[Big-Daddy]

Consider a gas sample of N2O5 with n moles of gas. This then reacts via 2 N2O5  4 NO2 + O2.

When a fraction f of the N2O5 has reacted, calculate the total pressure of the sample in terms of the initial pressure (P₀) and f.

Originally I wrote that when f of N2O5 has reacted, there will be n*(1-f) mol of N2O5 left, 2nf mol of NO2 created, and (1/2)nf mol of O2 created. So x[N2O5]=(n*(1-f))/((n*(1-f))+(2nf)+((1/2)nf))=(1-f)/(1+(3/2)f). Now x[N2O5]=P[N2O5]/P[total], so P[total]=P[N2O5]*(1+(3/2)f)/(1-f).

Where do we go from here? We need to express P[N2O5] in terms of P₀.

[Borek]

You started with N₂O₅ only?

[Big-Daddy]

You started with N₂O₅ only?

Yes. There were n moles of N2O5 to begin with, nothing else. Does that help?

I have inferred something else. n_total=n*(1+(3/2)f)) where n is the starting number of moles of N2O5 i.e. n=n₀. I get the feeling that, seeing as P₀ came solely from n₀ (N2O5 was the only gas at the beginning), if n_total=n₀*(1+(3/2)f)) then P_total=P₀*(1+(3/2)f)). But I have no idea how to prove this quantitatively.

[Borek]

You started with N₂O₅ only?

Yes. There were n moles of N2O5 to begin with, nothing else. Does that help?

What does it tell you about the relationship between P₀ and initial P_N2O5?

[Big-Daddy]

You started with N₂O₅ only?

Yes. There were n moles of N2O5 to begin with, nothing else. Does that help?

What does it tell you about the relationship between P₀ and initial P_N2O5?

P₀=P₀[N₂O₅]. Yes I knew this before too, but now how to reach P[N2O5] in terms of P₀[N₂O₅]?

[Corribus]

Total pressure, P_t, at any time is equal to the partial pressures of O₂, NO₂ and N₂0₅.

Or: P_t = P(O₂) + P(NO₂) + P(N₂0₅)

Now, how do you relate each of the quantities on the right hand side of the equation in terms of the starting pressure of the reactant at time zero and the fraction of starting material reacted at time t? Is there an expression for f in terms of the boldened quantities that would be helpful?

[Big-Daddy]

The second bold term is just f itself, whereas the first bold term is P₀.
But no, I'm not sure how to convert from the change in moles, which I can work out in terms of f, to the change in partial pressures.

[Corribus]

Ok, let me rephrase, because I see what I wrote was confusing.

What I meant was: Now, how do you relate each of the quantities on the right hand side of the equation in terms of the starting pressure of the reactant at time zero and the pressure of the reactant at time t?  Then you can worry about substitution for f. 

(Don't worry about moles - that's just confusing things.  Everything can be done with pressures, which are your concentrations.)

[Big-Daddy]

Ok, let me rephrase, because I see what I wrote was confusing.

What I meant was: Now, how do you relate each of the quantities on the right hand side of the equation in terms of the starting pressure of the reactant at time zero and the pressure of the reactant at time t?  Then you can worry about substitution for f. 

(Don't worry about moles - that's just confusing things.  Everything can be done with pressures, which are your concentrations.)

I'm not sure. I could manage this relation in terms of moles but I don't get how to draw the relation in terms of pressures (since the reaction equation occurs in terms of moles).

My thoughts are still like this: n_t=n₀*(1+(3/2)f). I can't help but feel this means P_t=P₀*(1+(3/2)f), but the question is how to quantitatively prove that the n relationship is equivalent to the P relationship? If it's harder then we can leave that aside and focus on the pressures for now, but I'm not really sure where to start with them.

Edit: I see. What I should have mentioned before is that this reaction occurs in a closed-volume container.

With this in mind, any relation I can find between moles also applies to pressures (as I can replace n in any situation with PV/(RT), assuming ideal gases, and then divide through by V/(RT) to leave an expression where P has simply replaced n, as V is constant). So P_t=P₀*(1+(3/2)f) is therefore proven.

It's also quite easy to show that if we start with n₀[N2O5], n₀[O2] and n₀[NO2], then: (subscript 0 means initial, subscript t means at time t, [total] means total) n_t[total]=n₀[N₂O₅]*(1+(3/2)f)+n₀[NO2]+n₀[O2]. This translates into P_t[total]=P₀[N₂O₅]*(1+(3/2)f)+P₀[NO2]+P₀[O2]. If we then substitute P₀[component]=P₀[total]*n₀[Component]/n₀[total], we end up with:

P_t[total]=(P₀[total]*((n₀[N₂O₅]*(1+(3/2)f))+n₀[NO₂]+n₀[O₂]))/(n₀[N₂O₅]+n₀[NO₂]+n₀[O₂]). I should think this formula is correct particularly since it reduces to my last one when n₀[NO₂]=n₀[O₂]=0.

Thanks for the help. Can you think of any extension questions I might use to go further?

[Big-Daddy]

Actually I have one of my own:

Let's say we want to model the reaction with time rather than with the  fraction of N₂O₅ which has reacted.

Considering that n₀[N2O5]=n[N2O5] reacted + n_t[N2O5], and f=n[N2O5] reacted/n₀[N2O5], could we write f=(n₀[N2O5]-n_t[N2O5])/n₀[N2O5], substitute in a rate law integrated expression for n_t[N2O5] (e.g. n_t[N2O5]=n₀[N2O5]*e^(-kt), if the reaction were first-order overall and in N2O5), and then substitute this expression for f into the P_t[total] equation I derived in my last post? (Interestingly, all n₀[N2O5] terms seem to drop out of f if the reaction is first or second order.)

[Borek]

P=nRT/V

that means that - as long as volume & temperature are constant

P=k×n

Pressure and number of moles are directly related, this is Avogadro's hypothesis.

As usual, you are building elaborate models not seeing or not knowing the most basic things about the system you are describing.

[Corribus]

I agree with borek; I think you are really overcomplicating what's effectively just a simple stoichiometry problem.

If you knew a rate law expression I see no reason why you couldn't express the concentrations at any time in terms of the rate constant.

[Big-Daddy]

P=nRT/V

that means that - as long as volume & temperature are constant

P=k×n

Pressure and number of moles are directly related, this is Avogadro's hypothesis.

As usual, you are building elaborate models not seeing or not knowing the most basic things about the system you are describing.

I would ask whether the calculations in my last post were correct or not, but clearly you have not read it. I already wrote this (using PV=nRT) down there. What is giving you the idea that I do not know or see the most basic things about this system?

When you see the mass of calculations, if you start reading them you may find that sometimes I do know what I'm doing. And I don't think this model is 'elaborate' at all ...

To summarize, your k=RT/V and so, if we convert every n to P and divide through by RT/V, we are left with the analogous expression now with P as we had before with n.

I don't see any overcomplication anywhere. The method I'm following is exactly that which brings (what seems to be) the right answer to me. Prove it in terms of moles, then prove that any mole relationship also holds in terms of pressures, thus proving the same relationship in terms of pressures.

[Corribus]

You're overcomplicating it because the convention is to express molar gas concentrations in terms of partial pressures.  Unless you are given specific information which would lead you to abandon this convention, anything else is just additional and needless work.  You're not wrong, mind.  But sometimes adding too much information can lead you to be penalized in an examination.  And overthinking a problem CAN lead you to a wrong answer in many situations.  Case in point - all your talk of moles and n's caused me to just skim over your posts because I knew the answer was fairly simple to derive and I saw this mass of posts and handwringing about mole conversions and so forth. .I didn't even realize you had gotten pretty much the right answer in your second post - you just didn't express it as the question asked. You might say: well that's my problem for not reading thoroughly.  Maybe you're right, but if I'm grading your exam then you're still the one who loses points. 

So... maybe there's a lesson to be learned in there for both of us, eh?

[Big-Daddy]

You're overcomplicating it because the convention is to express molar gas concentrations in terms of partial pressures.  Unless you are given specific information which would lead you to abandon this convention, anything else is just additional and needless work.  You're not wrong, mind.  But sometimes adding too much information can lead you to be penalized in an examination.  And overthinking a problem CAN lead you to a wrong answer in many situations.  Case in point - all your talk of moles and n's caused me to just skim over your posts because I knew the answer was fairly simple to derive and I saw this mass of posts and handwringing about mole conversions and so forth. .I didn't even realize you had gotten pretty much the right answer in your second post - you just didn't express it as the question asked. You might say: well that's my problem for not reading thoroughly.  Maybe you're right, but if I'm grading your exam then you're still the one who loses points. 

So... maybe there's a lesson to be learned in there for both of us, eh?

OK, so as an exercise then let's try and do it the simple way without using moles. Because the reactions themselves happen in terms of moles, how do we express the partial pressures of N₂O₅, O₂ and NO₂ after the fraction f moles of N₂O₅ has reacted? Key insight is since V is constant so is V/(RT) and this means that thinking of the reaction equation in terms of moles is the same as thinking about it in terms of partial pressures.

Well, again, PV=nRT so n=P*(V/(RT)) and we can always just divide everything through by V/(RT) so P_t[N₂O₅]=P₀[N₂O₅]*(1-f); P_t[NO2]=P₀[NO2]+2*f*P₀[N₂O₅]; P_t[O2]=P₀[O2]+(1/2)*f*P₀[N₂O₅]. P_t[total]=P₀[N₂O₅]*(1-f)+P₀[NO2]+2*f*P₀[N₂O₅]+P₀[O2]+(1/2)*f*P₀[N₂O₅]=P₀[N₂O₅]*(1-f+2*f+(1/2)*f)+P₀[NO2]+P₀[O2]=P₀[N₂O₅]*(1+(3/2)*f)+P₀[NO2]+P₀[O2]. And there is our solution in terms of f and the initial pressures! Hopefully this would get me full marks in the exam. In the special case of P₀[NO2]=P₀[O2]=0, then P_t[total]=P₀[N₂O₅]*(1+(3/2)*f), and since now P₀[N₂O₅]=P₀[total], there is the simple P_t[total]=P₀[total]*(1+(3/2)*f) which the question was looking for.

Mainly it may seem I put up a lot of maths unnecessarily, I guess that's down to my not having any reliable source to check my method/math against except you guys! (and of course myself but I can't be trusted)