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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

Expert verified
To ensure the complete filling of the balloon, \( 5337.5 \: kg \) of iron splints and \( 10488.5 \: kg \) of 98% sulfuric acid are needed.

Step by step solution

01

Calculate the total hydrogen gas volume required

Given that the volume of the balloon is 4800 m³ and the loss of hydrogen gas during filling is 20%, we will calculate the total volume of hydrogen gas required to fill the balloon completely. Let V_total be the total volume of hydrogen gas needed. V_total = 4800 m³ / (1 - 0.20)
02

Convert the volume to Liters

We need to convert the volume from m³ to Liters since we'll use the ideal gas law later. V_total (L) = V_total * 1000
03

Calculate the moles of hydrogen gas using the Ideal Gas Law

Using the Ideal Gas Law, PV = nRT, where P is pressure, V is volume, n represents the moles of hydrogen gas, R is the gas constant, and T is the absolute temperature. We are given the volume V_total(L), the pressure P=1.0 atm, and the temperature T=0°C. Convert the temperature to Kelvin: T(K) = 0 + 273.15 The gas constant, R, for L. atm/mol. K is given as 0.0821. Now, we solve for n to calculate the moles of hydrogen gas: n = (PV) / (RT)
04

Determine the moles of iron splints and sulfuric acid needed

In the balanced equation, one mole of iron reacts with one mole of sulfuric acid to produce one mole of hydrogen gas. Therefore, the number of moles of iron and sulfuric acid required is equal to the number of moles of hydrogen gas obtained in the previous step. moles_Fe = moles_H₂ moles_H₂SO₄ = moles_H₂
05

Calculate the mass of iron and sulfuric acid

Now, we will find the mass required using the molar masses of iron (Fe) and sulfuric acid (H₂SO₄). The molar mass of Fe is 55.85 g/mol, and the molar mass of H₂SO₄ is 98.08 g/mol. Let m_Fe represent the mass of iron and m_H₂SO₄ be the mass of sulfuric acid. We will calculate the masses as follows: m_Fe = moles_Fe * 55.85 g/mol m_H₂SO₄ = moles_H₂SO₄ * 98.08 g/mol As the sulfuric acid is given as 98% by mass, we need to determine the total mass of diluted sulfuric acid required. Let m_diluted_H₂SO₄ represent the mass of diluted sulfuric acid: m_diluted_H₂SO₄ = m_H₂SO₄ / 0.98 Now we have the mass of iron splints and diluted sulfuric acid needed to ensure the complete filling of the balloon.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Ideal Gas Law
The Ideal Gas Law is a fundamental equation in chemistry and physics that relates the pressure (P), volume (V), temperature (T), and amount in moles (n) of an ideal gas. The equation, given by PV = nRT, where R is the universal gas constant, is critical for predicting the behavior of gases under various conditions.

In our balloon-filling exercise, we use the Ideal Gas Law to determine the amount of hydrogen gas needed to fill the balloon at a given temperature and pressure. To apply the Ideal Gas Law, we first convert the volume to liters and the temperature to Kelvin, as these are the standard units. Then, we solve for n, the number of moles of gas, which further allows us to calculate the necessary reactants for the chemical reaction. This law assumes that the gas behaves ideally, meaning the gas particles have negligible volume and no intermolecular forces.
Chemical Reactions in Stoichiometry
Chemical reactions are processes where reactants transform into products. A balanced chemical equation represents the stoichiometry of the reaction, indicating the proportions in which substances react. In the scenario of filling a balloon with hydrogen, we look at the stoichiometric equation provided:
\[\mathrm{Fe}(s) + \mathrm{H}_2\mathrm{SO}_4(aq) \longrightarrow \mathrm{FeSO}_4(aq) + \mathrm{H}_2(g) \]
This equation tells us that iron (Fe) reacts with sulfuric acid (H2SO4) to produce iron sulfate (FeSO4) and hydrogen gas (H2). Stoichiometry allows us to understand the exact ratios and amounts of each substance needed. Since the reaction produces hydrogen gas in a 1:1 ratio with the reactants, we use the moles of hydrogen calculated through the Ideal Gas Law to determine the amounts of iron and sulfuric acid required.
Molar Mass and Its Role in Stoichiometry
Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a bridge between the macroscopic world of grams and the microscopic world of moles. Determining the molar mass of the reactants in a chemical reaction is essential for converting between mass and moles, allowing us to perform stoichiometric calculations.

In the example of the balloon expedition, molar mass helps us calculate the mass of iron splints and sulfuric acid needed to produce the desired volume of hydrogen gas. We use the molar masses of iron (55.85 g/mol) and sulfuric acid (98.08 g/mol) to convert the moles of each reactant required into grams. It is also important to account for the purity of the reactants; for instance, the sulfuric acid is 98% pure, which we must consider to obtain the total mass of the diluted acid required. Remember, accurate stoichiometry depends on correct molar mass values.

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Most popular questions from this chapter

Methane \(\left(\mathrm{CH}_{4}\right)\) gas flows into a combustion chamber at a rate of 200. L/min at \(1.50 \mathrm{~atm}\) and ambient temperature. Air is added to the chamber at \(1.00 \mathrm{~atm}\) and the same temperature, and the gases are ignited. a. To ensure complete combustion of \(\mathrm{CH}_{4}\) to \(\mathrm{CO}_{2}(\mathrm{~g})\) and \(\mathrm{H}_{2} \mathrm{O}(g)\), three times as much oxygen as is necessary is reacted. Assuming air is 21 mole percent \(\mathrm{O}_{2}\) and \(79 \mathrm{~mole}\) percent \(\mathrm{N}_{2}\), calculate the flow rate of air necessary to deliver the required amount of oxygen. b. Under the conditions in part a, combustion of methane was not complete as a mixture of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\) was produced. It was determined that \(95.0 \%\) of the carbon in the exhaust gas was present in \(\mathrm{CO}_{2}\). The remainder was present as carbon in \(\mathrm{CO}\). Calculate the composition of the exhaust gas in terms of mole fraction of \(\mathrm{CO}, \mathrm{CO}_{2}, \mathrm{O}_{2}, \mathrm{~N}_{2}\), and \(\mathrm{H}_{2} \mathrm{O} .\) Assume \(\mathrm{CH}_{4}\) is completely reacted and \(\mathrm{N}_{2}\) is unreacted.

You have a helium balloon at \(1.00\) atm and \(25^{\circ} \mathrm{C}\). You want to make a hot-air balloon with the same volume and same lift as the helium balloon. Assume air is \(79.0 \%\) nitrogen and \(21.0 \%\) oxygen by volume. The "lift" of a balloon is given by the difference between the mass of air displaced by the balloon and the mass of gas inside the balloon. a. Will the temperature in the hot-air balloon have to be higher or lower than \(25^{\circ} \mathrm{C}\) ? Explain. b. Calculate the temperature of the air required for the hotair balloon to provide the same lift as the helium balloon at \(1.00\) atm and \(25^{\circ} \mathrm{C}\). Assume atmospheric conditions are \(1.00\) atm and \(25^{\circ} \mathrm{C}\).

A sealed balloon is filled with \(1.00 \mathrm{~L}\) helium at \(23^{\circ} \mathrm{C}\) and \(1.00\) atm. The balloon rises to a point in the atmosphere where the pressure is 220 . torr and the temperature is \(-31^{\circ} \mathrm{C}\). What is the change in volume of the balloon as it ascends from \(1.00\) atm to a pressure of \(220 .\) torr?

A \(15.0\) -L rigid container was charged with \(0.500\) atm of krypton gas and \(1.50\) atm of chlorine gas at \(350 .{ }^{\circ} \mathrm{C}\). The krypton and chlorine react to form krypton tetrachloride. What mass of krypton tetrachloride can be produced assuming \(100 \%\) yield?

Urea \(\left(\mathrm{H}_{2} \mathrm{NCONH}_{2}\right)\) is used extensively as a nitrogen source in fertilizers. It is produced commercially from the reaction of ammonia and carbon dioxide: $$ 2 \mathrm{NH}_{3}(g)+\mathrm{CO}_{2}(g) \underset{\text { Pressure }}{\text { Heat }}{\mathrm{H}}_{2} \mathrm{NCONH}_{2}(s)+\mathrm{H}_{2} \mathrm{O}(g) $$ Ammonia gas at \(223^{\circ} \mathrm{C}\) and \(90 .\) atm flows into a reactor at a rate of \(500 . \mathrm{L} / \mathrm{min}\). Carbon dioxide at \(223^{\circ} \mathrm{C}\) and 45 atm flows into the reactor at a rate of \(600 . \mathrm{L} / \mathrm{min} .\) What mass of urea is produced per minute by this reaction assuming \(100 \%\) yield?

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