[morten925]

Hi chemists/physicists/mathematicians/students

I am attending on calculating the half lives for the reactants A + B, respectively, in the reaction A + B  P, with k_r = 4.5 · 10^-3 dm³ mol^-1 s^-1, after 3600 s [A] = 0.045 mol dm^-3, [B ] = 0.020 mol dm^-3 and initially [A₀] = 0.0750 dm^-3 and  [B₀] = 0.05 dm^-3,

I don't think it is possible to make use of the fact that t_½ = ([A₀] · k_r)^-1 because I think this equation accounts only for reactions of type: A --> P, but I could be greatly mistaken (using this equation was my initally attempt, but I realised it led me to incompleteness). I have heard rumours that the half lives for the first type of reaction are undefined but I can hardly believe them, because Atkins/Julio de Paula does/do come up with the final answers but I can't reproduce those answers. For additional information I refer to Atkin's Physical Chemistry, 10th edition, page 830, Brief illustration 20B.1.

Help is undoubtly appreciated, thanks!

Best regards, Morten V

(mod edit to sort out [ B] "bug")

[mjc123]

I don't think it is possible to make use of the fact that t½ = ([A0] · kr)-1 because I think this equation accounts only for reactions of type: A --> P

Yes, if the reaction is second order. It also works for A + B  P if [A]₀ = [B ]₀, so that [A] = [B ] throughout. When that is not the case, it is a bit more complicated, but still doable. Write [B ] = [A] - c, where c is a constant, and solve the differential equation for [A]
As a check, I get 6161 s and 2557 s for the half lives of A and B respectively. Does Atkins agree?

[morten925]

Thank you very much, mcj123. In the book they write:

Anwer: t_1/2(A) = 5.1 · 10³ s, t_1/2(B) = 2.1 · 10³ s

Maybe there is a miscalculation

[mjc123]

Maybe, I'll check - I did it very quickly!

[Corribus]

I got the same answer mcj123 got. So, either we're both making the same methodological mistake, or the book is wrong.

@morten925

The reason the half life is "undefined" for this type of reaction is that it depends on what reactant you are talking about. The time it takes for A to deplete by half is different from the time it takes for B to deplete by half.

[mjc123]

Having rechecked, I get the same answer. I also get the given result for 3600 s, which seems to me inconsistent with their half lives. I wonder if they used a k value of 5.5e-3 rather than 4.5e-3 to get the half lives?

[shafaifer]

I got the same answer mcj123 got. So, either we're both making the same methodological mistake, or the book is wrong.

@morten925

The reason the half life is "undefined" for this type of reaction is that it depends on what reactant you are talking about. The time it takes for A to deplete by half is different from the time it takes for B to deplete by half.

Having rechecked, I get the same answer. I also get the given result for 3600 s, which seems to me inconsistent with their half lives. I wonder if they used a k value of 5.5e-3 rather than 4.5e-3 to get the half lives?

Thank you! I do not see through your methodology because all initial and time-dependent concentrations are known, the change in concentration is the same for both reactants. This is why I am unfortunately not capable at the moment to understand what is meant by solving for the time-dependent concentration, A, because it's value is already given after 3600 s.

[mjc123]

But not at any other time

[morten925]

Thanks, sorry for the typo "it's" it should without doubt be "its"

Is it too much of me to ask you to specify the differential equation in question for this second order reaction?   I have no reasonable guess of your intention, I might have learned the  techniques but their application in this problem is blurred and hidden from me (of course I have dealt with first order diff. equations (the reaction is first order in A and B, hence second order overall) before but mostly within mathematical isolation and not in connection with practical application You can see that I believe in the results that you and Corribus got and that they made a systematical error in their book.

[Corribus]

You don't have to solve any differential equations, necessarily, if you are willing to look up the integrated rate law, which for a 2nd order equation is:

[tex]\frac {[A]}{[B ]} = \frac {[A_0]}{[B_0]} \exp(([A]_0-[B ]_0)kt)[/tex]

Just use the substitution recommended by mjc123 ([B ] = [A] - c, where c = 0.025) and solve for t under the conditions appropriate for determining a half life (the concentration of the reactant is half the initial concentration). You'll have to do this twice, once for A and once for B.

[mjc123]

But if you want to, (omitting square brackets for clarity)
dA/dt = dB/dt = -kAB = -kA(A-c)
I didn't recall (or look up) the rate law, but my solution was
ln(A/A₀) - ln(B/B₀) = ckt
which is equivalent to the equation quoted by Corribus

[Plontaj]

I am attending on calculating the half lives for the reactants A + B, respectively, in the reaction A + B  P, with kr = 4.5 · 10-3 dm3 mol-1 s-1, after 3600 s [A] = 0.045 mol dm-3, [B ] = 0.020 mol dm-3 and initially [A0] = 0.0750 dm-3 and  [B0] = 0.05 dm-3

I don't want to cling but I hate examples like this, because they are stirring a little bit. How many are reactions which stoichiometry agrees with the rate law? only few! I know that it's school example but later people see the reaction like that and they automatically say that it's second order (first to A and B). How do you know that? Did you make an experiment?