The average size of a house in Texas is 2000 square feet with a ceiling height o

Question

The average size of a house in Texas is 2000 square feet with a ceiling height of 8 ft. Assume that on a hot summer day, you turn on the A/C unit and cool down the house from 90 °F to 75 °F.

(a) What volume of air needs to come into the house from outside to maintain a constant pressure of 1 atm (assume air is an ideal gas)

(b) Given that the heat capacity of an ideal gas is 5/2 R, how much energy is required to cool down the air inside the house?

(c) Considering the cost of electricity in Texas is about 11 cents per (kW * hour), how much does it cost to cool down the air in the house?

Explanation / Answer

Volume of house = square feet x ceiling height

= 2000 ft2 x 8 ft

= 16000 ft3

Part a

Volume of air = volume of house = 16000 ft3

Part b

Volume V = 16000 ft3 x 28.317 L/ft3 = 453072 L

P = 1 atm

Gas constant R = 0.0821 L-atm/mol·K

Temperature T1 = 90°F

T1 in K = (90 + 459.67) x (5/9) = 305.37 K

Moles of air = PV/RT = 1 atm x 453072 L / (0.0821 L-atm/mol-K x 305.37 K)

= 18071.65 mol

T2 = 75°F = 297.1 K

Energy required = mol x heat capacity x (T1-T2)

= 18071.65 mol x 2.5 x 8.314 J/mol·K x (305.37 - 297.1)K

= 3106371.158 J

Part c

Cost of electricity = ( 11 cents / kWh) x (1kWh/3.6*10^6J)

= 3.055 x 10^-6 cents / J

Cost to cool down the air = Energy required x Cost of electricity

= 3106371.158 J x 3.055 x 10^-6 cents / J

= 9.5 cents

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