Problem $1$

The following gas$-$phase elementary reactions are involved in the conversion of species A

and $BC$ to species ABC,

$A+BCAB+C,k_1$

$A+BCharrAB{C}^{**},k_2,k_{-2}$

$AB{C}^{**}AB+C,k_3$

$AB{C}^{**}+M$ABC$+M,k_4$

Here, $M$ represents any species in the mixture and $AB{C}^{**}$ represents an energized

intermediate that is present in small concentrations and does not accumulate. You may

consider that "species $M"$ is an inert and is present in high concentrations such that the

total molar density of the gas mixture is approximately equal to the concentration of $M.$

Part a$.$ Derive an equation for the rate of ABC production in terms of the concentrations

of stable molecules and the rate coefficients given.

Part b$.$ Derive equations for the effective rate coefficients of ABC production in the limits

of low and high total pressure, where the total pressure is proportional to the concentration

of species M$.$

Part c$.$ For reaction at low pressure determine equations for the effective rate coefficients

in the limits of low and high temperature. The temperature dependence of the elementary

rate coefficients is given by the Arrhenius equation and $E_{-2}>E_3$ and $A_{i}i{H}_2{D}_e=458\mathrm{.}3k\frac{J}{m}ol,r_e=0\mathrm{.}74=1\mathrm{.}83{}^{-1}{V}_M(r)=D_e$exp$(-2(r-r_e))-2D_e$exp$(-(r-r_e))V(r)=A+B(r-r_e)+C(r-r_e{)}^2+D(r-r_e{)}^3{A}_{-2}$ and $A_i$

are the activation energy and pre$-$factor for reaction $i.$