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## Chemistry library

### Course: Chemistry library > Unit 17

Lesson 2: Relationship between reaction concentrations and time- First-order reactions
- First-order reaction (with calculus)
- Plotting data for a first-order reaction
- Half-life of a first-order reaction
- Half-life and carbon dating
- Worked example: Using the first-order integrated rate law and half-life equations
- Second-order reactions
- Second-order reaction (with calculus)
- Half-life of a second-order reaction
- Zero-order reactions
- Zero-order reaction (with calculus)
- Kinetics of radioactive decay
- 2015 AP Chemistry free response 5

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# First-order reactions

The integrated rate law for the first-order reaction A →

*products*is ln[A]_*t*= -*kt*+ ln[A]_0. Because this equation has the form*y*=*mx*+*b*, a plot of the natural log of [A] as a function of time yields a straight line. The rate constant for the reaction can be determined from the slope of the line, which is equal to -*k*. Created by Jay.## Want to join the conversation?

- Could anyone please show the derivation of the function for the first-order reactions?(3 votes)
- This is grade-12/college-level but if you're curious I will show you below.

So for a first order reaction -- we have the reaction equals the rate constant times the concentration of the (only) reactant --> R = k[A]

1. Then we choose to re-write R as -Δ[A]/Δt

and we get -Δ[A]/Δt = k[A]

2. Then we bring -Δt to the right side

Δ[A] = -k[A]Δt

3. Then we bring [A] to the left side

Δ[A]/[A] = -kΔt

4. Then we integrate (the left side with respect to A and the right side with respect to t)

∫Δ[A] 1/[A] = -k∫Δt

Ln[A] = -kt

5. Then we evaluate both integrals from 0 to t

Ln[A]ₜ - Ln[A]₀ = -kt-(-k0)

6. Then we bring Ln[A]₀ to the right

Ln[A]ₜ = -kt -0 + Ln[A]₀

7. Finally, we have our answer:

Ln[A]ₜ = -kt + Ln[A]₀

8.Notes:

i. For the connection to y=mx+b

The natural log of the concentration of A at a given time t --> Ln[A]ₜ is basically Y, and is equal to the natural log of the initial concentration of A --> Ln[A]₀ which is basically b, minus the rate constant -->k (basically m, aka the slope of the line) multiplied by time (basically x). So we get a linear graph of the form Y=mx+b

ii. The reason the it is negative at the beginning -Δ[A]/Δt and at -kt is because the rate is positive, but the change in reactant is negative because it is decreasing, so we build in a negative sign to cancel it and make the rate positive.(12 votes)

- why does the natural log of pressure have no unit?(4 votes)
- It still has the original unit of pressure because all we've done to it by taking the ln(P) is a mathematical operation. We haven't added a new unit or eliminated the original unit. So technically the unit of ln(P) should still be Torr and they should have indicated the unit on graph.

Hope that helps.(2 votes)

- What does he mean by the "natural log" at0:43?(2 votes)
- what does the t subscript mean in ln[A]_t(2 votes)
- In the same way that ln[A]_0 is concentration at time 0, ln[A]_t is concentration at time t where t can be any time desired(2 votes)

- hi,

at2:25, Jay said if the coefficient of A is 2, then -kt will become -2kt, so why don't we generalize it as -akt, where a is the coefficient of reactant A?

then according to this new convention, won't the half life equation be [At] = [Ao]e^(-akt) and t1/2 = ln2÷ak?

thank you!(2 votes)- Yeah, the calculus works out to that.(1 vote)

- please what does calculus and log mean(1 vote)
- Calculus is a more advanced branch of math that people usually learn after algebra and geometry. The reason we use calculus here is because the rate law defining the disappearance of the reactant, R = -Δ[A]/Δt, can be written with derivatives. Derivatives are one of the main concepts in calculus which essentially are a more precise way of imagining slope or rate of change. So, R = -Δ[A]/Δt is rate of change of a reaction using algebra, but R = -d[A]/dt, is rate of change using calculus where the ‘d’ stands for derivative. And the derivative form is technically what is being used here (even though it’s not being shown).

The equation at the bottom of the screen at0:00is known as the integrated rate law which uses the two equations above to derive. This requires using another concept in calculus known as the integral.

Without getting into the details, all you have to know is that the integrated rate law for a certain order is the result of calculus related math. They omit the actual calculus here so as not to confuse people with the math so they can focus on the chemistry.

Log, short for logarithm, is a way to express exponential problems and solve them. An exponential problem is one where the unknown to be solved for, the variable, is an exponent of another number called the base. So a problem like: 10^(x) = 100, is an exponential problem; we’re trying to find which power 10 must be raised to to get 100. This can be rewritten with logs as: log10 (100) = x. Logs have a base written as a subscript in the lower right side which we want to raise by a certain power to get the number in parentheses. Here we know x is 2 because 10^(2) = 100, but logs allow us to solve any exponential problem no matter how difficult they get.

The most common bases for logarithms are 10 and e. Log10 is so common that it is the default base and most calculators will express it as simply log. e is a mathematical constant known as Euler’s number equal to approximately 2.718, so a logarithm using base e would look like loge. Again it’s so commonly used that it gets its own symbol ‘ln’ which stands for natural logarithm from the Latin logarithmus naturalis.

Hope that helps.(3 votes)

- I’m having trouble finding how you got -2.08 x 10 ^-4. When I add up the y and x numbers to do m= change in y divided by change in x, I get -16.189 divided by - 39800 = 4.067x10^-4(1 vote)
- The slope of a line is defined as the change in the y-direction divided by the change in the x-direction (rise over run). As a formula looks like m = Δy/Δx; where m is the slope, Δy is the change in y, and Δx is the change in x. Change here being a difference (subtraction) between two point's x and y coordinates. So we can also write the slope formula as m = (y2-y1)/(x2-x1); where x1 and y1 are the coordinates for the first point and x2 and y2 are the coordinates for the second point.

Now assuming it is a perfectly straight line, the slope should be constant at all points on the line and so we can pick any two points to calculate the slope. I'll choose the first and last points; (0,6.219) and (15000,3.109). So x1 is 0, y1 is 6.219, x2 is 15000, and y2 is 3.109. Substituting these into the previous formula yields: m = (3.109 - 6.219)/(15000 - 0) = -2.07 x 10^(-4). Which is reasonably close to what Jay got in the video. The discrepancy between my answer and Jay's is due to him using a graphing computer of some sort which takes into account all the coordinate points and also shows that the line is not perfectly straight. At any case even doing it by hand our answers should agree for the most part.

With your calculation I'm not sure why or what you added together, but having a change in y of -16.189 and a change in x of -39800 is wildly wrong. Additionally if you divide those two numbers you get a positive slope (dividing a negative by a negative) and judging solely off the graph of the line it should have a negative slope.

Hope that helps.(2 votes)

## Video transcript

- [Instructor] Let's say we
have a hypothetical reaction where reactant A turns into products and that the reaction is
first-order with respect to A. If the reaction is first-order
with respect to reactant A, for the rate law we can write
the rate of the reaction is equal to the rate constant K times the concentration
of A to the first power. We can also write that
the rate of the reaction is equal to the negative of
the change in the concentration of A over the change in time. By setting both of these
equal to each other, and by doing some calculus, including the concept of integration, we arrive at the integrated rate law for a first-order reaction, which says that the natural
log of the concentration of A at some time T, is equal to negative KT, where K is the rate constant plus the natural log of the
initial concentration of A. Notice how the integrated rate law has the form of Y is equal to mx plus b, which is the equation for a straight line. So if we were to graph the
natural log of the concentration of A on the Y axis, so let's
go ahead and put that in here, the natural log of the concentration of A, and on the X axis we put the time, we would get a straight line and the slope of that straight line would be equal to negative K. So the slope of this line, the slope would be equal to the negative of the rate constant K, and the Y intercept would
be equal to the natural log of the initial concentration of A. So right where this line meets the Y axis, that point is equal to the natural log of the initial concentration of A. The conversion of methyl
isonitrile to acetonitrile is a first-order reaction. And these two molecules
are isomers of each other. Let's use the data that's
provided to us in this data table to show that this conversion
is a first-order reaction. Since the coefficient in
front of methyl isonitrile is a one, we can use this form
of the integrated rate law where the slope is equal to the negative of the rate constant K. If our balanced equation
had a two as a coefficient in front of our reactant, we
would have had to include 1/2 as a stoichiometric coefficient. And when we set our two
rates equal to each other now and go through the calculus,
instead of getting negative KT, we have gotten negative two KT. However for our reaction we
don't have a coefficient of two. We have a coefficient of one and therefore we can use this form of
the integrated rate law. Also notice that this form
of the integrated rate law is in terms of the concentration of A but we don't have the
concentration of methyl isonitrile in our data table, we have the pressure of methyl isonitrile. But pressure is related to concentration from the ideal gas law,
so PV is equal to nRT. If we divide both sides by V, then we can see that pressure is equal to, n is moles and V is volumes, so moles divided by
volume would be molarity, so molarity times R times T. And therefore pressure
is directly proportional to concentration, and for a
gas it's easier to measure the pressure than to
get the concentration. And so you'll often see data for gases in terms of the pressure. Therefore, we can imagine this form of the integrated rate law as the
natural log of the pressure of our gas at time T
is equal to negative KT plus the natural log of the
initial pressure of the gas. Therefore, to show that this reaction is a first-order reaction we
need to graph the natural log of the pressure of methyl
isonitrile on the Y axis and time on the X axis. So we need a new column in our data table. We need to put in the natural log of the pressure of methyl isonitrile. So for example, when time is equal to zero the pressure of methyl
isonitrile is 502 torrs. So we need to take the natural log of 502. And the natural log of
502 is equal to 6.219. To save time, I've gone ahead and filled in this last column here, the natural log of the
pressure methyl isonitrile. Notice what happens as
time increases, right, as time increases the
pressure of methyl isonitrile decreases since it's being
turned into acetonitrile. So for our graph, we're
gonna have the natural log of the pressure of methyl
isonitrile on the y-axis. And we're gonna have time on the X axis. So notice our first point here when time is equal to zero seconds, the natural log of the
pressure as equal to 6.219. So let's go down and
let's look at the graph. All right, so I've
already graphed it here. And we just saw when time
is equal to zero seconds, the first point is equal to 6.219. And here I have the other data
points already on the graph. Here's the integrated rate
law for a first-order reaction and I put pressures in there
instead of concentrations. And so we have the natural
log of the pressure of methyl isonitrile on the y-axis and we have time on the X axis, and the slope of this line should be equal to the negative of the rate constant K. So there are many ways to
find the slope of this line, one way would be to use
a graphing calculator. So I used a graphing calculator and I put in the data from the data table and I found that the slope of this line is equal to negative 2.08 times 10 to the negative fourth. And since if I go ahead and
write y is equal to mx plus b, I need to remember to take
the negative of that slope to find the rate constant K. Therefore K is equal to positive 2.08 times 10 to the negative fourth. To get the units for the rate constant, we can remember that slope is equal to change in Y over change in X. So change in Y would be the
natural log of the pressure, which has no unit, and X
the unit is in seconds. So we would have one over
seconds for the units for K. And finally, since we got a straight line when we graphed the natural log
of the pressure versus time, we know that this data is
for a first-order reaction. And therefore we've proved
that the transformation of methyl isonitrile to acetonitrile is a first-order reaction.