Write the rate equation for the following reactions:
$1.$ $2 N_2 O_{5(g)} \rightarrow 4 NO_{2(g)} + O_{2(g)}$
$2.$ $2 HI_{(g)} \rightarrow H_{2(g)} + I_{2(g)}$

The rate of a reaction is defined as the change in concentration of reactants or products per unit time,divided by their respective stoichiometric coefficients.
For reaction $1$: $2 N_2 O_{5(g)} \rightarrow 4 NO_{2(g)} + O_{2(g)}$
Rate $= -\frac{1}{2} \frac{d[N_2O_5]}{dt} = \frac{1}{4} \frac{d[NO_2]}{dt} = \frac{d[O_2]}{dt}$
For reaction $2$: $2 HI_{(g)} \rightarrow H_{2(g)} + I_{2(g)}$
Rate $= -\frac{1}{2} \frac{d[HI]}{dt} = \frac{d[H_2]}{dt} = \frac{d[I_2]}{dt}$