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.

Short answer

To ensure the complete filling of the balloon, approximately 14,351,563 g of iron splints and 25,709,669 g of 98% H₂SO₄ solution are required.

Step-by-step solution

Calculate the volume of hydrogen gas required for filling the balloon

As 20% of hydrogen gas is lost during the filling, we need to prepare 100% + 20% = 120% hydrogen gas for the given volume of the balloon. Hence, Required volume of hydrogen gas = 4800 m³ × 1.20 = 5760 m³ Since 1 m³ = 1000 L Required volume of hydrogen gas = 5760000 L.

Convert the given temperature to Kelvin

Next, we need to convert the given temperature in Celsius to Kelvin: Temperature in Kelvin (T) = 0 + 273.15 T = 273.15 K

Use the ideal gas law to find the moles of hydrogen gas

We will use the ideal gas law, PV=nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature. Given: Pressure (P) = 1.0 atm Volume (V) = 5760000 L Temperature (T) = 273.15 K Ideal gas constant (R) = 0.0821 L atm K⁻¹ mol⁻¹ Now, rearrange the equation to find the number of moles (n) of hydrogen gas. n = PV/RT n = (1.0 atm × 5760000 L) / (0.0821 L atm K⁻¹ mol⁻¹ × 273.15 K) n ≈ 256963 mol H₂ gas

Calculate the mass of iron required for the reaction

Using the balanced chemical reaction: Fe(s) + H₂SO₄(aq) → FeSO₄(aq) + H₂(g) We can see that 1 mole of iron reacts with 1 mole of H₂SO₄ to produce 1 mole of H₂ gas. Therefore, the moles of iron required = moles of hydrogen gas = 256963 mol To find the mass of iron, we multiply the moles by its molar mass (55.85 g/mol): Mass of iron = 256963 mol × 55.85 g/mol Mass of iron ≈ 14351563 g

Calculate the mass of H₂SO₄ required for the reaction

Since 1 mole of H₂SO₄ reacts with 1 mole of Fe to produce 1 mole of H₂ gas, the moles of H₂SO₄ required = moles of hydrogen gas = 256963 mol Now we need to find the mass of 98% H₂SO₄ needed. First, let's find the mass of pure H₂SO₄ needed. Molar mass of H₂SO₄ = 2 × 1.01 + 32.07 + 4 × 16 = 98.08 g/mol Mass of pure H₂SO₄ = 256963 mol × 98.08 g/mol Mass of pure H₂SO₄ ≈ 25195656 g Since 98% H₂SO₄ is used, we can find the mass of the 98% H₂SO₄ solution needed as follows: Mass of 98% H₂SO₄ solution = (Mass of pure H₂SO₄) / 0.98 = 25195656 g / 0.98 Mass of 98% H₂SO₄ solution ≈ 25709669 g

Present the final answer

To ensure the complete filling of the balloon, the mass of iron splints required is around 14351563 g and the mass of the 98% H₂SO₄ solution required is around 25709669 g.

Key concepts

Chemical Reactions

Chemical reactions involve the transformation of substances through the breaking and forming of chemical bonds, resulting in the production of new substances with different properties. In the exercise, the chemical reaction that occurs is between iron (Fe) and sulfuric acid (H₂SO₄), producing iron sulfate (FeSO₄) and hydrogen gas (H₂). This is a redox reaction, where iron is oxidized as it loses electrons to become a cation, while the hydrogen ions in sulfuric acid gain electrons, forming hydrogen gas. Understanding this fundamental process is crucial because it allows us to predict the products and calculate how much of each reactant we need. One important aspect of chemical reactions is the stoichiometry, which refers to the quantitative relationship between reactants and products. By interpreting the balanced chemical equation, we can deduce that one mole of iron reacts with one mole of sulfuric acid to yield one mole of hydrogen gas. This stoichiometric relationship guides us in determining the quantities of reactants necessary to produce the desired amount of product.

Moles Calculation

Moles calculation is a core concept in chemistry that provides a bridge between the atomic world and the macroscopic scale we can measure. In our scenario, the objective was to find how many moles of hydrogen gas were needed to fill the balloon. The ideal gas law, given by the formula \( PV = nRT \), is an essential tool here. This formula relates pressure (P), volume (V), and temperature (T) of a gas with the number of moles (n) and the ideal gas constant (R), which is 0.0821 L atm K⁻¹ mol⁻¹.
By rearranging the ideal gas law formula to solve for moles \( n = \frac{PV}{RT} \), it becomes straightforward to calculate the number of moles if pressure, volume, and temperature are known. After converting the given volume from cubic meters to liters and the temperature from Celsius to Kelvin, you plug these values into the equation to find the moles of hydrogen needed. In our exercise, this amounted to approximately 256,963 moles of hydrogen. This step highlights the importance of understanding moles and using the ideal gas equation to transition between the macroscopic and microscopic views in chemistry.

Mass Calculation

Mass calculation in chemistry is crucial for preparing reactions properly and efficiently. It involves converting moles, which are a measure of quantity at a molecular level, into mass, which is tangible and measurable. In the exercise, we started with the moles of iron and sulfuric acid determined from the moles of hydrogen gas requirement, due to the stoichiometric relationships in the chemical equation.
To calculate mass from moles, use the formula:

• Mass = Moles × Molar Mass

For iron, with a molar mass of 55.85 g/mol, the calculation gives a mass of about 14,351,563 grams. For sulfuric acid, the molar mass is calculated as the sum of atomic masses of its elements, resulting in 98.08 g/mol. Since only 98% sulfuric acid is used, adjust the mass calculation by dividing the mass of pure acid by 0.98, totaling approximately 25,709,669 grams of the solution.
Understanding these conversions allows us to perform experiments with precision, ensuring that all chemicals react completely without excess or deficit, which is especially important in large-scale reactions.