Question

The hydrogen-filled dirigible Hindenburg had a volume of \(1.99 \times 10^{8} \mathrm{~L}\). If the hydrogen used was produced by the reaction of carbon with steam, how many kilograms of carbon would have been needed to produce enough hydrogen to fill the dirigible at \(20^{\circ} \mathrm{C}\) and \(740 \mathrm{~mm}\) pressure? $$ \mathrm{C}(s)+\mathrm{H}_{2} \mathrm{O}(g) \longrightarrow \mathrm{CO}(g)+\mathrm{H}_{2}(g) $$

Short answer

Approximately 96700 kg of carbon is needed.

Step-by-step solution

Determine the moles of hydrogen

Use the Ideal Gas Law to determine the number of moles of hydrogen gas. The Ideal Gas Law is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin. Convert the given conditions to the proper units: \( P = 740 \text{ mmHg} = 0.974 \text{ atm} \) (by dividing by 760 mmHg/atm), \( V = 1.99 \times 10^8 \text{ L} \), \( T = 20^{\circ} \text{C} = 293 \text{ K} \). \( R \) is 0.0821 L·atm/mol·K. Solving for \( n \), we have \( n = \frac{PV}{RT} = \frac{(0.974 \text{ atm})(1.99 \times 10^8 \text{ L})}{(0.0821 \text{ L·atm/mol·K})(293 \text{ K})} \approx 8.05 \times 10^6 \text{ mol} \) of hydrogen.

Calculate the moles of carbon needed

According to the balanced reaction \( \mathrm{C}(s) + \mathrm{H as a}{_2} \mathrm{O}(g) \rightarrow \mathrm{CO}(g) + \mathrm{H_2}(g) \), each mole of hydrogen requires one mole of carbon. Therefore, the moles of carbon needed are the same as the moles of hydrogen, which is \( 8.05 \times 10^6 \text{ mol} \).

Convert moles of carbon to mass

The molar mass of carbon is 12.01 g/mol. To find the mass of carbon, multiply the moles of carbon by its molar mass: \( 8.05 \times 10^6 \text{ mol} \times 12.01 \text{ g/mol} = 9.67 \times 10^7 \text{ g} \). Convert grams to kilograms by dividing by 1000, which gives \( 9.67 \times 10^4 \text{ kg} \) of carbon.

Key concepts

Stoichiometry

Stoichiometry is the part of chemistry that helps us understand the relationships between the amounts of reactants and products in a chemical reaction. Imagine you're baking a cake: stoichiometry is like figuring out how much flour and sugar you need for a certain number of cakes. With stoichiometry, you can predict the amounts of substances consumed and produced. This is done using balanced chemical equations.

In the case of the Hindenburg, stoichiometry helps determine how much carbon is needed to produce the hydrogen gas to fill the dirigible. By analyzing the balanced chemical equation \( \text{C}(s) + \text{H}_2\text{O}(g) \rightarrow \text{CO}(g) + \text{H}_2(g) \), we can see that one mole of carbon reacts with water to produce one mole of hydrogen and one mole of carbon monoxide. This relationship allows us to calculate the exact amount needed for the desired output, which in this case, is filling the large volume of the Hindenburg.

Chemical Reactions

Chemical reactions involve substances changing or reacting to form new substances. This is a fundamental concept in chemistry where elements or compounds interact and undergo transformation. In our problem, carbon and steam react to produce carbon monoxide and hydrogen gas, which is the reaction \( \text{C}(s) + \text{H}_2\text{O}(g) \rightarrow \text{CO}(g) + \text{H}_2(g) \).

These reactions can be understood through a balanced chemical equation, representing how much of each substance is involved. A key takeaway is that in chemical reactions, mass is conserved, which means the same atoms are present in the products as in the reactants, merely rearranged. This principle allows us to use stoichiometry effectively to calculate the necessary reactants for the desired amount of product.

Molar Mass

Molar mass is the weight of one mole of a given element or compound. It's like knowing how much one dozen apples would weigh if you only knew the weight of one apple. The molar mass of an element is its atomic weight in grams per mole.

In the context of the Hindenburg problem, knowing the molar mass of carbon is crucial to finding out how much carbon, in grams or kilograms, is needed. Carbon’s molar mass is 12.01 g/mol. By multiplying the moles of carbon required by its molar mass, we can convert the amount from moles to a more tangible mass amount in grams or kilograms, which in this case results in approximately 96,700 kg of carbon.

Gas Laws

Gas laws are mathematical formulas that describe the behavior of gases. The Ideal Gas Law is particularly useful, stating that \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is temperature in Kelvin.

For the hydrogen in the Hindenburg, we apply the Ideal Gas Law to determine how many moles of hydrogen fill the dirigible under given conditions of temperature and pressure. By rearranging the formula to solve for \( n \) (moles), \( n = \frac{PV}{RT} \), and substituting in the known values, you can calculate the moles of hydrogen gas required. Thus, the Ideal Gas Law allows us to link physical conditions with chemical quantities, which is essential for using stoichiometry to further solve this problem.