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Lecture 20. The Integrated Rate Law.

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Lecture 20. The Integrated Rate Law.
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UCI Chem 131C Thermodynamics and Chemical Dynamics (Spring 2012) Lec 20. Thermodynamics and Chemical Dynamics -- The Integrated Rate Law -- Instructor: Reginald Penner, Ph.D. Description: In Chemistry 131C, students will study how to calculate macroscopic chemical properties of systems. This course will build on the microscopic understanding (Chemical Physics) to reinforce and expand your understanding of the basic thermo-chemistry concepts from General Chemistry (Physical Chemistry.) We then go on to study how chemical reaction rates are measured and calculated from molecular properties. Topics covered include: Energy, entropy, and the thermodynamic potentials; Chemical equilibrium; and Chemical kinetics. Index of Topics: 0:00:41 Two Types of Reactions 0:02:19 Stoichiometric Reaction 0:07:14 Experimentally Determining the Rate Law 0:17:48 Use Integrated Rate Law to Define Half-Life 0:20:30 Second Order Reaction 0:21:56 Zero Order Reaction 0:23:56 Catalyzed Reactions 0:26:00 Common Integrated Rate Laws 0:27:06 Three Methods of Classifying a Reaction 0:30:34 Reversible Reactions
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Q so we're talking about Chapter
19 already talked about the stuff that we
talked about on Wednesday and then we're going to talk very briefly about reversible reactions we started to
talk about those on Wednesday
and then we're going to say a few words about a radius and his equations it's very exciting lecture today very exciting
operates so we mentioned on
Wednesday so we can know where you still be a chemical reactions
that are written down like this I would classify these as 2 different types of data
stored in Manteca elementary stole the
metric but this reaction represents is the overall still geometry 2 moles of B reacts overall with 1 mole together 1 that's the only information that's conveyed by this equation the overall
still geometry of the reaction is no kinetic information conveyed by it at all OK no mechanistic
information reaction order is conveyed in other words a does not necessarily ever interact with 2 B directly
right it happens to some interactive mechanism that has multiple steps in it that we don't know about all we know
from the store geometric reaction is distorting commentary here but the reaction rate will have this general form but we don't know what else when they are we don't know anything about them from nests in particular date is not too well there's not 1 necessarily that
could be but they're not
necessarily people to be stalking metrical efficiencies
OK now you can't tell by
looking at this reaction whether stole Metroka a Elementary you have to be told but you
can't tell just by looking at it it's another important thing to know
about elementary reactions describe a single chemical event or reactive encounter right if this was an elementary reaction instead of a stogie metric reaction we could write this weight loss just by looking
at the reaction we can write the rate right the X volunteer
are going to correspond to those documented coefficients for the reactants for itself we know
what's in elementary reactions we can write down the differential rate while right away just by looking at it OK so there's 2 types of reactions stowed metric that don't convey any kinetic information the tells about the stole geometry reactants reactants and products any
elementary reaction for elementary reactions are a lot can be written down immediately upon inspection if this is our stoking much of this is our elementary reaction rather it's a you molecular reaction all that means is that there's a 1 in front of his age the call UT molecular writer differential rate while BBS where there's a 1 exponent BA because
there's a 1 in front of the if this is our elementary reaction then the differential rate last gonna look like this it there's a wonder ex-foreign on the a and B because there's a 1 start magical coefficient in front of the day in the BU get the
idea I can look at this
reaction and write down this differential rate lock without any problem this is this is a buyer molecular reaction when I say biomolecular reaction in the overall order
of the reaction is to
1 plus 1 right this is also by molecular reaction in which the overall order is to in this case there just too way instead of an a and B and this is what the differential a lot would look like for that now I notice how we normalize the rate using stored metrical efficient here expressing the rate in terms of a N in terms of the right I think you can see the berated the apparent rates going to be twice as
fast for a at as it is for me because 2 is reacting to give me in this case and so to compensate for
that we multiplied the rate with respect to a by one-half right and in general we multiplied by 1 over whatever that still Demetra coefficient is to get a rate that is invariant independent of which reacted the product we
decide to use to express the rate make sense
but remember this documentary coefficient is negative for reactants that's why there's a minus sign here rights of these 2 rates whether Rice who express rate with respect to be or expressed the rate with respect to a I'm going to get the same rate
if I don't get the same rate as somehow done that recalculation along but
so let me emphasize this is a differential rate like it does not allow for comparison of a and B is a function of time I think you can see that this is the time rate of change of B. This is the time rate of change of a bright and this is what that's equal to I can't create a plot of AD is a function of
time or B is a function of time directly from this equation can but it's a differential rate loss not an integrated rate locked the fight integrated than I can OK so if we perform
immigration let's say we choose to use a I move this 1 half over the right hand side and I die with that but that Interpol looks like it's just 1 over a minus 1 over a 0 that's an equal minus 2 times Katie because I'm integrating from 0 here and so this is going to be my integrated right Lauer right well this is not the
1 that's in your book but if you look at this
reaction book its reaction 19 points 13 be handled like this but it doesn't look like there's and the reason is that this true by convention is always rolled up together with the case all right that too was always incorporated into the case the 2 get devoured by the
case but the constant includes so it's not
there will talk about the rate constant we're talking about this case that includes this to you will think about
it that way but now we have to be
able as chemist to
experimentally determined what this rate lot years we've got a reaction vessel we take a sample from it we inject that onto a GC MS we get some peaks it tells us that there are 3 reactants in our reaction vessels at the beginning of the reaction before we initiate the reaction somehow maybe with the temperature jumped the by stirring it whatever but we know this 3 reactants but we don't know but the stoking metrical fashions are 1 the terminal the lies for the reaction that's occurring between these 3
reactors a B and C. but stables took a major corporations are alpha beta and gamma and forming some products right how do we as chemists figure
out what the actual differential rate light years but what's the Stallkamp still geometry with which these reactants are reacting with 1 another this is a very important the issue for us is chemists but we know different
realize going to have this fall army yes assuming this is an elementary reactions and I've indicated here that it is because you can't tell by looking at it OK so there's 3 methods for doing this the 1st is called method of initial rates the basic idea is very simple we wanted
designers reactions so that the rate overall rate of the reaction only depends on 1 reactant at a time that's the key but so to do that we're
going to make it 2 of these other reactants large much larger than a horror how we know what it's much larger than that say that it is 1 million dollars right we might start
off by making B and C 1 Muller the factor a thousand larger then what we do we measure the reaction rate as a function of time and then we changed insane right if we if we go from 1 molar to 1 . 1 molar or from 1 mullah 0 . 9 Moeller there shouldn't be any change in the reaction rate because the
reaction rate should only depend on OK so experimentally we can tell whether we've isolated area not right if we have an
isolated area we need to make the concentrations a B and C 2 smaller or 5 molar however large we can go before the solubility limit possibly maybe it's impossible to isolate it but we won't know until we start to do some experiments all right but we can tell In the laboratory whether we succeeded if we change the concentrations of being seeing the reaction rate does not change we know
where isolated with respect that's the limit that we have to be and this will isolate react and air OK so once we've done that there's a new effective rate constant K Prime that includes B and C b and c are not going to change appreciably as a reacts but cost CEO
reacting to the concentrations are to be going down but the change in the reacting to the change in the in the concentrations of B and C are going to be so small that the overall concentrations a B and C. Art in significantly affected by this OK so we've got 1 million of a all of the 8 could react and we're going to have a lot by 0 . 9 9 9 more upbeat if we start off with 1 molar right so
b is going to be the same to within 3 6 figs it's not going to be affected by this reaction and we can assume its constant and we can roll it up into this Sudan L order rate constant OK so you see this case Prime what they make these guys big I can roll them up together with K performed escape I'm and now my apparent reaction rate looks like that's This is my apparent differential
brought rate equations OK then
what we do so no before isolation overall reaction was Alford was available as Gamal but after isolation the parent appearance overall reaction orders not just Alpha here just elephant here it's palpable speedup was gamut my beating us to death by going to measure offered to do this we measure the reaction rate as a function of alpha 0 the
initial concentration but a for Fazio rather OK so
I take the logarithm of this guy I get log the rate that's a lot of the crime was Alpha time slot of a and so on dues on interchange the concentration of a instead of using 1 molar one-million-dollar I'm used to Millie Moeller and three-million-dollar informally
mall like all of those concentrations are still small compared to 1 b and c Wright so I'm still isolated a presumably and in fact you
see how this is a and this is a 0 the reason I'm going use the initial concentration of areas so that in case there's a reverse reaction that occurs I will not be
influenced by that because at the beginning of the experiment there's no we can't act to cause there to be a reverse reaction right and so there's no panic willing to measure the rate of the forward reaction and if there is a reverse reaction I don't know about perhaps it won't influence what I'm trying to determine trying to determine the stoic image of coefficient of 8 right that's why we're gonna use despite the initial rate methods
look aviation industry possible influence of a reverse reaction cases there will have to think about it OK so here's what that data looks like fears formerly Muller 3 million dollars to Millie Moeller and 1 million molar of a are right and I'm measuring initial rate which is the slope of this red line and drawn here all right we don't know what happens to a during the rest of this reaction we don't have an integrated weight
loss for a right all going
determine is the initial rate 4 different concentrations of a yes that's what the slope of his line is In the plot that lot the rate versus log of a 0 we should get a straight line the
slope of a straight line is helpful it's beautifully simple it can't fail it works hesitate to say it works every time all right it it's a very simple and straightforward way to determine what the stoic metrical fission of each react and is is the problem Asia gotta do a lot of experiments invented for
experiments here right that's just to get the stock metric and ovarian and there's B and C. a
lot of experiments are involved OK
that's the 1st method method of initial rates the 2nd method is to calculate the integrated rate Libor integrate the differential rate loss when you do that you get an integrated rate lot that now explicitly predicts what it will be as a function of time the integrated rate law
allows you to model the concentration of any reacting to a product is a function of time in your computer you can just but the equation the XL you can
set your experimental data using the rate constant initial concentration right and when you do that you should be able to match up you
predicted versus time curve with the experimental data and that tells you that whatever
differential rate lie you assume for a first-order 2nd order 0 water but whatever differential rely you assume is correct if you can set your experimental data as a function of time and you predicted curve overlays experimental data y you've got whatever molecular ready first-order 2nd orders awarded 3rd order that you've assumed is what you actually have a new experiment this is so In this case were propped plotting a lot of 8 on this axis versus time and intercepts obviously going to be logically 0 and slope is going to be minus escape a you don't have to create a linear ice form of this equation the way that these textbooks do that work written 23 years ago
25 years ago OK it's not necessary to do that anymore necessarily
although it is convenient to be able to skip a cave on the slope of this equation near the says this gives
you a good feeling for whether the Fed your
1st order reaction with your first-order reaction fits the data or not but you can
just plodded directly also faded most pressing problems programs will do that automatically now for you OK so the
integrated rail line it is the 2nd Way to determine what the differential rate equation it's
better than the initial rate lot in the sense that you're fitting over a range of times over a range of reacted product concentrations and so if you get a good fit it really tells you unequivocally that that reaction first-order Over a range of times in a range of concentrations for reactants and that's not true for the initial rate love of the initial so method the initial
method of initial rates soaring OK so there is a 3rd method we already talked about it on Wednesday and that is what you got your integrated right law there's a shortcut that you can take which is
not to do anything it's just to determine what the half-life of the reaction is new determined that by plugging
0 over to infer grabbed just plight is 0 over to infer a and when I do that the 8 0 stenciling at 1 over to and that's a miner's lot of 2 and so the half-life of the reaction is just gonna the lot to overtake the help like this
because is the time when the concentration is dropping half of its initial value obviously and so this is yet a
3rd way just involved taking integrated rate not substituting a 0 over 2 4 a that time is now called the half-life console for that and c if what you measure
conforms To what you expect all right so this is the
half-life a first-order reaction and the half-life stays the same right the 2nd half life is the same as the 1st of half-life is the same as the 2nd what you're interested in here is the progression of half wives as a function of
Tom the 1st half life of that compared to the 2nd half like others that
compared to the 3rd have like to do that you've essentially got a map of out his green curve here so it's not really a time-saver but it does allow you to just look at the data and say
Hey this is obviously first-order data right without having a computer handy to fit it you can just look at it and say right I
start up a point 2 years one-half flight number .period certified that 2nd half life yet it's very started so it's handy to
be able to do that without having to get into your computer and get into Excel and transfer call slope and do everything else that you would normally need to
OK so there are 2 types of 2nd order reactions we already mentioned this two-way goes to products that reaction is overall second-order we can work out the integrated rail offer that we really did that keeping in mind that this to gets rolled up into the case but the half-life in this case is different than it is for a first-order reaction proportional over a 0 and so now the half-life doubles every successive half-light is twice as long as the 1 before 1st second-order reaction right I measure this half-life and then I call that 0 measure second-half light it's twice as long on measure the a case it's inversely proportional to a 0 if this goes down by a factor of 2 the half-life will double and then quadrupled
everyone see that so once again even if you're taking this
data and you look at a computer screen you can see immediately you're starting a point to as your 1st half like cycle once twice as long looks like it could be
a 2nd order reaction with respect and but number is the
2nd time a secular reaction isn't there that has this differential rate locked and the integrated rely on this case is a little more complicated it's worked out for you on page 8 0 to your boss sorry but suffice it to say there is no simple T 1 half for a 2nd reaction and involved 2 reactants where there is no
simple closed form expression for to the 1 half for this case when it really only works for two-way those 2 products so
here you actually have to fit your integrated rate what you have to figure experimental data with your
integrated waited equation to see whether it's really 2nd order
OK we've mentioned 1st and 2nd order reactions these the most common times but we left out 1 other important reaction that we talked about in your quiz today which is a 0 or a reaction in which a ghost products but and as you order reaction is characterized by different for a lot that doesn't depend on the concentration of that's a little odd all right the rate doesn't depend on any day if I make a small I get a rate the unified make a bed and at the same rate it's all but
something similar happens in then Haber process the rate doesn't appear and necessarily on the hydrogen and nitrogen concentrations how can that
happen that's right the rate of reaction is independent of a well if you work out the math This is the integrated realize that you get is proportional or I should say decreases linearly as a function of time and so far about the half-life I just plug in a over 2 4 8 and simplify this equation the half-life is equal to Asia over to take stability what that looks like now the half-life get shorter as a function of time the reaction rate is Lanier 4 0 order reaction so used to if you're looking at the data on your computer you can see immediately that is your reaction because you're
you're reacting concentrations are decreasing in a straight line proceed immediately only is your reaction does that but the fact is for the
half-life the 2nd half .period half as long as the 1st one the 3rd half-life is half as long as the 2nd 1 and so on the half-life get shorter as a function of time for Use your order reactions what kind of reaction does this catalyzed reactions have 0 order for example to NO x reacts to give X 0 2 plus into we have catalytic converters that made up the NO and the CEO In the hydrocarbons by catalyzing these reactions to these benign products so we used to think C O 2 was a benign price write the catalyst is your catalytic converter right the rate of their reaction doesn't depend on the concentrations of these guys what it depends on is how much ruled the rate on the catalyst
particles the catalyst particle is where this reaction
is occurring you reacted with Seana lacks comes in and it's sticks to this catalyst particle right here all right and then the chemistry occurs all right the only thing that matters is how much room there is on this catalyst particle that will
determine the rate might not the concentration of NO x
there's plenty of NOX even a really low concentrations to saturate the surface coverage Of these tiny metal particles OK so what
you observed experimentally is the rate at which the Analects is getting eaten up by the catalyst doesn't depend on what the elects concentration and is it does depend
on how much of a catalyst you have more catalyst faster rate makes sense right something similar happens in the Haber process same idea and passes you have an iron particles right and in the case of these reactions here in general the particles are made of
go platinum alloy so kind of expensive or but trying to figure out how to get rid of the platinum these catalysts OK so there
are some many great realize that are very common 0 awarded by the rate of any reaction in a 0 or reaction just increases linearly as a function of time Bill straight lines see
straight line you know immediately 2 0 order reactions
a log a miners log a 0 COS miners Katie that's a first-order integrated rate last here the lot of a decreases linearly as a function of time not a directly In the 2nd order reactions we have 1 ElBaradei equals 1 or a 0 was Katie sold 1 over 80 as a function of time that increases in a we merely
His it's 1 over it write a
3rd reactions that the
good test question for the term too OK so in reality we have
3 methods 3 minutes for classifying react with respect to its order method 1 method of initial rates isolate 1 reactant make the other 2 big measure the initial
rate as a function of the initial concentration of the isolated reactant plot your data determine what the stock metrical fishing method to
drive the integrated rate equation feature data over a wide range of with
different integrated rate equations to find out which 1 fits your data then that tells you it's 1st at 2nd order its 3rd order because that's the 1 that fits it's tedious to do that but that's what she did there isn't a shortcut and finally
reaction half-life you take on a greater rate equation In that integrated rate equation you
plug in a a 0 over 2 8 simplify the equation for the half-life and that's the F equation when you
compare that with what you observe four-year experimental data as a function of time over
multiple half flights that's the key you gotta see if the half-life stays the same if you get shorter author gets longer but you have to look at the data over multiple have to seal the half-life evolves as the reaction is proceeding
I hear this shows what that looks like 0 order reaction straight line half-life get shorter according to this equation right here the first-order reaction 2nd reaction look the same qualitatively don't they just a curve right you can't tell 1st order a 2nd order just by looking at it but if you look at the half lives here the are not changing here the half-life is doubling right that's the 2nd order reaction at a first-order reaction it's a 0 or a reaction to you can tell by
looking at the half lights so
S O 2 C L 2 what is that I have no idea his effort 2 policy Ltd With this rate constant right here at 593 Greek Calvin what per cent will decompose after 1 hour assuming this is an elementary reaction that a written down here in other words it's really you know right it's really first-order if that's true here's the integrated weight loss OK so it's basically Clinton shot from that point a sulfur a over 80 0 I plug in the rate constant here's the amount of time that we're talking about 1 hours 30 600 seconds I get 92 . 4 per cent is
left so the mother decomposes is 100 -minus that right along for
half of it to decompose at this temperature right here while I consulted the half-light This is the normal equation for the half-life of a first-order reaction that's the rate constant log to 30 100 31 thousand 500 seconds that's 8 hours and 45 minutes that's the halfway
through the reaction now we talked
a little bit about reversible reactions on Wednesday in what we said was locked you can treat a reversible reaction like an irreversible reaction if you are able to remove the
RIA removed the product as its form right if I
remove the Stage as its form in the rate of this reverse reaction here will be 0 because there will be any
react and present for this reverse reaction if
I get rid of it In this case it was study reaction electoral removing his Nunavut so rate while would just be the rate like that you would expect the forward reaction assuming this is an elementary reaction but in
general you don't remove the reactant you don't remove the product rather as its form you need to know what the reaction rate is with the product in there so what about
this more general case but look at the
simplest example of this write a forms be
and isn't it's a reversible reaction there's a forward rate constant and the reverse rate constant for this equilibrium it's an
equilibrium isn't OK so we
can write a differential rate loss with respect to a for example there's going to be the forward rate like a half time minus the reverse way right which is KB times that's going to represent the overall rate at which a disappears on the left-hand side so let's say for the sake of argument that we allow the reaction to go until we don't measure
any change in the concentrations of a and B anymore we allow the reaction to go all the way to but then the Tyree a
change of a 0 because but the line it's not changing any more and if that's the case then this expression is equal to 0 but we have to remember we're talking now about particular concentrations of a and B the equilibrium concentrations right OK
and so in other words that has to be equal to that at equilibrium and so if I deride the equilibrium constant by putting be over it that's the expression for equilibrium constant it becomes apparent that that equilibrium constants also equal to the ratio of these 2 rate constants right playoff over TB this is an important thing to remember so the equilibrium constant is the ratio of the forward the reverse way constants yet the higher
the forward rate constant the larger the case equilibrium the greater the extent of products are favored with respect to reactants it all makes intuitive sense OK 4 I'll
just continue this on Monday I have a good weekend