Solid fuel used in rocket is a mixture of $\mathrm{Fe}_{2} \mathrm{O}_{3}$ and $\mathrm{Al}$ (in ratio 1 : 2). The heat evolved $(\mathrm{kJ})$ per gram of the mixture is ____________. (Nearest integer)

Given: $$\Delta \mathrm{H}_{\mathrm{f}}^{\theta}\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right)=-1700 \mathrm{~kJ} \mathrm{~mol}^{-1}$$

$$\Delta \mathrm{H}_{\mathrm{f}}^{\theta}\left(\mathrm{Fe}_{2} \mathrm{O}_{3}\right)=-840 \mathrm{~kJ} \mathrm{~mol}^{-1}$$

Molar mass of Fe, Al and O are 56, 27 and 16 g mol$^{-1}$ respectively.

<p>First, let&#39;s consider the reaction : <br/><br/>$$2\mathrm{Al}(s) + \mathrm{Fe}_2\mathrm{O}_3(s) \rightarrow \mathrm{Al}_2\mathrm{O}_3(s) + 2\mathrm{Fe}(s)$$</p> <p>The heat change for this reaction $\Delta H^0$ can be calculated from the heats of formation of the reactants and the products : <br/><br/>$$\Delta H^0 = [\Delta H_f^0(\mathrm{Al}_2\mathrm{O}_3) + 2\Delta H_f^0(\mathrm{Fe})] - [2\Delta H_f^0(\mathrm{Al}) + \Delta H_f^0(\mathrm{Fe}_2\mathrm{O}_3)]$$</p> <p>Assuming the elements in their standard states have zero enthalpy of formation, i.e., <br/><br/>$\Delta H_f^0(\mathrm{Al}) = \Delta H_f^0(\mathrm{Fe}) = 0$, we can simplify this to : <br/><br/>$$\Delta H^0 = \Delta H_f^0(\mathrm{Al}_2\mathrm{O}_3) - \Delta H_f^0(\mathrm{Fe}_2\mathrm{O}_3)$$</p> <p>Substitute the given heats of formation into the equation : <br/><br/>$$\Delta H^0 = (-1700\ \mathrm{kJ/mol}) - (-840\ \mathrm{kJ/mol}) = -860\ \mathrm{kJ/mol}$$</p> <p>This is the heat of reaction for the above reaction. However, we are asked to find the heat evolved per gram of the mixture. </p> <p>To find this, we need to determine the molar mass of the reactants in the reaction. The molar mass of $\mathrm{Fe}_2\mathrm{O}_3$ is $2 \times 56 + 3 \times 16 = 160\ \mathrm{g/mol}$ and the molar mass of 2 moles of $\mathrm{Al}$ is $2 \times 27 = 54\ \mathrm{g/mol}$. The total molar mass of the mixture is $160 + 54 = 214\ \mathrm{g/mol}$. </p> <p>So, the heat evolved per gram of the mixture is $\frac{-860\ \mathrm{kJ/mol}}{214\ \mathrm{g/mol}} = -4.0187\ \mathrm{kJ/g}$</p> <p>Rounded to the nearest integer, this value is approximately -4 kJ/g. The negative sign indicates that the heat is evolved (exothermic reaction).</p>