1.48 Assume that the air volume in a small automobile tire is constant and equal to the volume between two concentric cylinders 13 cm high with diameters of 33 cm and 52 cm. The air in the tire is initially at 25 °C and 202 kPa. Immediately after air is pumped into the tire, the temperature is 30 °C and the pressure is 303 kPa. What mass of air was added to the tire? What would be the air pressure after the air has cooled to a temperature of 0 °C?

a) 0.018 kg

b) 262 kPa

Explanation:

The volume of the concentric cylinders would be:

V = π/4 * h * (D^2 - d^2)

V = π/4 * 13 * (52^2 - 33^2) = 16500 cm^3 = 0.0165 m^3

The state equation of gases:

p * V = m * R * T

Rearranging:

m = (p * V) / (R * T)

R is 287 J / (kg * K) for air

25 C = 298 K

m0 = 202000 * 0.0165 / (287 * 298) = 0.039 kg

After pumping more air the volume remains about the same, but temperature and pressure change.

30 C = 303 K

m1 = 303000 * 0.0165 / (287 * 303) = 0.057 kg

The mass that was added is

m1 - m0 = 0.057 - 0.039 = 0.018 kg

If that air is cooled to 0 C

0 C is 273 K

p = m * R * T / V

p = 0.057 * 278 * 273 / 0.0165 = 262000 Pa = 262 kPa