A motorist filled his tires to a pressure of 30.0 pounds per square (psi), he that the pressure had built up to 34.2 psi. Estimate the temperature (in °F) of the air in the tires. Assume that the tire Gage reads relative pressure, that the volume of the tires remains constant, and that the ambient pressure is 1.00 atmosphere.
To estimate the temperature of the air in the tires, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure in atmospheres
V = volume in liters
n = number of moles
R = ideal gas constant (0.0821 L.atm/mol.K)
T = temperature in Kelvin
Since the volume and the number of moles remain constant, we can rewrite the equation as:
P1/T1 = P2/T2
Where:
P1 = initial pressure (30.0 psi + 1 atm = 2.07 atm)
T1 = initial temperature in Kelvin (let's assume 300K)
P2 = final pressure (34.2 psi + 1 atm = 2.23 atm)
T2 = final temperature in Kelvin
Now we can plug in the values and solve for T2:
(2.07 atm) / 300K = (2.23 atm) / T2
2.07T2 = 669
T2 = 323.4K
Finally, we convert the temperature to Fahrenheit:
T(°F) = T(K) * 9/5 - 459.67
T(°F) = 323.4 * 9/5 - 459.67
T(°F) = 584.12°F
Therefore, the estimated temperature of the air in the tires is approximately 584.12°F.