# Question #cecb0

Aug 13, 2017

Write on species mole basis

#### Explanation:

Let us denote each components using the following letters
$A = {\left(N {H}_{4}\right)}_{2} S {O}_{4}$
$B = N {H}_{4}^{+}$
$C = S {O}_{4}^{2 -}$
$D = N {H}_{3} \left(a q\right)$
$E = {H}^{+}$
$F = {H}_{2} O$
$G = H S {O}_{4}^{-}$
$H = O {H}^{-}$

Then we have
$A \setminus \underset{{k}_{2}}{\setminus \stackrel{{k}_{1}}{\setminus r i g h t \le f t h a r p \infty n s}} 2 B + C$
$B \setminus \underset{{k}_{4}}{\setminus \stackrel{{k}_{3}}{\setminus r i g h t \le f t h a r p \infty n s}} D + E$
$C + F \setminus \underset{{k}_{6}}{\setminus \stackrel{{k}_{5}}{\setminus r i g h t \le f t h a r p \infty n s}} G + H$

If you are writing a component wise mass balance then start with components that take part in a single reaction

$\frac{{\mathrm{dC}}_{A}}{\mathrm{dt}} = {k}_{2} {C}_{B}^{2} {C}_{C} - {k}_{1} {C}_{A}$
$\frac{{\mathrm{dC}}_{D}}{\mathrm{dt}} = {k}_{3} {C}_{B} - {k}_{4} {C}_{D} {C}_{E}$
$\frac{{\mathrm{dC}}_{E}}{\mathrm{dt}} = {k}_{3} {C}_{B} - {k}_{4} {C}_{D} {C}_{E}$

Generally water molecule is present in excess. Hence we don't write a balance for that. And hence the third reaction is generally treated as pseudo first order reaction.

$\frac{{\mathrm{dC}}_{G}}{\mathrm{dt}} = {k}_{5} {C}_{C} - {k}_{6} {C}_{G} {C}_{H}$
$\frac{{\mathrm{dC}}_{H}}{\mathrm{dt}} = {k}_{5} {C}_{C} - {k}_{6} {C}_{G} {C}_{H}$

Now onto the other 2 components

$\frac{{\mathrm{dC}}_{B}}{\mathrm{dt}} = {k}_{1} {C}_{A} - {k}_{2} {C}_{B}^{2} {C}_{C} - {k}_{3} {C}_{B} + {k}_{4} {C}_{D} {C}_{E}$
$\frac{{\mathrm{dC}}_{C}}{\mathrm{dt}} = {k}_{1} {C}_{A} - {k}_{2} {C}_{B}^{2} {C}_{C} - {k}_{5} {C}_{c} + {k}_{6} {C}_{G} {C}_{H}$

Then for each component do a mass balance as
${v}_{0} {\left({C}_{i}\right)}_{o} - v \left({C}_{i}\right) + V \left({r}_{i}\right) = V \frac{{\mathrm{dC}}_{i}}{\mathrm{dt}}$

where $i \setminus \in \left\{A , B , C , D , E , G , H\right\}$

If you are writing an overall balance then we have the overall reaction as
$A + F \setminus \underset{{k}_{2}}{\setminus \stackrel{{k}_{1}}{\setminus r i g h t \le f t h a r p \infty n s}} B + D + E + G + H$

Then we can write balances for A,B,D,E,G,H given we have ${k}_{1}$ and ${k}_{2}$.