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).

Short answer

The energy released when enough water vapor condenses to form 0.5 inches of rain over an area of 1 square mile is \(8.035 \times 10^{13} \mathrm{~kJ}\). To provide the same amount of energy, approximately \(1.913 \times 10^{7}\) tons of dynamite would be needed.

Step-by-step solution

Convert Units

First, we need to convert all relevant units to their SI equivalent. 0.50 inches of rain is equivalent to 0.0127 meters (1 inch = 2.54 cm = 0.0254 m). 1 square mile is equivalent to 2.58999 x 10^6 square meters (1 mile = 1,609.34 m).

Calculate Volume of Water

Calculate the volume of water represented by 0.50 inches of rain over 1 square mile. Volume = Area x Depth Volume = (2.58999 x 10^6 m^2)(0.0127 m) = 3.289 x 10^4 m^3

Convert Volume to Mass

Next, we convert the volume of water to mass. The density of water is 1000 kg/m^3. Mass = Volume x Density Mass = (3.289 x 10^4 m^3)(1000 kg/m^3) = 3.289 x 10^7 kg

Convert Mass to Moles

Now, we will convert the mass of water to moles. The molar mass of water is 18.015 g/mol or 0.018015 kg/mol. Moles = Mass / Molar Mass Moles = (3.289 x 10^7 kg) / (0.018015 kg/mol) = 1.826 x 10^9 mol

Calculate Energy Released

Now that we have the moles of water, we can calculate the energy released when the water vapor condenses. Energy = Moles x Heat of Vaporization Energy = (1.826 x 10^9 mol)(44.0 x 10^3 kJ/mol) = 8.035 x 10^13 kJ

Calculate Equivalent Tons of Dynamite

Finally, we calculate the number of tons of dynamite needed to provide the same amount of energy as the rainstorm. Tons of Dynamite = Energy of Rainstorm / Energy per Ton of Dynamite Tons of Dynamite = (8.035 x 10^13 kJ) / (4.2 x 10^6 kJ) = 1.913 x 10^7 tons So, approximately 19,130,000 tons of dynamite would be needed to provide the energy associated with 0.5 inches of rain over an area of 1 square mile.

Key concepts

Enthalpy of Phase Change

The enthalpy of phase change, also known as the latent heat, refers to the amount of energy per unit mass or mole required to change the state of a substance without changing its temperature. In the context of our exercise, the focus is on the heat of vaporization, which is the energy needed to convert a substance from a liquid into a vapor (or the energy released when vapor condenses back into a liquid).

This enthalpy change is an essential concept in chemistry when understanding how matter interacts with energy and undergoes phase transitions. The heat of vaporization is unique to each substance; for water, as stated in the exercise, it is 44.0 kJ/mol. This heat release during condensation is responsible for the significant energy associated with the weather phenomenon such as rainstorms.

Energy Calculations in Chemistry

Energy calculations in chemistry play a vital role in understanding reactions and physical processes. The calculation of energy involves understanding and manipulating various units of measurement, such as joules, kilojoules, or in our example, tons of dynamite. When we calculate the energy involved in the phase change, we use the formula:
\( \text{Energy} = \text{Moles} \times \text{Heat of Vaporization} \)
In practical terms, these calculations can relate the abstract concept of energy to something tangible, like the comparison between a rainstorm's energy and that of an explosive, illustrating the vast power of natural processes compared to human-made ones.

Molar Mass Conversion

Molar mass conversion is fundamental in chemistry and is used to convert between the mass of a substance and the number of moles. Since moles represent a certain number of molecules or atoms, converting to moles allows us to use Avogadro's number to relate physical mass to chemical proportions.

For our problem, we use the molar mass of water, which is approximately 18.015 g/mol, to convert the mass of the condensed water rain into moles. The formula used for this conversion is:
\( \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \)
Calculating the moles is a crucial step for us to then determine the energy released during the condensation phase change, bridging the gap between the mass of a substance and the associated chemical energy.