The world’s electrical demand is typically 1.4 x 10^13 W. If all the H2O(g)in the atmosphere suddenly
condensed to liquid, and all the enthalpy of this process was captured
and converted to electrical
energy, for how long would this energy power the world? Assume the atmosphere is 2% H2O(g)by volume.
To calculate the amount of electrical energy that can be generated from the condensation of water in the atmosphere, we need to consider the enthalpy change during the phase transition from water vapor to liquid water.
First, let's find the number of moles of water in the atmosphere. Since the atmosphere is 2% water vapor by volume, we can convert this to a molar fraction.
Assuming atmospheric pressure and temperature, we can use the ideal gas law to calculate the number of moles of water vapor (H2O(g)). The ideal gas law equation is:
PV = nRT
Where:
P is the pressure of the gas
V is the volume of the gas
n is the number of moles of the gas
R is the ideal gas constant
T is the temperature of the gas
Since we know the volume fraction of water vapor in the atmosphere is 2%, we can assume that 2% of the total volume is occupied by water vapor.
Let's assume a constant temperature of 300K and atmospheric pressure (~1 atmosphere). The value of the ideal gas constant (R) is approximately 0.0821 L.atm/(mol.K).
Since the question doesn't provide the specific volume of the atmosphere, we can assume it to be one cubic meter (1 m^3) for simplicity.
Using the ideal gas law, we can calculate the number of moles of water vapor:
n(H2O(g)) = (0.02 * 1) / (0.0821 * 300)
n(H2O(g)) ≈ 0.00081 moles
Next, we need to calculate the enthalpy change during the condensation of this amount of water. The enthalpy of vaporization (ΔHvap) of water is approximately 40.7 kJ/mol.
The total energy released during the condensation of this amount of water can be calculated using the equation:
Energy = ΔHvap * n(H2O(g))
Energy = 40.7 kJ/mol * 0.00081 moles
Energy ≈ 0.033 kJ
Now, we can calculate the time this energy would power the world by dividing the total electrical demand by the energy generated:
Time (seconds) = Energy (kJ) / Power (W)
Since we have the electrical demand in watts (1.4 x 10^13 W), we need to convert the energy from kJ to joules:
Energy (J) = Energy (kJ) * 1000
Time (seconds) = Energy (J) / Power (W)
Time (seconds) = (0.033 kJ * 1000) / (1.4 x 10^13 W)
Time (seconds) ≈ 2.36 x 10^-12 seconds
Therefore, if all the water in the atmosphere condensed to liquid and its enthalpy change was captured, it would only power the world for approximately 2.36 picoseconds.