A driver measures her "cold" tire gauge pressure to be 1.95 105 Pa. The measurement is made before driving, when the tires are at the temperature of the surroundings, 21.7° C. After driving several miles, she checks the gauge pressure again and finds that it has increased to 2.18 105 Pa. Find the final temperature of the air in the tires, assuming negligible change in volume.
To find the final temperature of the air in the tires, we can use the ideal gas law equation:
PV = nRT
Where:
P = pressure (in Pascal)
V = volume (which is assumed to be constant)
n = number of moles of gas
R = ideal gas constant
T = temperature (in Kelvin)
In this case, since the volume is assumed to be constant, we can rewrite the equation as:
P1/T1 = P2/T2
Where:
P1 = initial pressure
T1 = initial temperature
P2 = final pressure
T2 = final temperature
Let's plug in the given values:
P1 = 1.95 * 10^5 Pa
T1 = 21.7°C = 21.7 + 273.15 = 294.85 K (conversion to Kelvin)
P2 = 2.18 * 10^5 Pa
T2 = unknown (what we are trying to find)
Now we can solve for T2 using the equation:
P1/T1 = P2/T2
Rearranging the equation:
T2 = (P2 * T1) / P1
Substituting the values we have:
T2 = (2.18 * 10^5 * 294.85) / (1.95 * 10^5)
Calculating this expression will give us the final temperature of the air in the tires.