Question

Both Jacques Charles and Joseph Louis Guy-Lussac were avid balloonists. In his original flight in 1783 , Jacques Charles used a balloon that contained approximately \(31,150 \mathrm{~L}\) of \(\mathrm{H}_{2}\). He generated the \(\mathrm{H}_{2}\) using the reaction between iron and hydrochloric acid: $$ \mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ How many kilograms of iron were needed to produce this volume of \(\mathrm{H}_{2}\) if the temperature was \(22^{\circ} \mathrm{C} ?\)

Short answer

$$ \begin{aligned} n &= \frac{(1 \mathrm{~atm})(31,150 \mathrm{~L})}{(0.0821 \mathrm{~L~atm / (K~mol)})(295.15 \mathrm{~K})} \\ n &\approx 1365.4 \mathrm{~mol} \end{aligned} $$ #tag_title#Step 2: Find the number of moles of Fe needed#tag_content# Now that we know the number of moles of H₂ produced, we can use the balanced chemical equation to find the number of moles of Fe needed. The equation is: $$ \mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ Since there's a 1:1 ratio of Fe to H₂, we need the same number of moles of Fe, which is 1365.4 mol. #tag_title#Step 3: Find the mass of Fe in kilograms#tag_content# Finally, we need to convert the number of moles of Fe to mass in kilograms. The molar mass of Fe is 55.845 g/mol. First, we'll find the mass in grams: $$ \begin{aligned} m_\mathrm{Fe} &= (1365.4 \mathrm{~mol}) \cdot (55.845 \mathrm{~g/mol}) \\ m_\mathrm{Fe} & \approx 76,305.8 \mathrm{~g} \end{aligned} $$ Now, let's convert the mass to kilograms: $$ \begin{aligned} m_\mathrm{Fe} &= 76,305.8 \mathrm{~g} \cdot \frac{1 \mathrm{~kg}}{1000 \mathrm{~g}} \\ m_\mathrm{Fe} &\approx 76.3 \mathrm{~kg} \end{aligned} $$ #Answer# Thus, the amount of iron needed is approximately \(76.3 \mathrm{~kg}\).

Step-by-step solution

Find the number of moles of H₂ produced

First, we need to find the number of moles of hydrogen gas produced from the given volume. We can do this using the Ideal Gas Law, which is: $$ PV = nRT $$ where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature. We're given the volume V, which is 31,150 L, and the temperature T, which is 22°C or 295.15 K (remember to convert to Kelvin). Since the problem doesn't provide a specific pressure, we assume it is produced under standard pressure conditions, which is approximately 1 atm. The ideal gas constant R is 0.0821 L atm / (K mol). Now we can solve for n: $$ \begin{aligned} (1 \mathrm{~atm})(31,150 \mathrm{~L}) &= n \cdot (0.0821 \mathrm{~L~atm / (K~mol)}) \cdot (295.15 \mathrm{~K}) \\ n &= \frac{(1 \mathrm{~atm})(31,150 \mathrm{~L})}{(0.0821 \mathrm{~L~atm / (K~mol)})(295.15 \mathrm{~K})} \end{aligned} $$ Let's calculate the number of moles.

Key concepts

Jacques Charles

Jacques Charles was an influential figure in the development of ballooning, an activity that fascinated many during the late 18th century. As a physicist and chemist, he made significant contributions to our understanding of gases. His connection to gas laws, particularly the one named after him, is noteworthy. Charles's Law describes how gases tend to expand when heated and is mathematically represented as \( V \propto T \), or \( \frac{V}{T} = \text{constant} \), where \( V \) is volume and \( T \) is temperature.

In Charles's time, exploring the atmosphere using balloons was not just an adventurous pursuit but also a scientific endeavor. He achieved fame for his pioneering flight in 1783, using hydrogen—a gas much lighter than air—to lift his balloon. This demonstrated the practicality of using lightweight gases to achieve flight—a concept that balloonists embraced and pursued further. It marked an era where scientific innovation was directly applied to real-world experiences, bridging the gap between theoretical knowledge and practical application.

Iron and Hydrochloric Acid Reaction

The reaction between iron and hydrochloric acid is a classic example of a single displacement reaction in chemistry. In this reaction, iron (Fe) reacts with hydrochloric acid (HCl) to produce iron(II) chloride (FeCl₂) and hydrogen gas (H₂). The chemical equation for this reaction is:

• \( \mathrm{Fe}(s) + 2 \mathrm{HCl}(aq) \rightarrow \mathrm{FeCl}_{2}(aq) + \mathrm{H}_{2}(g) \)

Steps of the Reaction

• Iron is a solid metal that reacts with hydrochloric acid, which is present in aqueous form.
• During the reaction, iron atoms give up electrons to the hydrogen ions in the hydrochloric acid, thus forming iron ions and hydrogen gas.
• Hydrogen gas forms bubbles, which can be captured and observed in laboratory settings.

This reaction is exothermic, releasing energy as the bonds in the reactants are broken and new bonds are formed in the products. Knowing how to handle reactions safely and understand their outcomes is essential for experimental chemistry, as demonstrated by Jacques Charles in his ballooning experiments.

Moles of Gas

Understanding the concept of moles is fundamental when dealing with gases and chemical reactions. The mole is a standard unit of measure in chemistry that helps us quantify atoms, molecules, or ions in a sample of matter. One mole is equivalent to Avogadro's number, approximately \(6.022 \times 10^{23}\), which is the number of constituent particles in a given amount of substance.

In gas calculations, the Ideal Gas Law is indispensable. It relates the pressure, volume, temperature, and number of moles of a gas:

• \( PV = nRT \)

Calculating Moles

• The Ideal Gas Law can be rearranged to solve for moles \( (n) \):
  • \( n = \frac{PV}{RT} \)
• In the example of Jacques Charles's balloon, using the given volume (\(31,150 \) L) and temperature (295.15 K), along with known constants like pressure (1 atm) and the gas constant (R = 0.0821 L atm / K mol), we can calculate the exact number of moles of hydrogen gas produced.
• This provides a bridge from theoretical calculations to practical applications in chemical engineering and physics.

Balloonists

Ballooning during the 18th and 19th centuries was seen as both a daring adventure and a method for scientific exploration. People like Jacques Charles, often referred to as balloonists, played a pivotal role in the evolution of this field. Their experiments sought to understand atmospheric conditions and allowed them to witness firsthand the effects of altitude on the environment and themselves.

The initial curiosity about flight and atmospheric exploration led balloonists to innovative approaches using gases. Hydrogen, in particular, became the go-to choice due to its lighter-than-air properties, allowing balloons to float above the ground. However, choosing hydrogen also came with risks, given its flammability, which balloonists had to carefully manage.

• The volume of gas required to lift balloons to considerable heights forced balloonists to understand and utilize gas laws practically, forming the basis of modern aerostatics.
• As nineteenth-century technology progressed, so did the materials and designs of balloons, enhancing safety and efficiency.
• These early flights contributed significantly to the meteorological observations and atmospheric studies that followed.

The legacy left by early balloonists like Jacques Charles underscored the importance of inquiry and experimentation, driving many scientific advancements.