Writing rate of disappearance and rate of appearance using rate law [closed]

Question

I have studied that rate of disappearance and rate of appearance is the change in concentration of reactants and products (respectively) with respect to time. Thus,

$$\text{ROD} = -\frac{\Delta c_R}{\Delta T}$$

and

$$\text{ROA} = \frac{\Delta c_P}{\Delta T}$$

and

$$\text{rate of reaction} = % % \frac{\text{ROD}} {\text{stoichiometric coefficient}} % % = \frac{\text{ROA}} {\text{stoichiometric coefficient} }$$

Then writing rate of reaction using rate law was taught.

And according to rate law rate of reaction can be expressed as:

$$\text{rate of reaction} = k[\ce{A}]^x[\ce{B}]^y$$

here $x + y = \text{overall order of reaction}$, with $\ce{A}$ and $\ce{B}$, the reactants.

Here is my confusion: Now they wrote rate of disappearance and appearance using rate law. How using rate law they wrote expression for ROA and ROD.

For a reaction: $\ce{aA -> bB}$ (elementary reaction), it was stated that: $$\text{ROD}_\text{A} = k_d [\ce{A}]^a$$

So how ?

Using rate law we can write rate of reaction.. (OK) but how can we write ROA and ROD ??!!

Hope you understood what I meant to say.

Answer 1

You overthink it. It is just matter of eventual multiplication or division with respective stoichiometric coefficients.

If there is an elementary reaction

$$\ce{a A -> b B}$$

then for respective reaction, appearance and disappearance rates:

$$\begin{align} R_\mathrm{r} &= k_\mathrm{r}[A]^a\\ R_\mathrm{a,B} = \frac {\mathrm{d[B]}}{\mathrm{d[t]}}=bR_\mathrm{r} &= bk_\mathrm{r}[A]^a = k_\mathrm{a,B}[A]^a\\ R_\mathrm{d,A} = -\frac {\mathrm{d[A]}}{\mathrm{d[t]}} = aR_\mathrm{r} &= ak_\mathrm{r}[A]^a = k_\mathrm{d,A}[A]^a \end{align}$$

.. and backwards:

$$\begin{align} R_\mathrm{r} = \frac{R_\mathrm{a,B} }{ b} &= \frac 1b \frac {\mathrm{d[B]}}{ \mathrm{d[t]}}\\ R_\mathrm{r} = \frac{R_\mathrm{d,A} }{ a} &= -\frac 1a \frac {\mathrm{d[A]}}{\mathrm{d[t]}} \end{align}$$