The total volume of hydrogen gas needed to fill the Hindenburg was \(2.0 \times 10^{8} \mathrm{L}\) at 1.0 atm and \(25^{\circ} \mathrm{C}\). Given that \(\Delta H_{\mathrm{f}}^{\circ}\) for \(\mathrm{H}_{2} \mathrm{O}(l)\) is \(-286 \mathrm{kJ} / \mathrm{mol},\) how much heat was evolved when the Hindenburg exploded, assuming all of the hydrogen reacted to form water?

To find the initial number of moles of hydrogen gas, we use the Ideal Gas Law: \(PV = nRT\). We are given the volume (\(V= 2.0 \times 10^8 L\)), pressure (\(P = 1.0\space atm\)), and temperature (\(T = 25^{\circ}\mathrm{C} \approx 298\space K\)). The gas constant, \(R = 0.0821 \frac{L \cdot atm}{mol \cdot K}\). Rearranging the formula to solve for n, we get \(n = \frac{PV}{RT}\). \(n = \frac{(1.0\space atm)(2.0 \times 10^{8}\space L)}{(0.0821\space L\cdot atm/mol\cdot K)(298\space K)} \approx 8.1 \times 10^6\space mol\)

Since all the hydrogen gas reacts to form liquid water, the enthalpy change of the reaction can be calculated using the number of moles of hydrogen gas and the enthalpy of formation for water. We are given that the enthalpy of formation of water, \(\Delta H_{\mathrm{f}}^{\circ}\), is -286 kJ/mol. To determine the total heat evolved, we multiply the number of moles of hydrogen gas by the enthalpy of formation of water: \(q = n \cdot \Delta H_{\mathrm{f}}^{\circ}\) \(q \approx (8.1 \times 10^6 \space mol)\cdot(-286 \space kJ/mol) \approx -2.3 \times 10^9\space kJ\) Since the value is negative, this indicates that heat was evolved (exothermic reaction) during the explosion. The total amount of heat evolved is approximately \(-2.3 \times 10^9\space kJ\).