Chemistry - Chemical Kinetics Question with Solution | TestHub

The decomposition of dimethylether (DME) is a first-order reaction.

CH₃OCH₃(g) --> CH₄(g) + H₂(g) + CO(g)

Initial moles (t=0): 1 mol DME, 0 mol products.

After 29 min:

The half-life ($t_{1/2}$) is 14.5 min.

Since 29 min = $2\times t_{1/2}$, the amount of DME remaining is $(1/2{)}^2=1/4$ of the initial amount.

Moles of DME remaining = $1\times (1/4)=0.25$ mol.

Moles of DME reacted = $1-0.25=0.75$ mol.

From the stoichiometry, 1 mole of DME produces 1 mole of CH₄, 1 mole of H₂, and 1 mole of CO.

So, 0.75 mol of DME reacted produces:

Moles of CH₄ = 0.75 mol

Moles of H₂ = 0.75 mol

Moles of CO = 0.75 mol

Total moles in the mixture after 29 min = Moles of DME + Moles of CH₄ + Moles of H₂ + Moles of CO

Total moles = $0.25+0.75+0.75+0.75=2.5$ mol.

The initial molar mass of the gas (DME) is $M_{DME}=2\times 12.01+6\times 1.01+16.00=46.08$ g/mol.

Since the reaction is carried out in a constant volume container, the total mass of the gas remains constant.

Initial mass = $1mol\times 46.08g/mol=46.08g$.

The average molar mass of the mixture ($M_{avg}$) is the total mass divided by the total number of moles.

$M_{avg}=\frac{Totalmass}{Totalmoles}=\frac{46.08g}{2.5mol}=18.432g/mol$.

Vapour density (V.D.) of the mixture is $V.D.=\frac{M_{avg}}{2}$.

$V.D.=\frac{18.432}{2}=9.216$.

The problem asks for Z, where $Z=vapourdensityofmixture\times 10$.

$Z=9.216\times 10=92.16$.

Rounding to the nearest whole number, $Z=92$.