Problem 141 - Chemistry An Atoms First Approach

Question

The total volume of hydrogen gas needed to fill the Hindenburg was \(2.0 \times 10^{8} \mathrm{L}\) at 1.0 atm and \(25^{\circ} \mathrm{C}\). Given that \(\Delta H_{\mathrm{f}}^{\circ}\) for \(\mathrm{H}_{2} \mathrm{O}(l)\) is \(-286 \mathrm{kJ} / \mathrm{mol},\) how much heat was evolved when the Hindenburg exploded, assuming all of the hydrogen reacted to form water?

Step-by-Step Solution

Answer

The total amount of heat evolved when the Hindenburg exploded, assuming all of the hydrogen reacted to form water, is approximately \(-2.3 \times 10^9\space kJ\).

Step 1: Find the initial number of moles of hydrogen gas

To find the initial number of moles of hydrogen gas, we use the Ideal Gas Law: \(PV = nRT\). We are given the volume (\(V= 2.0 \times 10^8 L\)), pressure (\(P = 1.0\space atm\)), and temperature (\(T = 25^{\circ}\mathrm{C} \approx 298\space K\)). The gas constant, \(R = 0.0821 \frac{L \cdot atm}{mol \cdot K}\). Rearranging the formula to solve for n, we get \(n = \frac{PV}{RT}\). \(n = \frac{(1.0\space atm)(2.0 \times 10^{8}\space L)}{(0.0821\space L\cdot atm/mol\cdot K)(298\space K)} \approx 8.1 \times 10^6\space mol\)

Step 2: Determine the amount of heat evolved during the explosion

Since all the hydrogen gas reacts to form liquid water, the enthalpy change of the reaction can be calculated using the number of moles of hydrogen gas and the enthalpy of formation for water. We are given that the enthalpy of formation of water, \(\Delta H_{\mathrm{f}}^{\circ}\), is -286 kJ/mol. To determine the total heat evolved, we multiply the number of moles of hydrogen gas by the enthalpy of formation of water: \(q = n \cdot \Delta H_{\mathrm{f}}^{\circ}\) \(q \approx (8.1 \times 10^6 \space mol)\cdot(-286 \space kJ/mol) \approx -2.3 \times 10^9\space kJ\) Since the value is negative, this indicates that heat was evolved (exothermic reaction) during the explosion. The total amount of heat evolved is approximately \(-2.3 \times 10^9\space kJ\).

Key Concepts

Moles of Hydrogen

The concept of moles is a fundamental part of understanding how chemical reactions work. In this exercise, we're dealing with hydrogen gas, and we use the Ideal Gas Law to calculate the number of moles. The Ideal Gas Law is represented by the equation \(PV = nRT\), where:

• \(P\) is the pressure of the gas in atm
• \(V\) is the volume of the gas in liters
• \(n\) is the number of moles
• \(R\) is the ideal gas constant, approximately 0.0821 \(\frac{L \cdot atm}{mol \cdot K}\)
• \(T\) is the temperature in Kelvin

To find the number of moles \(n\), we rearrange the formula to \(n = \frac{PV}{RT}\).
By substituting the given values from the exercise: volume \(V = 2.0 \times 10^8\space L\), pressure \(P = 1.0\space atm\), and temperature \(T = 298\space K\), we calculate approximately \(8.1 \times 10^6\space moles\) of hydrogen. Understanding moles helps in determining how much of the substance is reacting in a chemical process.

Enthalpy Change

Enthalpy change is a crucial concept in chemical reactions, indicating how much heat energy is absorbed or released. It gives insight into the energy dynamics within a reaction.
In this exercise, we look at the formation of water from hydrogen gas. The enthalpy of formation, \(\Delta H_f^\circ\), provides the energy change as hydrogen reacts to form water. It is given as \(-286\space kJ/mol\), meaning it releases 286 kJ of energy per mole of water produced.
To find the total enthalpy change during the Hindenburg explosion when all the hydrogen reacts, we calculate \(q = n \cdot \Delta H_f^\circ\).
Substituting the values: \(n = 8.1 \times 10^6\space mol\) and \(\Delta H_f^\circ = -286\,\mathrm{kJ/mol}\), leads to \(q \approx -2.3 \times 10^9\space kJ\). This result indicates a large amount of heat is evolved, reaffirming the reaction's exothermic nature.

Exothermic Reaction

An exothermic reaction is one that releases energy, usually in the form of heat, into its surroundings. The Hindenburg disaster, used as a basis for this exercise, is a historical example of a powerful exothermic reaction.
When hydrogen gas reacts with oxygen to form water, the reaction releases energy, making it exothermic. This release is represented mathematically by a negative enthalpy value, \(-2.3 \times 10^9\) kJ in our case.
Exothermic reactions are integral in various processes:

• Combustion, such as burning fuels
• Cooking food, where heat transforms raw ingredients
• Industrial processes, like cement setting or the production of various chemicals

By studying these reactions, we get a clearer picture of how energy transformations affect both industrial applications and naturally occurring events. Understanding this concept helps in designing better energy-efficient systems, and predicting the outcomes of similar chemical reactions.