Problem $2$:

One of the major reasons for engine oil degradation is the oxidation of the motor oil. To retard the

degradation process, most oils contain an antioxidant $[$see Ind. Eng. Chem. $26,902(1987)].$ Without an

inhibitor to oxidation present, the suggested mechanism at low temperatures is

$I_2{k}_{0}2I*$

$I*+RH{k}_{i}R*+HI$

$R*+O_2{k}_{p1}R{O}_2*$

$R{O}_2*+RH{k}_{p2}\frac{R}{O}OH+R.$

$2R{O}_2*k_t$ inactive

Where $I_2$ is an initiator and $RH$ is the hydrocarbon in the oil.

When an antioxidant is added to retard degradation at low temperatures, the following additional

termination steps occur:

$R{O}_2*+AH{k}_{A1}\frac{R}{O}OH+A*$

$A*+R{O}_2*k_{A2}$ inactive

$($a$)$ Derive a rate law for the degradation of the motor oil in the absence of an antioxidant at low

temperatures

$($b$)$ Derive a rate law for the rate of degradation of the motor oil in the presence of an antioxidant for low

temperatures.

$($c$)$ How would your answer to part $($a$)$ change if the radicals I$-$ were produced at a constant rate in the

engine and then found their way into the oil?