Compressed air can be pumped underground into huge caverns as a form of energy storage. The volume of a cavern is 5.60 105 m3, and the pressure of the air in it is 8.40 106 Pa. Assume that air is a diatomic ideal gas whose internal energy U is given by U = 5/2 nRT. If one home uses 14.0 kWh of energy per day, how many homes could this internal energy serve for one day?
Well, PV=nRT, so U= 5/2PV.
That will be in Joules. To convert to kWh (kJ3600sec), divide by 1000, then divide by 3600. That will be the energy in kWh of stored energy.
To determine how many homes can be served by the internal energy stored in the cavern, we need to calculate the total internal energy stored in the compressed air and then divide it by the energy used by a single home in a day.
1. Calculate the total internal energy:
The internal energy (U) of an ideal gas is given by U = (5/2)nRT, where n is the number of moles, R is the gas constant, and T is the temperature in Kelvin.
Since we are given the volume and pressure of the air, we can use the ideal gas law to find the number of moles:
PV = nRT
n = PV / RT
Substitute the given values:
n = (8.40 * 10^6 Pa * 5.60 * 10^5 m^3) / (8.31 J/mol K * T)
We need to know the temperature to calculate the internal energy. If the temperature is not provided, we won't be able to calculate the total internal energy accurately.
2. Calculate the energy used by a single home:
The energy used per day by a single home is given as 14.0 kWh. Let's convert it to Joules:
1 kWh = 3.6 * 10^6 J
Energy used by a single home = 14.0 kWh * 3.6 * 10^6 J/kWh
3. Calculate the number of homes that can be served:
Divide the total internal energy by the energy used per home per day:
Number of homes served = Total internal energy / Energy used by a single home per day
Please provide the temperature (in Kelvin) to proceed with the calculations.