Write the differential rate expression for the following reactions and determine their order of reaction:
$1) \ 2 \ HI \rightarrow H_2 + I_2$
$2) \ 2 \ NO_{(g)} + O_{2(g)} \rightarrow 2 \ NO_{2(g)}$

(N/A) For a general reaction $aA + bB \rightarrow \text{products}$,the differential rate expression is given by $Rate = -\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt}$.
$1) \ 2 \ HI \rightarrow H_2 + I_2$
Differential rate expression: $Rate = -\frac{1}{2} \frac{d[HI]}{dt} = \frac{d[H_2]}{dt} = \frac{d[I_2]}{dt}$.
Order of reaction: This is a decomposition reaction which is experimentally found to be of $0^{th}$ order with respect to $HI$ at high concentrations,but generally,for such elementary reactions,it is considered $2^{nd}$ order. However,based on standard textbook examples,it is often cited as $2^{nd}$ order.
$2) \ 2 \ NO_{(g)} + O_{2(g)} \rightarrow 2 \ NO_{2(g)}$
Differential rate expression: $Rate = -\frac{1}{2} \frac{d[NO]}{dt} = -\frac{d[O_2]}{dt} = \frac{1}{2} \frac{d[NO_2]}{dt}$.
Order of reaction: This is a $3^{rd}$ order reaction ($2^{nd}$ order with respect to $NO$ and $1^{st}$ order with respect to $O_2$).