Chemistry Rant/Seeking Help with Kinetics

[MonkeySage]

This chapter has been a nightmare from the start. The math is, at best, very poorly explained. Even when I do the math correctly, I get answers that are not only wrong, but make zero sense.

Take for example the Arrhenius Equation; the book has done a horrible job explaining the math involved here. Maybe I'm just bad at math, but I haven't gotten a single correct answer yet. I spent 5 hours last night and 2 hours this morning trying to solve the same problem:

A reaction has a rate constant(k1) of 1.21 x 10-4/s at 302K(T1).
k2 is 0.226/s at 349K(T2).

In part one, I needed to find the activation barrier Ea, which took me 5 hours and 6 attempts before finally getting the answer.

In part two, I need to find the rate constant (k) at 288K. I've gotten six different answers and none of them make any sense whatsoever. Since I know this answer should decrease linearly from 0.226 to 1.21 x 10-4, it makes absolutely no sense for my answer to be 0.967 or 2.73 x 10-26.

[Kato]

Eh, I've never done any chemistry beyond my advanced classes in high school (or what might be the German equivalent) but applying maths and what Wikipedia says...

The 'k's are defined as A*exp(E/RT) (or k_B, depending on what unit you want for the energy, also, note, too lazy to put parantheses around RT, you know both are in the divisor)
k is the (temperature dependant) value describing the speed of the reaction (should be clear), A is some factor depending on... whatever kind of reaction it is, probably something to do with entropy and other fancy science things, E or E_a is the... activation energy? Wikis not totally clear on it, but I guess so. T is temperature in Kelvin (duh) and R is the gas constant or k_B is Boltzmann's constant.

So, what you got? Two values for k, missing A and E. First, let's throw them together! (I'll use R, but you can just replace it with k_B)
k_1=A*exp(E/RT_1)
k_2=A*exp(E/RT_2)
Well, okay, but those aren't equal. Dang. But what is equal? A! So let's modify them a little.
A=k_1/exp(E/RT_1) or k_1*exp(E/RT_1)
and because it's the same equation
A=k_2/exp(E/RT_2)
Those must be equal, because it's the same A, so
k_1/exp(E/RT_1)=k_2/exp(E/RT_2)
Now to get E from that with a bit of math
ln k_1 - E/RT_1 = ln k_2 - E/RT_2
- E /RT_1 + E/RT_2 = ln k_2 - ln k_1
E (-1/RT_1 + 1/RT_2) = ln k_2 - ln k_1
E = (ln k_2 - ln k_1)/(- 1/RT_1 + 1/RT_2)
Almost forgot the minus there...
Okay, looks a bit ugly, but plug in numbers and get results.

Also, now you can get A by plugging them into the equation above.




Uhm... unless you're Arrhenius equation is different from Wikis... nope, that's not linear. It's exponential. It really shouldn't be anything but putting in the right temperature. So something around... dunno, 10^-6/s?
I could run the numbers but... don't expect a linear process, unless I'm looking at the wrong equation?



edit: Typo fixed. Great job screwing up on every part you possibly could, Kato :smalltongue:


edit2: Since I should be doing more important things I instead ran the numbers, which obviously is waste of time if I used the wrong equation.
Anyway, the energy is -1.37e-5 J/mol, A is 4.23 1/s and for the second part my estimate way WAY off and I get only like 1.42e-46 which I'll admit looks weird but the equation isn't so hard I should have really screwed something up...

[weaseldust]

Would a similar but simplified example help you get a grip on the underlying mathematics?

Let's use the version of the Arrhenius equation ln(k) = ln(A) - E/RT. R is about 8, but let's say it's 10 instead for rounder numbers, since this is just an example. So the equation we're using is:


ln(k) = ln(A) - E/10T

Suppose the two data points we're given are that k=0.5 when T=300 and k=0.25 when T=200. We want to find E, and then to find k when T=100.

So, we have these two equations:


(1) ln(0.5) = ln(A) - E/3000
(2) ln(0.25) = ln(A) - E/2000

To find E, we first want to eliminate the ln(A) term because we don't know what it is yet. The obvious way to do this is to take (2) away from (1):


ln(0.5) - ln(0.25) = E/2000 - E/3000

Handily, ln(0.5) = - ln(2) and ln(0.25) = - 2ln(2), so the left hand side is just - ln(2) + 2ln(2) = ln(2).

Meanwhile, the right hand side is 3E/6000 - 2E/6000 = E/6000.

So we have:


ln(2) = E/6000

I.e., E = 6000ln(2).


If you want to find A, you can just plug this value for E into (1):


- ln(2) = ln(A) - (6000ln(2))/3000 = ln(A) - 2ln(2)

I.e.


ln(A) = - ln(2) + 2ln(2) = ln(2)

So A = 2 (but you don't need to know A for the equation, just ln(A), so if the numbers aren't nice, as in your question, you can skip a step here).


Now we know our equation is:


ln(k) = ln(2) - 6000ln(2)/10T = ln(2) - 600ln(2)/T

To find k when T=100, we just plug 100 into that equation:


ln(k) = ln(2) - 600ln(2)/100 = ln(2) - 6ln(2) = -5ln(2) = ln(2^-5)

So k = 2^-5 = 1/32 = 0.03125.


A few notes:


I made the numbers nice so they are easy to type and to read. You will have to use your calculator at various points because the numbers in your question aren't nice at all.
In particular, you won't get nice situations like ln(k) = -5ln(2). Instead, you'll probably have nasty things like ln(k) = -31ln(2.3 x 10^6) or whatever. The easiest thing for you is probably to work out the right hand side with your calculator. Say it comes to -317 (it doesn't), so you have ln(k) = -317. Then you just get k = e^-317.
Note that k isn't linear in T. Instead, ln(k) is linear in -1/T. You have values for T=349 and T=302, and you're looking for the value at T=288. Since you know that 1/288 is greater than 1/302 and 1/349, you're expecting ln(k) to be smaller (more negative) than in the first two cases. ln(x) doesn't change inequalities, so you also expect k to be smaller than in the first two cases. Several orders of magnitude smaller, in fact, because when ln(k) decreases linearly, k decreases exponentially.

[Knaight]

With your part 2, there are two big issues. One is that you're assuming a linear relation, and there isn't one. It's an exponential decay. The other is that you are finding a result for a temperature below the lower temperature given, and appear to be assuming that it is within the range of the two given. It absolutely isn't.


Would a similar but simplified example help you get a grip on the underlying mathematics?

Let's use the version of the Arrhenius equation ln(k) = ln(A) - E/RT. R is about 8, but let's say it's 10 instead for rounder numbers, since this is just an example. So the equation we're using is:

R is extremely unit dependent, and I suspect that the one in use here is in terms of atmospheres, moles, kelvin, and liters. Call it .1 L*Atm/(mol*K), if using 1 significant figure.

[thethird]

This chapter has been a nightmare from the start. The math is, at best, very poorly explained. Even when I do the math correctly, I get answers that are not only wrong, but make zero sense.

Take for example the Arrhenius Equation; the book has done a horrible job explaining the math involved here. Maybe I'm just bad at math, but I haven't gotten a single correct answer yet. I spent 5 hours last night and 2 hours this morning trying to solve the same problem:

A reaction has a rate constant(k1) of 1.21 x 10-4/s at 302K(T1).
k2 is 0.226/s at 349K(T2).

In part one, I needed to find the activation barrier Ea, which took me 5 hours and 6 attempts before finally getting the answer.

In part two, I need to find the rate constant (k) at 288K. I've gotten six different answers and none of them make any sense whatsoever. Since I know this answer should decrease linearly from 0.226 to 1.21 x 10-4, it makes absolutely no sense for my answer to be 0.967 or 2.73 x 10-26.

My personal piece of advice when doing chemistry, or math for the matter, is to always make sure your units make sense. To me that translates to always work on the SI. Unless of course you are able to not care about the units at all.

That said:

Ln K2 - Ln K1 = Ea / R (1/T1 - 1/T2)

Or: Ea / R = (Ln K2 - Ln K1) / (1/T1 - 1/T2)

If I use the two known sets of K and T

Ea / R = (Ln 0.226 - Ln 1.21 x 10-4) / (1/302 - 1/349) = 16891.71 (feel free to divide that by R to get Ea)

We can make that work for us in order to get the second part

Ln K3 = Ln 1.21 x 10-4 + 16891.71 (1/302 - 1/288) = -11.8 = ln (8 * 10-6)