In 1897 the Swedish explorer Andreé tried to reach the North Pole in a balloon. The balloon was filled with hydrogen gas. The hydrogen gas was prepared from iron splints and diluted sulfuric acid. The reaction is $$\mathrm{Fe}(s)+\mathrm{H}_{2} \mathrm{SO}_{4}(a q) \longrightarrow \mathrm{FeSO}_{4}(a q)+\mathrm{H}_{2}(g)$$ The volume of the balloon was \(4800 \mathrm{m}^{3}\) and the loss of hydrogen gas during filling was estimated at \(20 . \% .\) What mass of iron splints and \(98 \%\) (by mass) \(\mathrm{H}_{2} \mathrm{SO}_{4}\) were needed to ensure the complete filling of the balloon? Assume a temperature of \(0^{\circ} \mathrm{C},\) a pressure of \(1.0\) atm during filling, and \(100 \%\) yield.

The balloon has a volume of \(4800 \, m^{3}\) with a \(20\%\) loss during filling. To find the moles of hydrogen gas required, we can use the ideal gas law: \[PV=nRT\] where, P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. The given pressure is \(1.0 \, atm\), the volume is \(4800 \, m^{3}\) without accounting for the \(20\%\) gas loss, which is equivalent to \((1.2)4800 \, m^{3}\), and the temperature is \(273 \,K\) as it is given in Celsius. The ideal gas constant, R is \(0.0821 \frac{L\cdot atm}{mol\cdot K}\). The volume needs to be converted to liters for units to be compatible \((\) \(4800\,m^{3} = 4800\times1000\,L\) \()\). Now, we can find the moles of \(\mathrm{H}_{2}\) required: \[ (1.0\, atm) (1.2 \times 4800 \times 1000\,L) = n (0.0821 \frac{L\cdot atm}{mol\cdot K})(273\,K)\]

Solve the equation for n: \[n = \frac{1.0 \, atm \times (1.2) \times 4800 \times 1000\,L}{0.0821\frac{L\cdot atm}{mol\cdot K}\times 273\,K}\] \[n = 223671.24 \, mol\] Thus, \(223671.24\) moles of hydrogen gas are required to completely fill the balloon.

In the balanced chemical equation, \[\mathrm{Fe}(s) + \mathrm{H}_{2}\mathrm{SO}_{4}(aq) \longrightarrow \mathrm{FeSO}_{4}(aq) + \mathrm{H}_{2}(g)\] we can see that the molar ratio of Fe : H₂ is 1:1. Therefore, we also need 223671.24 moles of iron splints and \(\mathrm{H}_{2}\mathrm{SO}_{4}\) for the reaction.

Now, we can determine the weight of the iron splints and sulfuric acid required using their respective molar masses. The molar mass of the iron splints is \(55.845\,g/mol\) and the molar mass of \(\mathrm{H}_{2}\mathrm{SO}_{4}\) is \(98.079\,g/mol\). Mass of iron splints: \[223671.24\, mol \times 55.845\frac{g}{mol} = 12492844.01\,g\] Since the sulfuric acid is \(98\%\) by mass, we can adjust the molar mass of \(\mathrm{H}_{2}\mathrm{SO}_{4}\): \[98.079 \, g/mol \times \frac{1}{0.98} = 100.08 \, g/mol\] Mass of \(\mathrm{H}_{2}\mathrm{SO}_{4}\): \[223671.24\, mol \times 100.08\frac{g}{mol} = 22377163.56\,g\]

The mass of iron splints required is approximately \(12,492,844\,g\) and the mass of \(98\%\) sulfuric acid required is approximately \(22,377,163\,g\) to ensure the complete filling of the balloon.