A car tyre contains air at a gauge pressure of 2 atm and a temperature of 300 K . More air is pumped into the tyre, increasing its total mass by 20 % . After pumping and driving, the temperature of the air inside the tyre rises to 330 K . Assuming the volume of the tyre remains strictly constant and the atmospheric pressure is 1 atm , what is the final gauge pressure of the air in the tyre?

Step-by-step solution

The initial absolute pressure of the air in the tyre is P₁ = P_ gauge + P_ atm = 2 + 1 = 3 atm . From the ideal gas equation, PV = nRT , the absolute pressure is directly proportional to the product of the number of moles and the absolute temperature for a constant volume ( P nT ). Since the mass of the air increases by 20 % , the new number of moles is n₂ = 1.2 n₁ . The initial temperature is T₁ = 300 K and the final temperature is T₂ = 330 K . The final absolute pressure P₂ is given by: P₂ = P₁ ( n₂ n₁ ) ( T₂ T₁