- [Instructor] So in the last video,
we explored the rate law in pretty good detail.
We talked about the importance of the order of reaction
and we talked about the importance of the rate constant.
So let's go ahead and use the same reaction
where we had nitrogen plus three equivalents
of hydrogen becoming two NH threes,
and we said that the rate law for this reaction
is the rate, equals the rate constant
times the concentration
of nitrogen to some power,
we'll call it 'a' here, and the concentration
of hydrogen to a different power, b.
Okay?
Experimentally, we wanna be able to figure out a way
where we can figure out what a and b are,
and then calculate what the rate constant is,
so to have a complete rate law,
we need to know the orders
of the reaction with respect to each reactant
and we also need to know the value
for the rate constant, okay?
And there are a number of ways to do this,
but really kind of like the most universally accepted
and easily accessible is known as the method
of initial rates, okay?
It's known as the method of initial rates,
because we are measuring the rate of a reaction
at the very very beginning, right?
So we mix together a certain amount of our two reactants
and we monitor that rate of the reaction
at the very very beginning,
so it's a place in the reaction
where we have the most control on concentrations, okay?
So what I'm showing here is a table of data, right?
These are three different experiments that we did,
so Experiment One, Experiment Two,
and Experiment Three, and we measured the rate
of the reaction for each of these combinations of nitrogen
and hydrogen, and from this experimental data,
we're gonna be able to determine
the complete rate law including orders with respect
to nitrogen and hydrogen,
and the value of the rate constant, okay?
So the trick to this is,
basically comparing the rate law of two different reactions,
so what we're gonna do is we're gonna take the rate law
for the first reaction,
divided by the rate law for the second reaction, okay?
So in this case, the rate law is,
right, so the rate of the first reaction
is 300, divided by 1200, okay,
so that's rate of the first reaction divided by the rate
of the second reaction.
This equals k, right, so our rate constant,
divided by the rate constant for the second reaction,
which is the same number,
times the concentration of nitrogen for the first reaction
which is three, okay, and this is to the a power,
divided by three to the a power.
Now, the concentration of hydrogen to the b power,
divided by the concentration
of hydrogen to the b power, okay?
So each piece here, right, so the top of this,
we have rate equals k times the concentration
of nitrogen to the a power times the concentration
of hydrogen to the b power.
We divide it by the rate law for the second reaction,
for the second experiment.
And the really nice thing about doing this
is that we can quickly see that this cancels,
the rate constants cancel,
and three to the power of a cancels.
It's the same on top and on the bottom,
so what we're able to do
is to simplify this to 3/12, which is,
so this is 1/4 equals 2/4 or 1/2 to the power of b, okay?
So you may be able to quickly figure out
that, if you take 1/2 squared,
you'll end up with 1/4, but let's go ahead
and go through and mathematically prove this.
So the easiest way to do this
is to take the log of both sides,
okay, so I'm gonna go ahead and move over here,
so I have a little more space.
I'm gonna take a log of both sides.
So the log of 1/4 equals,
when I take a log of this term right here,
this exponent comes out,
so equals b times the log of 1/2, okay?
And then this becomes b equals the log
of 1/4, divided by the log of 1/2.
Plug this into your calculator,
and you will see that b equals two, okay?
So this works really nicely
because we are able to cancel out so many terms
and be left with a mathematical expression
with only one unknown, okay?
Using the rules of logarithmic functions,
we can solve for b and figure out that b is two.
So up here, b is equal to two.
We can do the same thing comparing reactions two
and reactions three, because in this case,
the concentration of hydrogen is the same,
so we're gonna see that that term cancels out
and we'll be able to determine a,
because the concentration
of nitrogen is different in the two reactions.
So let's go ahead and do this, okay?
I'll change colors here, so we're going purple.
So we're gonna do the rate
of the second reaction divided by the rate law
of the third reaction.
So I have 1200 divided by 2400, equals the rate constant
divided by the rate constant,
okay, I have the concentration
of nitrogen to the a power divided by the concentration
of nitrogen to the a power,
times the concentration of hydrogen,
oops, the concentration of hydrogen which is four,
to the b divided by four to the b.
And we see here that our rate constants cancel
and four to the power of b cancels,
giving us 1/2 equals 1/2 to the power of a.
So in this case it's pretty clear that a equals one.
So in our rate law, a equals one.
We now know the order with respect to nitrogen
and the order with respect to hydrogen,
so the last step is to solve for the rate constant,
okay, and this is really straightforward.
Pick any one of the reactions,
either Reaction One, Reaction Two, or Reaction Three.
Pick any of them, and plug the values into the rate law
that includes what we now know is one for the order
of nitrogen and two for the order of hydrogen.
So I'm just gonna pick the first reaction here, okay?
Actually, I'll pick the third one because I can see it.
So our new rate law, rate equals k times the concentration
of nitrogen to the first power,
the concentration of hydrogen to the second power,
okay, my rate is 2400 and be careful with units here now.
So this is molar per minute, right?
Did you notice that up here, the rate is molar per minute?
So this equals the rate constant times the concentration
of nitrogen which is six molar,
this is to the first power,
and the concentration of hydrogen,
which is four molar to the second power, okay?
So we can plug these values into our calculator,
we have 2400 divided by six divided by four squared,
and, oops,
and we're gonna see that our rate constant
is equal to 25 and now,
if we're careful with units, what you'll see
is that the rate constant ends up having the units of molar
to the negative two minutes to the negative one,
and this equals one over molar squared times
one over a minute, okay?
So the rate constant is 25 per molar squared, per minute.


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