Finding out the value of rate constant on titration

The question goes as follows


Consider the reaction $$\ce{X(aq) \longrightarrow Y(aq) + Z(aq)}$$


The above reaction is titrated with a reagent $A$ and all of $X, Y, Z$ reacts with $A$ with the ratio of 'n' factors of $X, Y, Z=1:2:3$


here n factor refers to change in oxidation number of substance per mole of the substance on titration with $A$


The following data was obtained. 


![enter image description here](https://i.stack.imgur.com/bqM6Y.jpg)


The question is to find out the value of rate constant considering the titration reaction to be first order. 


The value of rate constant is given by $\frac{0. 693}{t\_{1/2}}$ So I have to find out half life of the reaction . I am facing trouble finding out since the rate of voume of A required depends on the amount of substance present at that instant. Any help shall be highly appreciated. Thanks. 




From a mole balance, we find $C = n\_x(t) + n\_y(t) + n\_z(t) = n\_x(t) + 2n\_y(t)$, where the second equality follows from the stoichiometry of the reaction. In particular, our boundary conditions are $C = n\_x(0) = 2n\_y(\infty).$ 


We assume that the reaction is first-order, so $n\_x(t) = n\_x(0)\exp(-kt) = C\exp(-kt).$ The two measurements give us the set of equations 
\begin{align\*} 
80D &= n\_x(5) + 5n\_y(5) &&= C\exp(-5k) + \frac{5}{2}C(1-\exp(-5k)) \\
100D &= 5n\_y(\infty) &&= \frac{5}{2}C,
\end{align\*}
where $D$ is an undetermined proportionality constant for the titration, and the coefficient of 5 in $n\_y$ accounts for the fact that 2 equivalents of $A$ react with $n\_y$ and 3 equivalents with $n\_z$. Dividing one equation by the other produces the final result $$\frac{4}{5} = \frac{2}{5}\exp(-5k)+1-\exp(-5k),$$ which can be solved for $k$ and hence $t\_{1/2}$.


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This is quite a tricky problem, and I'm not sure I have it all. The amounts of $X,Y, Z$ at time $t$ are $X=X\_0\exp(-kt), ~Y=X\_0(1-\exp(-kt)), ~ Z=X\_0(1-\exp(-kt))$ and the measurement is volume A. At infinite time $X\_\infty=0$ and so using the ratios given $2Y\_\infty+3Z\_\infty=5\cdot 100$ is equivalent to $2X\_0$ (by adding up $X,Y,Z$ at $t=\infty$) and at $t=5, $ $X\_5+2Y\_5+3Z\_5=6\cdot80$ is equivalent to $X\_0(2-\exp(-5k))$ so you should have all that you need if you now take ratios as $(500/480)=2X\_0/(X\_0(2-\exp(-5k)))$


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