Lake Erie contains about 4.8 x 10^17 grams of fresh water. Assume that the lake has frozen solid and that the ice starts off at -3 C. How much energy would the lake need to absorb from the Sun to cause it to evaporate completely?
Each gram of water will require 1.5 + 80 + 100 + 540 = 721.4 calories to heat up, melt and evaporate. That equals 3109 joules. Multiply that by the mass of water in the lake.
Make that 721.5 calories. (typing error)
To calculate the energy required for Lake Erie to completely evaporate, we need to consider the heat needed to raise the temperature of the ice to its melting point, then melt the ice into water, and finally heat the water to its boiling point and convert it into vapor.
Let's go step by step:
1. Heating the ice to its melting point:
To raise the temperature of ice from -3°C to its melting point of 0°C, we need to calculate the heat using the specific heat capacity of ice. The specific heat capacity of ice is 2.09 J/g°C.
The formula to calculate the heat needed is:
Energy = mass × specific heat capacity × temperature difference
Given:
Mass of ice = 4.8 x 10^17 grams
Specific heat capacity of ice = 2.09 J/g°C
Temperature difference = 0 - (-3) = 3°C
Energy required to heat the ice to its melting point:
Energy = (4.8 x 10^17 grams) × (2.09 J/g°C) × (3°C)
2. Melting the ice into water:
To melt the ice at its melting point of 0°C into water, we need to calculate the energy using the heat of fusion for ice. The heat of fusion for ice is 333.55 J/g.
The formula to calculate the heat needed is:
Energy = mass × heat of fusion
Given:
Mass of ice = 4.8 x 10^17 grams
Heat of fusion for ice = 333.55 J/g
Energy required to melt the ice into water:
Energy = (4.8 x 10^17 grams) × (333.55 J/g)
3. Heating the water from its melting point to its boiling point:
To raise the temperature of the water from its melting point of 0°C to its boiling point of 100°C, we need to calculate the heat using the specific heat capacity of water. The specific heat capacity of water is 4.18 J/g°C.
The formula to calculate the heat needed is:
Energy = mass × specific heat capacity × temperature difference
Given:
Mass of water = mass of ice
Specific heat capacity of water = 4.18 J/g°C
Temperature difference = 100°C - 0°C = 100°C
Energy required to heat the water to its boiling point:
Energy = (4.8 x 10^17 grams) × (4.18 J/g°C) × (100°C)
4. Vaporization of water to convert it into vapor:
To convert the water at its boiling point of 100°C into vapor, we need to calculate the energy using the heat of vaporization for water. The heat of vaporization for water is 2260 J/g.
The formula to calculate the heat needed is:
Energy = mass × heat of vaporization
Given:
Mass of water = mass of ice
Heat of vaporization for water = 2260 J/g
Energy required for water to evaporate completely:
Energy = (4.8 x 10^17 grams) × (2260 J/g)
To find the total energy required, we add up the energies calculated in each step:
Total energy = Energy (heating ice) + Energy (melting ice) + Energy (heating water) + Energy (vaporization of water)
Total energy = [ (4.8 x 10^17 grams) × (2.09 J/g°C) × (3°C) ] + [ (4.8 x 10^17 grams) × (333.55 J/g) ] + [ (4.8 x 10^17 grams) × (4.18 J/g°C) × (100°C) ] + [ (4.8 x 10^17 grams) × (2260 J/g) ]
By performing the calculations, you can find the total energy required for Lake Erie to completely evaporate.