A small room with dimensions 15 ft X 15 ft X 10 ft high contains air at 68 deg.F

Question

A small room with dimensions 15 ft X 15 ft X 10 ft high contains air at 68 deg.F and 14.7 psia. It is the camera stage of a television broadcasting studio, and contains many bright lights for illumination. The room is closed, sealed, and insulated to isolate the performers from outside distractions. Find the temperature in the room 0.5 hour after six 800-w lights are turned on. Assume air is an ideal gas, and there is no ventilation or air conditioning and ignore the effect of people in the room (1 watt = 3.412 btu/hr) can you show all work thanks

Explanation / Answer

14.7 psia = 101353 Pa,

68 deg F = 293 K

15 ft*15ft*10 ft = 2250 ft3 = 63.713 m3

pV = mRT where R = gas constant for air = 287 J/kg-K

101353*63.713 = m*287*293

m = 76.8 kg

Power input = 6*800*3.412 = 16377.6 Btu/hr

Energy input Q = 16377.6*0.5 = 8188.8 Btu = 8639642 J

Q - W = U2 - U1

W = 0 since closed volume.

So, U2 - U1 = 8639642

But U2 - U1 = m*Cv*(T2 - T1) where cv = sp. heat at constant volume for air = 718 J/kg-K

So, 76.8*718*(T2 - 293) = 8639642

T2 = 449.7 K = 349.8 deg F

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