$($c$)$ A gasoline is made of the blend defined in Table $2($c$)($i$)$ below.

Table $2($c$)($i$)$

$\backslash$table$[[$Component $\text{'}$i$\text{'},\backslash$table$[[$Flow Rate$],[($Barrels$/$day$)]],\backslash$table$[[$MON$],[($Motor Octane Number$)]]],[$Methanol $($MeOH$),2500,104],[$Thermally cracked gas oil $($TCGO$),4500,65\mathrm{.}1],[$Catalytically cracked gas oil $($CCGO$),17000,80\mathrm{.}2]]$

$($i$)$ Find the volume fractions of the various components in the blend $($use $4$ significant figures $(4$ s$.$f$.)),$ and calculate the Motor Octane Number $(\text{'}$MON$\text{'})$ using the mixing rule on the individual MONs $(4$ s$.$f$.).$

$[4$ marks$]$

$($ii$)$ Using the appropriate equations below, calculate Bloni, the blending indices of MON of each component $i$ in the mixture to $4$ s$.$f$.$

$[6$ marks$]$

For $11$ MON $76$

BloNi $=36\mathrm{.}01+38\mathrm{.}33(MO\frac{N}{100})-99\mathrm{.}8(MO\frac{N}{100}{)}^2+341\mathrm{.}3(MO\frac{N}{100}{)}^3-507\mathrm{.}02(MO\frac{N}{100}{)}^4+268\mathrm{.}64(MO\frac{N}{100}{)}^5$

For $76$ MON $103$

BloNi MON$/100)-1552\mathrm{.}9(MO\frac{N}{100}{)}^2+651(MO\frac{N}{100}{)}^3$

For $103$ MON $106$

BloNi $=2206\mathrm{.}3-4313\mathrm{.}64(MO\frac{N}{100})+2178\mathrm{.}57(MO\frac{N}{100}{)}^2$

$($iii$)$ Determine the blending index of the mixture using the mixing rule on the individual blending indices $($answer to $4$ s$.$f$.).$

$[2$ marks$]$

$($iv$)$ Using Table $2($c$)($ii$)$ below, determine the MON by mixing rule of the blending indices method $($answer to $3$ s$.$f$.),$ and calculate the relative $\%$ error $(2$ s$.$f$.)$ between the values obtained in $($i$)$ and $($iv$)$ based on the answer from $($i$).$

$[2$ marks$]$

Table $2($c$)($ii$)$

$\backslash$table$[[$BI$,58\mathrm{.}08,58\mathrm{.}16,58\mathrm{.}23,58\mathrm{.}31,58\mathrm{.}39,58\mathrm{.}47,58\mathrm{.}54,58\mathrm{.}62,58\mathrm{.}70],[$MON$,81\mathrm{.}4,81\mathrm{.}6,81\mathrm{.}8,82,82\mathrm{.}2,82\mathrm{.}4,82\mathrm{.}6,82\mathrm{.}8,83]]$