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

01

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.

02

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

03

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

04

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

05

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

06

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.

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