Question #44921

An industrial synthesis of a specialized azo dye requires the preparation of benzenediazonium chloride from $N$-phenylacetamide.

Step 1: $0.4\text{ mol}$ of $N$-phenylacetamide undergoes complete nucleophilic acyl substitution (hydrolysis) in aqueous $\ce{HCl}$ at $363\text{ K}$. The resulting aqueous mixture has a total heat capacity $C_{p1} = 2.0\text{ kJ K}^{-1}$.

Step 2: To shift the basicity-solvation competition and accelerate nitrosation, $500\text{ g}$ of $N,N$-dimethylformamide (DMF, specific heat capacity $s = 2.0\text{ J g}^{-1}\text{ K}^{-1}$) initially at $293\text{ K}$ is added to the hot mixture.

Step 3: The aprotic nature of DMF deprives the diazonium cation of hydrogen-bond stabilization, drastically lowering the activation barrier for $\ce{N2}$ loss. To prevent decomposition, the mixture is cooled to exactly $273\text{ K}$ by adding solid ice (at $273\text{ K}$).

Step 4: $0.4\text{ mol}$ of $\ce{NaNO2}$ is added, and diazotization ($\Delta H_{rxn} = -125\text{ kJ mol}^{-1}$) goes to completion at $273\text{ K}$.

Assuming no heat exchange with the surroundings and that the melted ice remains at $273\text{ K}$, calculate the total mass of ice required to achieve the cooling and maintain the system strictly at $273\text{ K}$ during the entire process.

(Given: $\Delta H_{fusion}$ of ice = $6.0\text{ kJ mol}^{-1}$, Molar mass of $\ce{H2O}$ = $18\text{ g mol}^{-1}$)