Question

The hydrogen-filled dirigible "Hindenberg" had a gas volume of \(1.4 \times 10^{8}\) liters. Assuming that it was filled to a pressure of \(1 \mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\), how much energy was released when it burned to gaseous products at a reaction temperature of \(25^{\circ} \mathrm{C}\) ? The standard heat of formation of water vapor at \(25^{\circ} \mathrm{C}\) is \(-57.8 \mathrm{~K}\) cal/mole.

Short answer

The energy released when the hydrogen-filled dirigible "Hindenberg" burned to gaseous products at a reaction temperature of \(25^{\circ} \mathrm{C}\) is approximately \(1.50 \times 10^{12}~\mathrm{J}\).

Step-by-step solution

1. Find the volume of hydrogen gas

Given that the volume of hydrogen gas in the dirigible is \(1.4 \times 10^{8}\) liters. To determine the number of moles of hydrogen present, we'll use the Ideal Gas Law formula: \( PV = nRT \) Where: P = Pressure of hydrogen gas (1 atm) V = Volume of hydrogen gas (\(1.4 \times 10^{8}\) liters) n = Number of moles of hydrogen gas R = Ideal Gas Constant (0.0821 L atm / K mol) - This constant will be used to work with the given units T = Temperature in Kelvin We know that \(0^{\circ} \mathrm{C}\) is the initial temperature of the gas. Convert it to Kelvin: \( T = 0 + 273.15 = 273.15 \mathrm{~K} \)

2. Calculate the number of moles of hydrogen gas

Rearrange the Ideal Gas Law formula to isolate n and plug in the values: \( n = \frac{PV}{RT} \) \( n = \frac{1 \times (1.4 \times 10^{8})}{0.0821 \times 273.15} \) \( n \approx 6.22 \times 10^6 \quad \mathrm{moles \, of \, hydrogen \, gas} \)

3. Find the energy released per mole during the reaction

The standard heat of formation of water vapor at \(25^{\circ} \mathrm{C}\) is given as \(-57.8 \mathrm{~K}\) cal/mole. This value is negative, which indicates that the reaction is exothermic (releases heat). We can convert this value to Joules (1 cal = 4.184 J) for consistency with the other units: \( -57.8 \, \mathrm{K} \frac{\mathrm{cal}}{\mathrm{mole}} \times \frac{4.184 \, \mathrm{J}}{\mathrm{cal}} \approx -241.83 \, \mathrm{K} \frac{\mathrm{J}}{\mathrm{mole}} \)

4. Calculate the total energy released during the reaction

Now we can find the total energy released by multiplying the energy released per mole by the total number of moles of hydrogen gas: \( \mathrm{Energy \, released} = \mathrm{Energy \, released \, per \, mole} \times \mathrm{Number \, of \, moles} \) \( \mathrm{Energy \, released} = (-241.83 \, \mathrm{K} \frac{\mathrm{J}}{\mathrm{mole}}) \times (6.22 \times 10^6 \, \mathrm{moles}) \) \( \mathrm{Energy \, released} \approx -1.50 \times 10^{12} \, \mathrm{J} \) The energy released when the hydrogen-filled dirigible "Hindenberg" burned to gaseous products at a reaction temperature of \(25^{\circ} \mathrm{C}\) is approximately \(1.50 \times 10^{12}~\mathrm{J}\).

Key concepts

Heat of Formation

Heat of formation is a fundamental concept in chemistry. It refers to the energy change that occurs when one mole of a compound is formed from its elements in their standard states. In our exercise, we deal with the formation of water vapor from hydrogen and oxygen. The heat of formation for water vapor at 25°C is given as

• -57.8 kcal/mole. This negative value indicates that heat is released in the process.

Understanding the heat of formation helps us predict whether a reaction is exothermic or endothermic. For the Hindenberg dirigible, the heat of formation helps to calculate the total energy released when hydrogen burns to form water vapor. To convert this energy to a more usable unit like Joules, we need to use the conversion factor:

• 1 cal = 4.184 J.

Calculating in such consistent units provides a better understanding of the energy changes happening in the reaction.

Exothermic Reaction

Exothermic reactions are those that release energy, typically in the form of heat. When hydrogen burns, it undergoes a chemical reaction with oxygen to form water, releasing energy in the process. The term 'exothermic' comes from the Greek word 'exo,' meaning outside, and 'thermic,' meaning heat. Thus, exothermic reactions release heat to the surroundings. In the context of our exercise:

• The burning of hydrogen gas in the Hindenberg is an exothermic reaction.
• Given that the heat of formation of water is negative, it confirms that the reaction releases heat.

It's essential to remember that in an exothermic reaction, the energy of the products is lower than the energy of the reactants. As a result, the surrounding environment becomes warmer. This energy release is crucial for converting stored chemical energy into different forms during the reaction.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In chemical reactions, particularly exothermic ones, stored chemical energy is converted into heat. The exercise helps illustrate this by considering the reaction of hydrogen with oxygen to form water.

• The energy stored in the chemical bonds of the hydrogen molecules is released as heat when these bonds are broken and new bonds form in water.

Given that the total energy release was calculated to be approximately \(-1.50 \times 10^{12}\) Joules, this demonstrates a significant conversion of chemical energy into thermal energy. Understanding this conversion is essential for applications like predicting how much fuel is needed to achieve a specific amount of work, or in this case, knowing how much energy was released when the Hindenberg burned.

Reaction Enthalpy

Reaction enthalpy, also known as the heat of reaction, is a measure of the heat change during a chemical reaction at constant pressure. It is symbolized by \(\Delta H\). In the given exercise, the enthalpy change is primarily due to the energy released as hydrogen combusts.

• The negative enthalpy value indicates that the reaction is exothermic, releasing heat to the surroundings.

To calculate the reaction enthalpy, we utilize the relationship:\[\Delta H = \text{(Number of moles)} \times \text{(Energy per mole)}\]This equation helps determine the total energy change for the reaction. With the given values, we find that:

• \(-241.83\ \text{J/mole}\) for the reaction.

This enthalpy change is essential in understanding how much energy is involved in the formation of products from reactants, which allows for predicting the effects on environments or systems in which these reactions occur.