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 \(0^{\circ} \mathrm{C}, \mathrm{a}\) pressure of \(1.0\) atm during filling, and \(100 \%\) yield.

Short answer

To find the mass of iron splints and the mass of diluted H₂SO₄ needed to completely fill the balloon, follow these steps: 1. Calculate the actual volume of hydrogen gas needed, considering the 20% loss: \(4800\,\text{m}^{3} / (1 - 20/100) = 6000\,\text{m}^{3}\) 2. Convert the volume to moles of hydrogen gas using the ideal gas law: \(n = PV / (RT) = (101325 \times 6000) / (8.314 \times 273.15) = 266,585.96 \, \text{mol}\) 3. Determine the moles of iron and H₂SO₄ needed, which are equal to the moles of hydrogen gas: 266,585.96 mol 4. Calculate the mass of iron needed: \((266,585.96 \, \text{mol}) \times (55.85 \, \text{g/mol}) = 14,889,473.56 \, \text{g}\) 5. Calculate the mass of the diluted H₂SO₄ solution needed: \((266,585.96 \, \text{mol}) \times (98.08 \, \text{g/mol}) / 0.98 = 26,673,301.56 \, \text{g}\) The mass of iron splints needed is approximately 14,889,473.56 g and the mass of the diluted H₂SO₄ is approximately 26,673,301.56 g.

Step-by-step solution

Calculate the volume of hydrogen gas needed to fill the balloon fully with a 20% loss

Since there's a 20% loss during filling, we need to calculate the actual volume of hydrogen gas needed to completely fill the balloon. Volume after filling = 4800 m³ Loss percentage = 20% Volume of hydrogen gas needed = (Volume after filling) / (1 - Loss percentage / 100) Volume of hydrogen gas needed = 4800 m³ / (1 - 20/100)

Convert the volume to moles of hydrogen gas

We need to convert the volume of hydrogen gas needed into moles using the ideal gas law formula. The ideal gas law formula is given by: PV=nRT Where: P = pressure = 1.0 atm (converted to Pa, 1 atm = 101325 Pa) V = volume of hydrogen gas needed (calculated in step 1) n = moles of hydrogen gas R = Universal gas constant = 8.314 J/(mol K) T = temperature = 0 °C (converted to K, 0 °C = 273.15 K) Rearranging the formula, we get: n = PV/(RT)

Calculate the moles of iron and H2SO4 needed

From the balanced chemical reaction, the mole ratio of Fe, H2SO4, and H2 is 1:1:1. Therefore, the moles of Fe and H2SO4 needed will be the same as the moles of hydrogen gas calculated in step 2. Moles of Fe = Moles of H2SO4 = Moles of H2 (calculated in step 2)

Calculate the mass of iron needed

The molar mass of iron (Fe) is 55.85 g/mol. We can use this to calculate the mass of iron by multiplying the moles of iron by its molar mass. Mass of Fe = Moles of Fe × molar mass of Fe

Calculate the mass of the diluted H2SO4 solution needed

The molar mass of H2SO4 is 98.08 g/mol. We can use this to calculate the mass of H2SO4 by multiplying the moles of H2SO4 by its molar mass. Mass of H2SO4 = Moles of H2SO4 × molar mass of H2SO4 Since we have a 98% by mass H2SO4 solution, we can calculate the mass of diluted H2SO4 solution needed as follows: Mass of diluted H2SO4 solution = (Mass of H2SO4) / (98%) These calculations will give you the mass of iron splints and the mass of diluted H2SO4 needed to completely fill the balloon.

Key concepts

Ideal Gas Law

The ideal gas law is a crucial concept in chemistry that allows us to calculate various properties of gases. When we talk about a gas, it often follows the equation \( PV = nRT \). This formula connects pressure \( P \), volume \( V \), number of moles \( n \), the gas constant \( R \), and temperature \( T \). For hydrogen gas in the balloon, we use these known values:

• Pressure \( P \) as 1.0 atm
• Calculated volume \( V \)
• Temperature \( T \) converted to Kelvin, which is 273.15 K at 0°C
• R is a constant that equals 8.314 J/(mol·K)

By plugging these values into the rearranged formula \( n = \frac{PV}{RT} \), we can find the number of moles of hydrogen gas necessary for filling the balloon, accounting for the 20% gas loss. It's a way to understand how much gas we need in terms of moles, which is vital for any stoichiometry calculations.

Chemical Reactions

Chemical reactions involve reactants transforming into products. In this scenario, iron reacts with sulfuric acid to produce iron sulfate and hydrogen gas: \[ \mathrm{Fe}(s) + \mathrm{H}_{2} \mathrm{SO}_{4}(aq) \longrightarrow \mathrm{FeSO}_{4}(aq) + \mathrm{H}_{2}(g) \]This balanced reaction tells us that one mole of iron reacts with one mole of sulfuric acid to form one mole of iron sulfate and one mole of hydrogen gas. Balancing reactions is crucial in stoichiometry because it helps us determine the exact amounts of each substance needed. In our case, the moles of hydrogen gas calculated earlier will be the same for iron and sulfuric acid due to the 1:1 reaction ratio.

Molar Mass

Molar mass is defined as the mass of one mole of a substance. It’s a vital aspect of stoichiometry, letting us convert between moles and grams. For our reaction:

• The molar mass of iron \( (\text{Fe}) \) is 55.85 g/mol.
• The molar mass of sulfuric acid \( (\text{H}_2\text{SO}_4) \) is 98.08 g/mol.

To find the mass of a substance, we multiply the number of moles by its molar mass. For instance, to uncover the mass of iron needed for our reaction, we calculate:\[ \text{Mass of Fe} = \text{Moles of Fe} \times \text{Molar mass of Fe} \]This calculation yields the actual grams of iron required. Similarly, knowing the mass of sulfuric acid needed is equally imperative for preparing the right solution concentration.

Gas Collection and Ballooning

Gas collection, particularly in ballooning, involves capturing a gas formed during a chemical reaction. In the case of Andrée's balloon, hydrogen gas served as the lifting agent. Filling a balloon requires consideration of any potential gas losses that occur during the process. The Swedish explorer needed to compensate for a 20% hydrogen gas loss, compelling calculations to ensure that the balloon reached its intended volume of 4800 m³. This involves calculating the volume of gas after a known loss and converting it into moles using the ideal gas law. By understanding the stoichiometry of the chemical reaction, we guarantee the correct amounts of reactants are used. This ensures that sufficient hydrogen gas is generated to counter the losses and fulfill the balloon's required lift capacity. It also emphasizes the importance of considering environmental factors like pressure and temperature during gas collection in practical applications such as ballooning.