Problem 69 - Chemistry An Atoms First Approach

Question

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.

Step-by-Step Solution

Answer

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.

Step 1: Find moles of hydrogen gas required

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: \[ (1.0\, atm) (1.2 \times 4800 \times 1000\,L) = n (0.0821 \frac{L\cdot atm}{mol\cdot K})(273\,K)\]

Step 2: Calculate moles of hydrogen gas

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.

Step 3: Determine moles of iron and sulfuric acid needed

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.

Step 4: Calculate the mass of iron splints and sulfuric acid required

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 \(\mathrm{H}_{2}\mathrm{SO}_{4}\): \[223671.24\, mol \times 100.08\frac{g}{mol} = 22377163.56\,g\]

Step 5: Final answer

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.

Key Concepts

Ideal Gas Law

In the realm of chemistry, the ideal gas law is a key principle used to relate the physical properties of gases. This law is represented by the formula: \ PV = nRT \ where \(P\) stands for pressure, \(V\) for volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is temperature. The ideal gas law allows us to calculate various properties of a gas if we know some values of others. For example, in the case of filling a balloon with hydrogen gas, though we know the pressure and temperature, we can compute the number of moles needed if we also know the balloon's volume. This law is crucial when gases follow near-ideal conditions, at relatively high temperatures and low pressures. Understanding this principle helps us predict how gases will behave under various conditions, making it invaluable for many chemical calculations, including the scenario of a hot air balloon.

Molar Mass

Molar mass plays an important role in stoichiometry, especially when connecting chemical reactions to measurable quantities. It is defined as the mass of a given substance (chemical element or chemical compound) divided by the amount of substance, typically expressed in g/mol. When dealing with the reaction of iron splints and sulfuric acid to produce hydrogen gas, knowing the molar masses of iron \(Fe\) and sulfuric acid \(H_2SO_4\) allows us to calculate the mass of each reactant required. For instance, iron has a molar mass of 55.845 g/mol, which helps us convert from moles to grams. This conversion is essential to determine how much solid iron and acid are needed to produce the exact number of moles of hydrogen gas needed for the balloon. Thus, the concept of molar mass bridges abstract chemical equations with real-world applications by allowing precise measurement of substances.

Chemical Reactions

Chemical reactions are processes where substances are transformed into different substances. They are represented by chemical equations that show the reactants converting to products. In our scenario, the reaction between iron \(Fe\) and sulfuric acid \(H_2SO_4\) produces iron(II) sulfate \(FeSO_4\) and hydrogen gas \(H_2\). \ \(\mathrm{Fe}(s) + \mathrm{H}_{2} \mathrm{SO}_{4}(aq) \rightarrow \mathrm{FeSO}_{4}(aq) + \mathrm{H}_{2}(g)\) \ Each part of the equation signifies a different component of the reaction. One important aspect is balancing the chemicals involved to abide by the law of conservation of mass. This law implies matter is not created or destroyed in a chemical reaction. Thus, the moles of reactants will equal the moles of products when balanced. In our case, for every mole of iron, one mole of hydrogen gas is produced. This balanced stoichiometry allows us to scale the reaction up or down as needed to produce the necessary amount of hydrogen gas to fill a balloon, illustrating how chemical equations practically guide us in lab settings and industrial applications.

Gas Laws

Alongside the ideal gas law, there are several other important gas laws that describe the behavior of gases under various conditions. These include Boyle's law, Charles's law, and Avogadro's law. These tools help provide a comprehensive understanding of gas behavior:

• **Boyle's Law**: Describes how pressure and volume are inversely proportional at constant temperature.
• **Charles's Law**: Explains that at constant pressure, the volume of a gas is directly proportional to its temperature in Kelvins.
• **Avogadro's Law**: States that at constant temperature and pressure, the volume of a gas is directly proportional to the number of moles.

Understanding these laws helps reinforce why the ideal gas law is so powerful; it combines these individual laws into a single formula. When dealing with the filling of balloons or any other application involving gases, it's crucial to consider these principles to make accurate predictions and calculations. The integration of gas laws enables chemists to anticipate changes in gas properties due to shifts in environmental conditions, ensuring successful application in practical situations.