[MrTeo]

Hello everybody!

Here's a problem I tried to solve, but I'm not sure about my work, could you check it?

— • —

Putting a certain amount of gaseous buta-1,3-diene (C₄H₆) in a sealed container, at the temperature of 600 K, and measuring at the same T its partial pressure (P), we notice that it changes over time following this law:



where P₀ is P(t=0) and k>0.

a) Justify this experimental effectiveness
b) Deduce the kinetic law written above
c) Find out the relationship between time and the total pressure of the gases

— • —

a) Buta-1,3-diene breaks homolytically due to high temperature into two free radicals (C₂H₃∙)

C₃H₆→2C₂H₃∙

b) A complete and accurate kinetic description of the phenomenon would require the solving of a really complex differential equation which should take into consideration both the direct and the inverse reaction:

$$ -kc+k'\left(c-c_0\right)^2=\frac{dc}{dt} /$$

Here k and k' are the kinetic constants of the direct and inverse reaction while c is the concentration of buta-1,3-diene at the time t and c₀ is the starting concentration.
I think this analysis goes beyond the intent of those who wrote this problem so I chose to neglect the inverse reaction and approximate the exponential trend using the first two term of the Taylor expansion. As usual we have:



and, solving, we find out the law:



which can be written approximately as:



but we also know that (from the Taylor series):




Knowing that P=cRT and putting k'=k/P₀ we have:





c) If we answer the last request using the same approximations we find quite a good relationship if we work with small t values, but an important evidence is lost: when all the buta-1,3-diene molecules are broken into radicals the original pressure is twice the starting one. To underline this fact I preferred to use an exponential function:

[MrTeo]

So? No ideas? C'mon... 

[AWK]

<sub>a) Buta-1,3-diene breaks homolytically due to high temperature into two free radicals (C2H3∙)

C3H6→2C2H3∙</sub>

Radicals are extremely unstable. In the reaction you will get stable species.

[MrTeo]

Ok, got it: polymerization.
If combining buta-1,3-diene molecules we get a polymer (or more chains) we don't have to consider an inverse reaction and the pressure in the container is equal to the partial pressure of butadiene (the vapor pressure of the polymer is equal to 0), so the law found in the second point is also the answer to the last one, right?

a) This should be the mechanism of the reaction (the termination is, as usual, caused by the combination of radicals)



b) The considerations regarding the reaction kinetics and the approximations used seem ok to me, even if the chemical interpretation of the phenomenon has changed.

c) As said above there is no need to take into consideration the contribution of the product(s) to the total pressure, so we can write:



What do you think?

Edit
Forgot to put an arrow in the mechanism (actually to be more precise I should've put also the arrows to show the interchanging resonance structures of the radicals but I am quite in a hurry): obviously the second electron form the double bond goes on the carbon as seen in the subsequent step.

[MrTeo]

Could someone double-check my answers, please?

[MrTeo]

Up
I really need some feedback...