvP
we'll talk about how to convert those
this is just a picture of equilibrium
okay you can start with pure product
pure these, and this is the equilibrium concentration
okay, folks
take a deep breath
it seemed so simple to me last night
so we're going back to the Nv2Ov4 and NOv2,
 
so
these our experimental data
okay
doesn't matter how good you are
in the lab
trying to get absolutely reproducible
results is really tough
so, in experiment number 1
you started with no NOv2
and
a bunch of Nv2Ov4
day this is molar concentration here
okay
now
is what we have gone situation
and that's
if u
because let's say in the old as we did
know we are doing
he's ocala what
uh... in this case could you start
looking
with into all four
it would be on the left side
than you have a coming over here to the
uh... the era would come to the right
side
to make the n_ o_ two
producto reactance
do you do that
in this experiment
that value for uh...
the equilibrium gives you a zero point
zero eight five
exterminator to we actually started
okay
we started this one wef
some ano two
and we started this one with some and
two oh four
all right
comes along here filed concentrations
are here
and look at the ratio here career what
was the rope
zero eight five two a point one
got another one point three point five
point zero four point six and this case
we start with only enero two
and distorted zero and this is what we
ended up at look at this
is this police been helpful
those numbers are changing
selwyn somebody not made figured out
if we used that quote patient
guess what
the corporation
product to react to this
it's the same number
say number
so if you forget that probation folks
you are in big trouble
coefficient like that it doesn't matter
what you start with
pier one peer or the other or a mix of
them
head doesn't matter
you're gonna come up with a concert
right there
that's one called the equipment conflict
this is true at one temperature
so these are all experiments are done
the same depth
so this is the reaction
how many greeted guests
how we would determine the case of c for
the case here
products
to the two
into a poor concentration how many agree
with that
graham we picked that off me
days before
all right folks
let's do this
was equal concentration
we start up with one atmosphere
atmospheres
you do the same way as as a
the more
if the case of c_o_ two k is equal to
let me let me just a minute let me say
that
let's make this one more
easier for
each one starts at one more
i've given you
so the first thing you want to do
is to figure out what is the reaction
quotient if that would
be what i would do
so what is q
did you remember what the equation look
like
is the kind of it was
and to or an ode to concentration
squared or into old
uh...
concentration
their each one
was one square
wats one square divide by one
so case of secret dot q
cincy is one
look at this
so it's much bigger
then that's i think that's gotta go
this direction contract
it's got to go before this way so we got
uh...
works and i said are not distraction
juice in the queue is bigger than case
in c
it's got to go back in this direction so
we gotta make some of this
into this
so how do we do that
anybody got an idea
shes going downhill quick
so we are teas that was direction it's
going to go right
this weekend
much much greater than that
so we know that it's gonna have to go
from the right side to the left side
cell
this is how you set it up
okay
you've got a one
saro-wiwa wonder what
so what's happening
to the enero two
is were going to lose some an oath to
you
because our are still was way too big
day so it's got a shift back to the left
so we're gonna take away some of the
anode to put it right here this is the
equation right here
we gotta lose some of this and make more
rest
okay because we're too big
okay
well he started to one
okay star one of each
okay
so if you start with one of these and
you have that equation right there are
just
you are not a question which is so you
start with
one squared over one
okay
how to get back guidelines one
how do you make that numbers walk
one of two things
you to make the denominator better
we make the numerator tomorrow
that's the only way you can do it
you can't just aren't really sound like
this one
these are dependent on each other
okay interdependent
you got to of these
back here
so i get an analogy last night
great because the class
place it was before class
they did not understand this
so i get an analogy
that that was a silly analogy
and but help about half the class
hollywood greedy
with me here where it really
the sandwich
has two slices of bread
in my world clubs
two slices of bread
at least for the next half hour
two pieces of bread
okay we get that
so ever folks i'd tell you
that we start with ten sandwiches
and twenty pieces of bread
tent sandwiches
and twenty pieces of bread
would ask you
it's gonna be
twenty
squared
over ten
which i think it's four
who doesn't get that
raise your hand please
so everybody in this classroom
understands that that feel is the goal
of forty
razor and if you don't get folks
okay you don't get it
the equation once again is kitty kat is
equal to
whatever this is squared
clover
if i say that you got won t
of these guys it's twenty squared
overtown
which is fourteen
that's like you
but let's say i say
that the paid
is supposed to be one
not forty
but i want to do this in your head
what is the concentration uh... jess
at equilibrium
kese cool wat
elaborate
and all pissy yaha this okay folks
this is this is stuff you should be able
to do with it
alito have a clue as to how to approach
it
one person's honest to people
three
right here folks
that is equation you gonna use
he replies sandwiches
and by here's my bread and here's my
sandwiches
the coalition itself
goes right there
is the two of these guys and one here
so what i'm saying quotes
is i want you to take
the number twenty pieces of bread
intense air which is
and rearrange that
you gotta make sandwiches
we got to take searches apart
if i had ten sandwiches folks
and twenty cases a brandeis h son of a
tiny twenty two pieces of bread
how they said you'd like to bookmark
one that's right because in my world
one sandwiches two pieces of bread yes
sawa
if i say argument that wanted to use the
bread
she can do it
font-weight older what am i going to
have here
five-star hotel
that's how it works
you taking one side boom of the other
okay so let's say in that world folks
in that world now
nine and twenty-two what is k
what is getting a lot of that
figure out what kato ste if i see that
equilibrium requiring his twenty till
at night
what his cat
wizard fifty-three fifty four
so one more one of the wrong direction
cuz i said i want to clear one
so if i have a cat one
what do i have to go to do i need to
make sandwiches
right now i've got a kate
when i got nine of these and twenty two
of these
i've got a case like fifty-something
i want a lot
how do you make set which is also it is
a part to give me acabo what
do we go
more sandwiches or less sandwiches
marcel teach us
when you get this bit here
does right now in setting at fifty
so i would make it better we need to
make a copy of your undecipherable
parenting is marked by the end this war
and or that bigger it's it's once again
on the other they gotta go one gives
burger will get smaller right because we
started with an intent
and this lady called part one sandwich
we lost one seven twenty two pieces of
bread
the equilibrium concentration for
sandwiches
with the cave equal to one
tiffany
cuisines
not much question commodity
sweeping over there
what is the concentration of sandwiches
if i wanna k of one
so i need to reduce nahi more sketches
yes
how they sandwiches luxury brand new
we get
is cracked your face when he should be
answered this question
must answer the question i don't know
the answer
might just make this up
so what we need folks is we need a
number here
left-right
fifteen feet dot sixteen sandwiches
sixty senators
if i sixteen sandwiches
how many pieces of bread too i have
application the bread bill i have
if i
changed a bunch of bread into extra
sandwiches
i've got sixteen sandwiches folks
how likely is it that way restricting
thirty-two
okay i started with forty pieces of
bread total
so what is probably six pieces of bread
or left
by a sixteen sandwiches
sixteen times to return to
i started with only four so i must
dedicate
so tally if that works
and sixteen what is k
was it
visit
i do know that
this guy says it's one
is that one
back outside of that
someone and i don't want for example a
salt best hopes its explain
sixty eight
this is a useful where
fact well
sixty-four over sixteen hoped his fault
not right now
what do i need to go jessica
is still too big for work to do
she wants more sandwiches
let's try
how about
seventeen sandwiches
hiking
seventeen or seventy try seventy
so this is going to be a seventeen-year
and this is going to be what
six square
hidden quotas
picture okay we have now
you see how this is the envelopes
it said it's kind of a game
now i did and there are so we have
actually do a mathematical
but this is how it works
so it went down and what is the eight
alexander to do i have with a k_ one
palace seventeen point eight
selects a
eighteen
basically sandwiches
cell that's eighteen and the denominator
and i left four bases abroad
four times four sixteen two sixteen or
eighteen is pretty darn close to one
that anybody or anything just now
i thought there tomorrow night classes
hot excel
that was just an exercise with an
exercise okay
let's not
so you remember
i said it
if you've got a ko
four point six three times in the minus
three
allergies that to millwork in eastern
rwanda
how do we end up
figure out what people are in
concentrations
we don't have pieces of bread of
sandwiches but that's exactly how you're
going to do it
communicate on this side and make these
or great listened to
okay
they all have one one one one one pw is
equal to one
try what
zero point zero zero four six
playing a much smaller sentences handed
much bigger
this has to be much smaller in order to
come up with that
and you know what folks
there's only so much adoration you can
do
you start with
one and go to
unknown point one whatever you want to
do
uh... but there's a a mathematical way
to solve this
for somebody you're gonna go artiste
this is so innocent
and some of your lead o
i i'd never seen this before
and and you're lying to yourself
okay so we know that we've got to go
k is
way too big
compared to what this is so we're gonna
have to come back we're gonna have to
make bread in the sandwiches
so gonna solve for acts
so we start with one
and how many for every sale to make how
many pieces of bread we lose
he agree with that
you lose two pieces of bread
every sandwich you made
and every tuesday like everything to
regulate begin one sandwich
corporation slopes
it's simple simple simple okay
this trial was that we would use any
tool to
four point three times and the one st
and we saw friends
flower
okay how do i get one minus two legs
because i told you i still want
okay
so i start with one
and i need to then subtract off from
that because it's too big
it's in the numerator i've got a reduced
this
and make this behavior
and i know that it takes to sandwiches
or to leno's attitudes
to make one of these
i struggled one i think await to hear
and i have one year
the corporation is always right there
products overreacts
how they can solve this problem if i say
that this is equal to
zero point zero zero four six three
you can do it
mccumber more do it
so folks i think
i think
this is keith great man
at least in the
not-so-good schools this and i went to
eight grade math for you folks use want
people
probably six craig
sixth-grader
ninth-grade gimme a break
would you go to school buddy
okay so
this is how you set up
duit solid
the events are added
all crap
jamaica wonderful
so don't don't say anything yet
you might just looked at the notes
so what's impulse this
your basic
i know why did this it may be
seventh-grade
coming to help me personally look
expanded said it was there are
sub
i thought i get a better job
cell here's another one side
here's the one side of the other side of
the war plus next time
uh... this is the other k
do you remember how to do this folks
one minus two x square
sixth-grade
there it is
see how we do it folks
what grade did you learn ascent
did you learn it
does remember when but he knows how to
do it
i don't really care when your word
okay this post is just
to meet like this it was third grade
math out
skeptical
priscilla were in kindergarten
i broke it down as simple as i could
you see how to do this
now what we do
all my god
what now
look at that
far from the quadratic
cell
how many of our governor quadratic
equation
what grade did your that'd
we were one nine three bodies
particularly after i traded
probably alka
so folks till the problem
here's a quadratic
and for those of you were born on start
i mean i i understand that there are
several u_n_ this class who loved laugh
and who who already know the answer this
at all
and and i'm sorry but
last night's
the bottle
has meet more evites
i'm not on the streets of the
the best of the brightest
how many of you
evade button on your calculated will do
this for you
i don't
i believe those out of those companies
are outlawed on the exam
okay i want to do it for you here
i went on this website they did it for
me
just type in a b and c
and it's amazing
dog to walk like i don't have that bus
now i can mess around a quadratic about
twenty minutes not get the same answered
twice
attitude that is true that if you can do
it okay
travel
in albert dot reviewed i'm still at
mid-june quadratic summer research
does that help
okay so
the website gave me to answer
that's how the plot radicals
so what's one of the eighties
the right answer
which one is the right answer folks
this is now we have to think about okay
what do i have
what is the equation that we're trying
to solved this
unlike due to some of what they want it
tactical agent
cell which
of the two value
point four five point five five
is going to make sense
which one makes sense point four five
rvp point five five how are you know
which one to use even if you push that
button on your calculator and he gives
you two answers
how do i know
how do i know
kuwait four five or point five five
so
nope nope
consultant k
we already have the whole thing here but
now look at this right now
is it possible to have a negative
concentration
so if i take that number one
license for activity lines axed
new taxes point four five
then what does this pop art deco
one minus point nine is equal to about
point one
have excuse
point five five
texas point five five
was this topic up
some stupid number
a negative number
not doesn't square process
dat elsewhere dot o_ r_ you can't have a
negative concentration
i give him a giggle away with your mouth
but there's no physical way to have a
negative concentration
began like anywhere
so folks
you look at the answers that you get
yet your answers
used look to see that makes sense so in
this case
the point four five works that way four
six now
everybody
the answer i said rats
is about point four six
plug it into this and see
what you get with a point four six six
four k
that's which all is one of the oz want
to see the jury had to make sense
and you get the right answer
void zero eighty cents
what do you get
four point three eight times tender
minus three
pretty close
okay
how many
dot not an essay question
priroda
i'm finished
