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?
To find out how many homes the internal energy of the compressed air could serve for one day, we need to calculate the total internal energy stored in the cavern and then divide it by the energy consumption of a single home.
First, let's calculate the internal energy U using the formula provided:
U = (5/2) nRT
Given:
Volume of the cavern (V) = 5.60 x 10^5 m^3
Pressure (P) = 8.40 x 10^6 Pa
R = Gas constant = 8.314 J/(mol·K)
Temperature (T) is not given, so we are assuming it to be constant.
We can use the ideal gas law to find the number of moles of air (n):
PV = nRT
Rearranging the equation, we get:
n = PV / RT
Substituting the given values:
n = (8.40 x 10^6 Pa) * (5.60 x 10^5 m^3) / (8.314 J/(mol·K) * T)
Now, we can substitute the value of n back into the equation for U to calculate the internal energy:
U = (5/2) * (8.314 J/(mol·K)) * T
To compare the internal energy to the energy consumption of a single home, we need to convert the internal energy from joules (J) to kilowatt-hours (kWh).
1 Joule (J) = 2.7778 x 10^-7 kilowatt-hours (kWh)
To find out how many homes the internal energy could serve for one day, divide the internal energy by the energy consumption of a single home:
Number of homes = U / (14.0 kWh)
Now we have all the necessary information to calculate the number of homes the internal energy of the compressed air could serve for one day. However, please note that without the given temperature (T), it is not possible to give a specific answer.
U=5/2PV
/1000
/3600
/14
=233333.33