[edge]

The stoichiometric equation of a hypothetical reaction is given by
aA -> bB + cC
If the rate of the decrease of A, -d[A]/dt, is equal to 2 times the rate of the increase of B, d/dt, and 3 times the rate of the increase of C, d[C]/dt, find the coefficients a, b, and c, which are integers. When the initial concentration of A is doubled, the initial value of
-d[A]/dt is increased by 4-fold. What is the order of the reaction with respect to A? What is the unit of the rate constant, k?

Thanks for any advice.

[Schrödinger]

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[edge]

lets see an example; N2O5 ---------> 2NO2 + 1/2 O2


here concentration of N2O5 decreases with time  and concentration of NO2 & O2 increases with time.Because these species have different coefficients in balanced equation , their concentration do not change at the same rate.When one mole of N2O5 decomposes,two moles of NO2 and one half mole of O2 are formed.this means


-d[N205] = d[NO2]/2 = d[O2]/1/2.


so rate of reaction now defined by dividing by change in time dt


-d[N205]/dt = d[NO2]/2dt = d[O2]/1/2dt


similary you problem gives you this


-d[A]/dt = 2d/dt = 3-d[C]/dt


=> a= 1 , b = 1/2 , c= 1/3


Now  When the initial concentration of A is doubled, the initial value of
-d[A]/dt is increased by 4-fold


lets say the initial rate of A = rate1


we know that  if A--------> Product


rate= k[A]^m where  m = order of reaction & k is rate constant


similary for this reaction


rate1 = k [A]^m


as concentration of A doubled the rate1 increase by 4 fold, so we calculate the ratio of the rates and concentration


4 rate1 = k [2A]^m


-----------------


rate1 = k[A]^m


4= [2A/A]^m


4 = 2^m


(2)^2 = 2^m   


=> m = 2


where m is the order ,


so order w.r.t A = second

Above is a solution that provided by my friend.
Unfortunately, I can't understand regarding the following answer:
"
-d[A]/dt = 2d/dt = 3-d[C]/dt
=> a= 1 , b = 1/2 , c= 1/3
"
I got a bit confusing why will get 1/3 and why the negative value in [C] is removed?

Thanks for any advice.

[Schrödinger]

The stoichiometric equation of a hypothetical reaction is given by
aA -> bB + cC

If that's the equation and the statement says

If the rate of the decrease of A, -d[A]/dt, is equal to 2 times the rate of the increase of B, d(B)/dt, and 3 times the rate of the increase of C, d[C]/dt,

then clearly, the equation is
-d[A]/dt = {d(B)/dt}/{1/2} = {d[C]/dt}/{1/3}

Which means the reaction will look like :
6A -> 3B +2C
after multiplying throughout by 6 (since you need integral co-efficients)

When the initial concentration of A is doubled, the initial value of
-d[A]/dt is increased by 4-fold.

We can only assume at this stage that the equation is not elementary and involves a multistep mechanism, because rate = k[A]^m and the integral co-efficient of 6 doesnt fit the data. Hence the order wrt A is 2.

[edge]

Hi Schrödinger,

Does you have any idea why 4 rate1 = k [2A]^m must divide rate1 = k[A]^m and generate 4= [2A/A]^m?

eg.
4 rate1 = k [2A]^m

-----------------

rate1 = k[A]^m

4= [2A/A]^m


Thanks for any advice.