Question

A workshop with well-insulated walls and containing 450 m³ of air at 305 K is heated at constant pressure (atmospheric). Consider air to be an ideal diatomic gas.

(a)Determine the energy (in kJ) required to increase the temperature of the air in the building by 2.90°C.

How does the amount of heat required depend on the molar specific heat at constant pressure and the desired change in temperature? What is the value for the molar specific heat at constant pressure? How can you determine the number of moles of the gas? kJ

(b)Determine the mass (in kg) this amount of energy could lift through a height 3.10 m.

Answer 1

Ideal gas law gives : PV = nRT,

where,

P = Atmospheric ressure = 1 atm ~ 10⁵ Pa,

(a) V = Initial volume = 450 m³,

n = Number of moles of air in the building,

R = Unoversal gas constant = 8.314 J / ( mol . K ),

T = Initial absolute temperature of air in the building = 305 K.

Hence, n = PV / RT = ( 10⁵ x 450 ) / ( 8.314 x 305 )

or, n = 1.77 x 10⁴.

Specific heat at constant pressure for a diatomic ideal gas is : C_P = 7R / 2.

Change intemperature = T = 2.9^o C = 2.9 K.

Hence, the energy required to increase the temperature of the air in the building by 2.90^o C is :

E = nC_PT = 1.77 x 10⁴ x ( 7 x 8.314 / 2 ) x 2.9 J

or, E = 1.4965 x 10⁶ J

Hence, energy required = 1496.5 kJ.

(b) Let the mass be m, in kg.

g = 9.8 m / s² be the gravitational acceleration.

Height raised = h = 3.1 m.

Hence, energy required = mgh = E

or, m = E / gh = ( 1.4965 x 10⁶ ) / ( 9.8 x 3.1 )

or, m = 4.93 x 10⁴ kg.