Problem 96 - Chemistry: The Central Science

Question

Comparing the energy associated with the rainstorm and that of a conventional explosive gives some idea of the immense amount of energy associated with a storm. (a) The heat of vaporization of water is \(44.0 \mathrm{~kJ} / \mathrm{mol}\). Calculate the quantity of energy released when enough water vapor condenses to form \(0.50\) inches of rain over an area of one square mile. (b) The energy released when one ton of dynamite explodes is \(4.2 \times 10^{6} \mathrm{~kJ} .\) Calculate the number of tons of dynamite needed to provide the energy of the storm in part (a).

Step-by-Step Solution

Answer

The energy released when enough water vapor condenses to form 0.50 inches of rain over an area of one square mile is \(8.031 \times 10^{10}\) kJ. The number of tons of dynamite needed to provide the energy of the storm in part (a) is approximately \(1.9121 \times 10^{4}\) tons.

Step 1: Change measurements to SI units

We are given that the rainwater is 0.50 inches deep and covers an area of one square mile. To convert these measurements to SI units (meters), we can use the following conversion factors: 1 inch = 0.0254 meters 1 square mile = 2.58999 x 10^6 square meters Depth of water: 0.50 inches = 0.50 * 0.0254 meters = 0.0127 meters Area of water: 1 square mile = 1 * 2.58999 x 10^6 square meters #Step 2#: Calculate the volume of the water

Step 2: Calculate the volume of water

Now we need to find the volume (V) of rainwater. To do this, we multiply the area (A) by the depth (D) of the water. V = A * D = (2.58999 x 10^6 square meters) * (0.0127 meters) = 32861.003 m^3 #Step 3#: Determine the mass of water

Step 3: Find the mass of water

To calculate the mass of water, we multiply the volume of water (V) by the density of water (ρ), which is 1000 kg/m^3. Mass = ρ * V = 1000 kg/m^3 * 32861.003 m^3 = 3.2861 x 10^7 kg #Step 4#: Calculate the number of moles of water

Step 4: Calculate the number of moles

To calculate the number of moles (n) of water, we divide the mass of water (m) by the molar mass of water (M), which is 18.015 g/mol or 0.018015 kg/mol. n = m / M = (3.2861 x 10^7 kg) / (0.018015 kg/mol) = 1.8253 x 10^9 moles #Step 5#: Calculate the energy released

Step 5: Calculate the energy released

We are given that the heat of vaporization of water is 44.0 kJ/mol. To find the energy (E) released during condensation, we multiply the number of moles (n) by the heat of vaporization (H). E = n * H = (1.8253 x 10^9 moles) * (44.0 kJ/mol) = 8.031 x 10^10 kJ #a# The energy released when enough water vapor condenses to form 0.50 inches of rain over an area of one square mile is 8.031 x 10^10 kJ. #b# Calculate the number of tons of dynamite needed to provide the energy of the storm in part (a) #Step 1#: Calculate the energy equivalent in dynamite

Step 6: Find dynamite energy equivalent

We are given that the energy released when one ton of dynamite explodes is 4.2 x 10^6 kJ. To find the equivalent amount of energy in terms of dynamite, we multiply the energy released during the storm (E) by the energy released per ton of dynamite (D). NumberOfTons = E / D = (8.031 x 10^10 kJ) / (4.2 x 10^6 kJ/ton) = 1.9121 x 10^4 tons #b# The number of tons of dynamite needed to provide the energy of the storm in part (a) is approximately 1.9121 x 10^4 tons.

Key Concepts

Heat of Vaporization

The heat of vaporization is the amount of energy needed to convert a substance from its liquid phase to its gaseous phase without changing its temperature. For water, this value is 44.0 kJ/mol.
This means that when 1 mole of water evaporates, or condenses, it absorbs or releases 44.0 kJ of energy respectively.

• When water vapor condenses to form rain, the heat of vaporization is released back into the atmosphere.
• This release of energy is significant, especially over large areas like a one square mile during a rainstorm as presented in our exercise.

In our calculations, we multiply the total number of moles of water by the heat of vaporization to find the total energy released through condensation. Understanding this principle helps to comprehend the massive energy exchange in weather phenomena like storms.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In the context of the rainstorm, gravitational potential energy obtained during evaporation is released in the form of thermal energy when vapor condenses into rain.
This principle is key in understanding weather events where large amounts of energy are transferred across different forms.

• The exercise illustrates a conversion between the heat of vaporization of water and a comparable amount of energy released by explosives like dynamite.
• By calculating the energy released by the rainstorm, we can demonstrate how natural processes equate to massive energy outputs that would otherwise require significant human-engineered sources.

Overall, this exercise helps us appreciate the scale of energy interactions in nature, revealing the power behind simple transformations like the water cycle.

Stoichiometry

Stoichiometry is a field of chemistry focused on the quantitative relationships between the reactants and products in chemical reactions. Although it may not seem apparent at first, stoichiometry is vital when calculating energy release in the exercise.
Here, stoichiometry helps to calculate:

• the number of moles of water from the total mass of rainwater using its molar mass.
• Subsequently, it allows the determination of energy released in the condensation process.

When dealing with energy calculations, stoichiometric relationships ensure every mole and molecule is accounted for, providing precise measures for energy release or absorption. Understanding these calculations supports more advanced thermodynamic concepts used in chemistry and meteorology.

Dimensional Analysis

Dimensional analysis is a technique used to convert units from one system to another, ensuring that calculations are consistent and precise. In our exercise, we use dimensional analysis to convert measurements into the International System of Units (SI), which provides a common framework for scientific calculations.

• Initially, the conversion of inches to meters and square miles to meters squared was done using appropriate conversion factors.
• This technique was crucial in calculating the correct volume and subsequent mass of the rainwater.

Without proper dimensional analysis, our calculations could lead to errors, especially in such large scales as those presented by the given storm. Mastering dimensional analysis is key to solving complex problems involving different units, particularly in fields such as physics and engineering.