Confusion regarding a chemical kinetics problem

[palaphys]

TL;DR Summary: cannot find out error in solution proposed.

[![question with rate laws][1]][1]

Now the rate law for the reaction (i.e reaction rate) can be written as:

$$ R= k[N_2O_5] $$
my main question is, WHAT is this reaction equal to?
what I mean here is, whether
$$k[N_2O_5]= -d[N_2O_5]/dt$$
or is it
$$k[N_2O_5]= -1/2 \frac{d}{dt} [N_2O_5] $$ ?
The latter seems to be more apt, as the reaction rate must be -1/2 (disappearance rate of N2O5), which adheres to the stoichiometry of the reaction.
Then I felt that the question would be solved as follows:
$$ -2k \int_{0}^{t} dt = \int_{P_{i,\text{total}}}^{P_{f,\text{total}}} \frac{d[N_2O_5]}{[N_2O_5]}$$
where $$P_{f,total} $$ and $$ P_{i,total}$$ represent the total pressure of the gasuous mixture.

However,
when we substitute the given values for k in this equation:
$$ k = -\frac{1}{2t} \int_{P_{i,\text{total}}}^{P_{f,\text{total}}} \frac{d[N_2O_5]}{[N_2O_5]}$$,

we get $$ k= −1.19×10^4 $$
which physically makes no sense.
Where have I gone wrong? What is the correct approach to solve this?

 

[mjc123]

Why do you use the total pressure? What is [N₂O₅] as a function of P_tot? Is it increasing?

 

[palaphys]

Why do you use the total pressure? What is [N₂O₅] as a function of P_tot? Is it increasing?

I used total pressure because the rate of the overall reaction was in picture.

 

[Borek]

I used total pressure because the rate of the overall reaction was in picture.

I feel like you have missed the point of mjc question. Look at two other questions he asked.

 

[palaphys]

I feel like you have missed the point of mjc question. Look at two other questions he asked.

AH okay I got it. so I gotta use something like an ICE table. like this:

that leaves me with this equation: 1/2 (100) ln( 0.5/0.476)= k, solving we get k= 2.46 x 10^-4 1/s, however the answer is given to be 4.92x 10^-4, which is surprisingly double of the k I have calculated..

 

[mjc123]

Referring to your initial post, I assume they intended the rate equation to be
k[N₂O₅] = -d[N₂O₅]/dt
rather than the version with a coefficient of 1/2.

 

[palaphys]

Referring to your initial post, I assume they intended the rate equation to be
k[N₂O₅] = -d[N₂O₅]/dt
rather than the version with a coefficient of 1/2.

WHY? this is confusing me. I strongly feel there should be a 1/2 factor at the front of the differential, as the stoichiometry is two molecules reacting. This should be the case if we represent the rate constant for the reaction as a whole. The model which you have proposed and so has the textbook seems to define a rate constant for the disappearance rate of N2O5 in specific.

 

[palaphys]

https://chemistry.stackexchange.com/questions/38167/do-coefficients-matter-in-rate-law/38168#38168

I think the first answer under this answers my question

 

[mjc123]

The model which you have proposed and so has the textbook seems to define a rate constant for the disappearance rate of N2O5 in specific.

Well, since the reaction is the disappearance of N₂O₅, that seems reasonable. I would instinctively say that the rate of that reaction was -d[N₂O₅]/dt.

However, the IUPAC definition would agree with you. I have said before (perhaps on chemicalforums) that I don't like it, and you should ALWAYS specify what you mean by "reaction rate" in terms of the rate of change of a specific reagent. But IUPAC has spoken...

In theory the IUPAC definition should be unambiguous. But what if I wrote the reaction as
N₂O₅ → N₂O₄ +1/2 O₂ ?
The definition doesn't specify "integers in lowest terms". It applies to the reaction as written, and it can be written in more than one way. (Your textbook would appear to be inconsistent, though.)

There is also the issue hinted at in the stackexchange, of what value of k you use in a formula like
t_1/2 = ln(2)/k
for a first-order reaction. You have to be careful about blindly using a formula without understanding its terms.

 

[palaphys]

Well, since the reaction is the disappearance of N₂O₅, that seems reasonable. I would instinctively say that the rate of that reaction was -d[N₂O₅]/dt.

However, the IUPAC definition would agree with you. I have said before (perhaps on chemicalforums) that I don't like it, and you should ALWAYS specify what you mean by "reaction rate" in terms of the rate of change of a specific reagent. But IUPAC has spoken...

In theory the IUPAC definition should be unambiguous. But what if I wrote the reaction as
N₂O₅ → N₂O₄ +1/2 O₂ ?
The definition doesn't specify "integers in lowest terms". It applies to the reaction as written, and it can be written in more than one way. (Your textbook would appear to be inconsistent, though.)

There is also the issue hinted at in the stackexchange, of what value of k you use in a formula like
t_1/2 = ln(2)/k
for a first-order reaction. You have to be careful about blindly using a formula without understanding its terms.

I agree that the IUPAC definition is kind of ambiguous. also the formula for half life, is derived using a stoichiometry of 1. I am sure that if we consistently took into account the stoichiometry in a rate expression, we would get the same answer even if I wrote the factor -1/2 (in half-life calculation)

But the big question now is, in school exams, which definition should I use? In almost 99% of the questions in my textbook they just bluntly mention: ''Calculate the rate constant''. I guess I will just go for the rate constant for the disappearance of the reactant.

Also you are right in saying that the reaction must be written in a different way, i.e $N_2O_5 → N_2O_4 + 1/2 O_2$ for the textbooks answer to be accurate.

 

[mjc123]

For school exams, ask your teacher, or the examining board. But "calculate the rate constant" should always prompt the question "rate of what?" In the absence of a more specific answer, it should be the rate of the reaction as written, using the IUPAC definition. But then, as I say, your textbook seems to be inconsistent.

 

[palaphys]

For school exams, ask your teacher, or the examining board. But "calculate the rate constant" should always prompt the question "rate of what?" In the absence of a more specific answer, it should be the rate of the reaction as written, using the IUPAC definition. But then, as I say, your textbook seems to be inconsistent.

https://goldbook.iupac.org/terms/vi...tions,or amount of substance) concentrations (
Wonder if I should follow the terms above.

Despite this, I will continue to use rate of disappearance, especially with reactions with a single reactant with a stoichiometric coefficient which is not one.

 

[DrDu]

Isn't this a bimolecular reaction?