Problem 58 - Chemistry: The Central Science

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

To produce 31,150 L of hydrogen gas at 22°C, approximately 72.045 kg of iron are needed.

Step 1: Convert temperature to Kelvin

First, convert the given temperature from Celsius to Kelvin by adding 273.15: \(T = 22^{\circ} \mathrm{C} + 273.15 = 295.15 \mathrm{K}\)

Step 2: Apply the Ideal Gas Law to find the moles of hydrogen gas

Now, we will use the Ideal Gas Law equation to find the moles of hydrogen gas produced: \(PV = nRT\) Where: P = Pressure (atmospheres) V = Volume (liters) n = Moles of gas R = Ideal Gas Constant (0.0821 L atm / mol K) T = Temperature (Kelvin) We will assume that the pressure is equal to 1 atm (standard atmospheric pressure). Plug in the given values into the Ideal Gas Law equation: \(1 \mathrm{atm} \times 31,150 \mathrm{L} = n \times 0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}} \times 295.15 \mathrm{K}\) Now, solve for n (moles of hydrogen gas): \(n = \frac{1 \mathrm{atm} \times 31,150 \mathrm{L}}{0.0821 \frac{\mathrm{L} \cdot \mathrm{atm}}{\mathrm{mol} \cdot \mathrm{K}} \times 295.15 \mathrm{K}}\) \(n \approx 1,290 \, \mathrm{moles \, of \, H}_{2}\)

Step 3: Use stoichiometry to find moles of iron

According to the balanced chemical equation, one mole of iron will produce one mole of hydrogen gas: $$ \mathrm{Fe}(s)+2 \mathrm{HCl}(a q) \longrightarrow \mathrm{FeCl}_{2}(a q)+\mathrm{H}_{2}(g) $$ Thus, the moles of iron needed for the reaction are equal to the moles of hydrogen gas: \(1,290 \, \mathrm{moles} \, \mathrm{of} \, \mathrm{Fe}\)

Step 4: Convert moles of iron to mass

Now, we need to convert the moles of iron to mass (in kilograms) by using the molar mass of iron: Molar mass of iron, \(Fe = 55.85 \, \frac{\mathrm{g}}{\mathrm{mol}}\) Mass of iron in grams = moles of iron × molar mass of iron Mass of iron in grams = \(1,290 \, \mathrm{moles} \, \mathrm{of} \, \mathrm{Fe} \times 55.85 \, \frac{\mathrm{g}}{\mathrm{mol}}\) Mass of iron in grams = \(72,045 \, \mathrm{g}\) Finally, convert grams to kilograms: Mass of iron in kilograms = \(\frac{72,045 \, \mathrm{g}}{1,000 \, \frac{\mathrm{g}}{\mathrm{kg}}\) = \(72.045 \, \mathrm{kg}\)

Step 5: Final answer

So, to produce 31,150 L of hydrogen gas at 22°C, approximately 72.045 kg of iron are needed.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation that relates the pressure, volume, temperature, and number of moles of a gas. In the textbook problem, we see it expressed as the formula:
\( PV = nRT \)
where \( P \) represents pressure in atmospheres, \( V \) is the volume in liters, \( n \) is the number of moles, \( R \) is the ideal gas constant \( (0.0821 \frac{L \cdot atm}{mol \cdot K}) \), and \( T \) is the temperature in Kelvin. This equation allows us to calculate any one of these variables as long as the other three are known. For the given problem, this law is applied to determine how many moles of hydrogen gas would fill a balloon to a certain volume at a known temperature and pressure. Always remember to convert the temperature to Kelvin and assume standard pressure (1 atm) if it is not provided, as these are common conditions for using this law in stoichiometry problems.

Molar Mass

Molar mass is the mass of one mole of a substance, typically expressed in grams per mole (g/mol). It is a physical property that is crucial for converting between moles and grams in chemical reactions. The molar mass of iron, for example, is \( 55.85 \frac{g}{mol} \), as used in the step-by-step solution. By knowing this value, we can determine how much mass of iron is needed in a reaction to produce a specific amount of a gas, such as hydrogen in the textbook problem. Whenever you need to perform such a conversion, always refer to the periodic table for the molar masses of the elements involved in your reaction.

How to Use Molar Mass:

• Identify the substance and find its molar mass (usually from the periodic table).
• Multiply the number of moles by the molar mass to find the mass in grams.
• If needed, convert grams to kilograms by dividing by 1,000.

Effective use of molar mass in stoichiometric calculations is essential to quantify the amounts of reactants or products in a chemical reaction.

Chemical Reaction

A chemical reaction is the process that leads to the transformation of one set of chemical substances to another. It involves breaking and forming bonds between atoms. In our exercise, the reaction is between iron (Fe) and hydrochloric acid (HCl) to produce iron(II) chloride (FeCl2) and hydrogen gas (H2). The equation has been balanced as:
\(\mathrm{Fe}(s) + 2\mathrm{HCl}(aq) \rightarrow \mathrm{FeCl2}(aq) + \mathrm{H2}(g)\)
Balancing a chemical equation is fundamental because it follows the Law of Conservation of Mass, which states that mass is neither created nor destroyed in a chemical reaction. Therefore, understanding the balanced equation helps to ensure that the correct stoichiometric proportions of reactants and products are used or produced.

Stoichiometric Calculations

Stoichiometric calculations are mathematical conversions between amounts of reactants and products in a chemical reaction using the balanced chemical equation. They are fundamental for scientists and engineers to predict how much reactants are needed or how much products are formed in a reaction. The concept of mole-to-mole conversion is particularly crucial, as seen in the textbook example. By knowing that one mole of iron produces one mole of hydrogen, we can directly relate moles of iron to moles of hydrogen produced. To improve your understanding of stoichiometry:

Practice Tips:

• Always start with a balanced chemical equation.
• Determine the mole ratios between reactants and products.
• Use unit conversion methods to switch between grams, moles, and liters (if dealing with gases at standard temperature and pressure).
• Check the unity of your calculation; each step should cancel out the previous unit, thus guiding you to the correct final unit.

Conceptual understanding and practice with stoichiometric calculations empower you to tackle a wide array of chemistry problems.