Problem 58 - Chemistry: The Central Science in SI Units, Global Edition

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 \(22{ }^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

The mass of iron needed to produce the given volume of \(H_2\) at the given temperature can be determined through the following steps: 1. Calculate the moles of \(H_2\) using the ideal gas law equation considering atmospheric pressure. 2. Determine the moles of iron required using stoichiometry, which is equal to the moles of \(H_2\). 3. Convert moles of iron into mass (in kg) by multiplying by the molar mass of iron (55.845 g/mol). After following these steps and calculations, the mass of iron required to generate \(31,150 L\) of \(H_2\) at \(22{ }^{\circ} \mathrm{C}\) is approximately \(17.23 kg\).

Step 1: Determine moles of \(H_2\) using the ideal gas law equation

First, we will need to use the ideal gas law equation, which is: \[PV = nRT\] Where: P = pressure (in atm) V = volume (in L) n = moles of the gas R = ideal gas constant (\(0.0821 \frac{\text{L atm}}{\text{mol K}}\)) T = temperature (in Kelvin) In this problem, the volume \(V = 31,150 L\), the temperature \(T = 22°C = 295 K\), and we assume the pressure to be atmospheric, \(P = 1 atm\). We need to find n, the moles of \(H_2\). Rearranging the equation, we get: \[n = \frac{PV}{RT}\]

Step 2: Calculate the moles of \(H_2\)

Substitute the given values and the assumption of atmospheric pressure into the equation from step 1: \[n = \frac{(1\,\text{atm})(31,150\, \text{L})}{(0.0821\, \frac{\text{L atm}}{\text{mol K}})(295\, \text{K})}\] Now, solve the equation to get the moles of \(H_2\).

Step 3: Use stoichiometry to determine moles of iron required

The balanced equation for the given reaction is: \[\mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g)\] We can see that \(1\) mole of iron reacts to produce \(1\) mole of \(H_2\). So, the moles of iron required are equal to the moles of \(H_2\) calculated in step 2.

Step 4: Convert moles of iron into mass

To convert moles of iron into mass (in kg), we multiply moles by the molar mass of iron: \[mass\: of\: iron = (moles\: of\: iron)(molar\: mass\: of\: iron)\] where the molar mass of iron = \(55.845 g/mol\). Now multiply the moles of iron we found in step 2 by the molar mass of iron and divide by 1000 to convert the grams into kilograms.

Step 5: Calculate the mass of iron in kilograms

Using the moles of iron from step 2 and the molar mass of iron, calculate the mass of iron needed in kilograms. This will be the final answer.

Key Concepts

Stoichiometry

Stoichiometry helps us understand the quantitative relationships in chemical reactions. It's like following a recipe, ensuring you have the right amount of ingredients for your dish. In chemical equations, stoichiometry tells us how much of each reactant is needed to form a desired amount of product.
In the exercise's reaction:

• One mole of iron (\(\mathrm{Fe}\)) reacts with two moles of hydrochloric acid (\(\mathrm{HCl}\)).
• This produces one mole of \(\mathrm{H_2}\) gas.

It’s crucial to balance the chemical equation first, as this shows the exact proportions needed.
For example, producing 31,150 liters of \(\mathrm{H_2}\) requires using the Ideal Gas Law first to find moles of \(\mathrm{H_2}\) and then using stoichiometry to connect it back to moles of iron. This ensures each component is measured correctly, much like making sure you don’t have too much or too little of any ingredient in a recipe.

Mole Concept

The mole concept is a way to count particles like atoms and molecules in chemistry. It gives us a bridge between the atomic scale and the real-world scale by using Avogadro’s number, which is \(6.022 \times 10^{23}\) particles per mole.
In the context of the exercise, you start by calculating the moles of \(\mathrm{H_2}\) gas using the ideal gas law: \(PV = nRT\).
Here:

• \(n\) is the number of moles.
• \(R\) is the ideal gas constant (\(0.0821 \text{ L atm/mol K}\)).

By inputting the volume, temperature, and pressure, you can solve for \(n\), giving you the moles of \(\mathrm{H_2}\).
This understanding enables you to convert between mass and moles, moving from the abstract atomic world to measurable quantities. Knowing how to manipulate these calculations is key in any chemistry task involving gases.

Chemical Reactions

Chemical reactions are transformations where substances change into new substances. This is shown using a balanced chemical equation. For the exercise, when iron reacts with hydrochloric acid:

• Iron (\(\mathrm{Fe}\)) and hydrochloric acid (\(\mathrm{HCl}\)) are the reactants.
• Iron chloride (\(\mathrm{FeCl_2}\)) and hydrogen gas (\(\mathrm{H_2}\)) are the products.

Every reaction involves the breaking of bonds in reactants and forming new bonds in products.
The balanced equation \(\mathrm{Fe}(s) + 2\,\mathrm{HCl}(aq) \rightarrow \mathrm{FeCl}_2(aq) + \mathrm{H}_2(g)\) shows the proportionate amounts of each substance involved.
This understanding is crucial because it reveals not just what substances will be formed but in what amounts, guiding you to connect observable changes back to molecular interactions.