Question

Compressed air can be pumped underground into huge caverns as a form of energy storage. The volume of a cavern is 4.40 105 m3, and the pressure of the air in it is 6.20 106 Pa. Assume that air is a diatomic ideal gas whose internal energy U is given by U = 5/2 nRT. If one home uses 22.0 kW · h of energy per day, how many homes could this internal energy serve for one day?

Answer 1

Answer:

86112 homes

Solution:

As per the question:

Volume, V = $4.40\times 10^{5}\ m^{3}$

Pressure, P = $6.20\times 10^{6}\ Pa$

Internal energy, U = $\frac{5}{2}nRT$

Energy usage of one home = 22.0 kWh

Now,

We know from the ideal gas equation:

PV = nRT

Thus we can write:

$U = \frac{5}{2}nRT = \frac{5}{2}PV$

$U = \frac{5}{2}PV = \frac{5}{2}\times 6.20\times 10^{6}\times 4.40\times 10^{5} = 6.82\times 10^{12} J$

Energy usage of one home = $22\times 1000\times 3600 = 7.92\times 10^{7} J$

Now,

No. of homes that could serve this internal energy for a day is given by:

$n = \frac{U}{Energy\ usage\ of\ one\ home}$

$n = \frac{6.82\times 10^{12}}{7.92\times 10^{7}} = 86112\ homes$