Equilibrium reaction ICE method to ideal gas law

[marellasunny]

QUESTION: Say I have the following equilibrium reaction
$$CO+\frac{1}{2}O_2\leftrightharpoons CO_2$$
The stoichiometric mixture of CO and $O_2$ in a closed vessel, initially at 1 atm and 300K, is exploded. Calculate the composition of the products of combustion at 2500K and the gas pressure.
Take $K_p$=27.5. Take $\alpha$ as the degree of dissociation.
$$$$
MY ATTEMPT AT AN ANSWER:
I use the ICE method to find out the reactant and product composition
$$CO+\frac{1}{2}O_2\leftrightharpoons CO_2$$
The reaction quotient can be given as Q=0/0.5=0. Therefore, the products must have plus sign.

Then I get the final composition(/concentration) as
$$CO=[1-\alpha] $$
$$O_2=[1-\alpha/2]$$
$$CO_2=[\alpha]$$

then, $$ K_p= \frac{\alpha}{[1-\alpha][1-\alpha/2]^{0.5} } $$
I would then use the IDEAL GAS LAW to find the product mixture

i.e $$ p_RV=n_RRT_R$$ $$p_pV=n_PRT_p$$

I would then substitute these values into the expression for $K_P$.

MY QUERY:
1.Is my calculation for $K_P$ correct? I take $CO_2$ as the product and CO and $O_2$ as the reactants.

2.I am not able to figure out what is the number of moles of the products $n_P$ and the number of moles of the reactants $n_R$.The book's solution gives $n_R$=3/2 and $n_P=1+\alpha/2$.

The author solves with the initial species as $CO_2$ and obtains the compositions $CO_2=(1-\alpha)$;$CO=\alpha$;$O_2=\alpha/2$, which is confusing given that the question states the reaction starts with CO and O2 in a closed vessel.

 

[Borek]

$$O_2=[1-\alpha/2]$$

If the mixture was stoichiometric, initial amount of oxygen was not 1. Unless I am missing something.

1.Is my calculation for $K_P$ correct? I take $CO_2$ as the product and CO and $O_2$ as the reactants.

See above. Other than that looks like you are on the right track.

The author solves with the initial species as $CO_2$ and obtains the compositions $CO_2=(1-\alpha)$;$CO=\alpha$;$O_2=\alpha/2$, which is confusing given that the question states the reaction starts with CO and O2 in a closed vessel.

Actually initial composition doesn't matter (as long as it is stoichiometric), as if the mass balance doesn't change, final equilibrium will be identical. Calling α degree of dissociation suggests CO₂ and its decomposition as a starting point. Doesn't mean your approach is wrong, it should yield the same result in terms of final pressures. But as α's are different, final result expressed using your α and using book's α won't look identical.

 

[marellasunny]

Borek,please find attached the problem I'm struggling with. I've highlighted the step which I do not understand also.

The K_p values were arrived from the temperatures empirically. But,what is the expression to the left in page 89?

 

[Borek]

Hard to say something specific without seeing eq 3.40. But in general it looks like equilibrium expressed using reaction degree (fraction) - as defined at the very top of the page. In a way similar to http://en.wikipedia.org/wiki/Ostwald_dilution_law

 

[marellasunny]

Oswald Dilution Law, very helpful insight.Since, concentration=n/V , this is not how it is expressed in the red highlighted area(attachment previous post). I first thought this was a relation between K_p and K_c,but the units didn't agree. It doesn't agree even in the Oswald dilution law.
$$\frac{1-\alpha }{\alpha (\alpha /2)^{0.5}}{\frac{n_p}{p_p}^{0.5}}=K_p $$
I also Attached 3.40 eq.