Riddle: A storm rains an average of 0.8 inches over an area of 135 square miles. How much latent heat of condensation was produced by the storm in order to produce this mass of rainfall? Final answer will be in Joules. Water density = 1 g/cm^3.
Answer to Riddle: In order to know how much Energy is produced we need to determine the mass of the rain that fell. Density = mass / volume. We know the density of water and we can find the volume of the water. Once we find the volume of the water we can solve the density formula for mass.
Volume of rain will be the area the rain fell on times the depth of the rain. The depth of the rain is 0.8 inches. In order to get the final answer in Joules we need to convert to the metric system for the area the rain fell on. Square miles will need to be converted the square centimeters and the rain depth will need to be in centimeters.
The following conversion factors are need to convert square miles to square centimeters:
1 mile = 5,280 feet
12 inches = 1 foot
1 inch = 2.54 centimeters
The conversion may end up the most challenging aspect of finding the answer. Units must be cancelled properly and if units are squared then the number must be squared also.
First convert 135 square miles to square feet
135 mi^2 * (5,280^2 ft^2 / 1 mi^2) = 3,763,584,000 ft^2
Next convert ft^2 to in^2
3,763,584,000 ft^2 * (12^2 in^2 / 1 ft^2) = 5.41956096 * 10^11 in^2
Now convert in^2 to cm^2 to get it into metric units
5.41956096 * 10^11 in^2 * (2.54^2 cm^2 / 1 in^2) = 3.496483949 * 10^12 cm^2
Since we have the area the rain fell over in square centimeters we can multiply by the rain depth to get the volume of rain in cm^3.
0.8 in (2.54 cm / 1 in) = 2.032 cm
Volume of rain = 3.496483949 * 10^12 cm^2 * 2.032 cm = 7.104855384 * 10^12 cm^3
The density formula can now be used to determine the mass of rain
Density = mass / volume
1 g/cm^3 = mass / 7.104855384 * 10^12 cm^3
mass = 7.104855384 * 10^12 cm^3 * 1 g/cm^3 = 7.104855384 * 10^12 grams
The latent heat of condensation has a known value of 2.5 * 10^6 Joules/kg
convert mass to kg since latent heat has kg in units
(7.104855384 * 10^12 grams) * (1 kg / 1,000 g) = 7.104855384 * 10^9 kg
Total energy is equal to mass of rain times the latent heat of condensation
2.5 * 10^6 Joules/kg * 7.104855384 * 10^9 kg = 1.776213846 * 10^16 Joules
written out and rounded this is 17,700,000,000,000,000 Joules
This is 17.7 quadrillion Joules which is an amazing amount of energy just from the condensation produced from a rain complex.