At the start of a trip, a driver adjusts the absolute pressure in her tires to be 2.84 ✕ 105 Pa when the outdoor temperature is 281 K. At the end of the trip she measures the pressure to be 2.95 ✕ 105 Pa. Ignoring the expansion of the tires, find the air temperature inside the tires at the end of the trip.
To solve this problem, we can use the ideal gas law equation:
PV = nRT
Where:
P = absolute pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature in Kelvin
In this case, we can assume that the volume and the number of moles of gas remain constant, since the problem states that we should ignore the expansion of the tires.
Therefore, the equation can be simplified to:
P₁/T₁ = P₂/T₂
Where:
P₁ = initial pressure
T₁ = initial temperature
P₂ = final pressure
T₂ = final temperature
Given:
P₁ = 2.84 × 10^5 Pa
T₁ = 281 K
P₂ = 2.95 × 10^5 Pa
T₂ = ?
Plugging in the values into the equation:
(2.84 × 10^5)/(281) = (2.95 × 10^5)/T₂
Cross multiplying:
(2.95 × 10^5) × (281) = (2.84 × 10^5) × T₂
Simplifying:
82795 × 10^5 = 269 440 × T₂
Dividing both sides by 269440:
T₂ = (82795 × 10^5) / 269440
T₂ ≈ 307 K
Therefore, the air temperature inside the tires at the end of the trip is approximately 307 K.
To find the air temperature inside the tires at the end of the trip, we can use the principles of ideal gas law. The ideal gas law states that the pressure of an ideal gas is directly proportional to its temperature and inversely proportional to its volume, given a constant amount of gas.
The formula for the ideal gas law is: PV = nRT
Where:
P = pressure of the gas
V = volume of the gas
n = number of moles of the gas
R = ideal gas constant
T = temperature of the gas
In this case, we can assume that the volume and number of moles of the gas remain constant. Therefore, we can rearrange the ideal gas law formula to solve for temperature (T):
T = PV / (nR)
Since the number of moles and the volume of the gas remain constant, we can simplify the equation to:
T1 = P1 / (nR)
T2 = P2 / (nR)
Where:
T1 = initial temperature of the gas
P1 = initial pressure of the gas
T2 = final temperature of the gas
P2 = final pressure of the gas
Now we can substitute the given values:
T1 = 281 K
P1 = 2.84 ✕ 10^5 Pa
P2 = 2.95 ✕ 10^5 Pa
We also need the value of the ideal gas constant, which is commonly denoted as "R". The value of the ideal gas constant depends on the units used for pressure, volume, and temperature. In this case, the pressure is given in Pascals (Pa), so we'll use the value of the ideal gas constant for Pascals:
R = 8.314 J/(mol⋅K)
Substituting the given values into the equation, we get:
T2 = (2.95 ✕ 10^5 Pa) / (n * 8.314 J/(mol⋅K))
Now we need to determine the number of moles of the gas. Since the volume and number of moles remain constant, the number of moles will also remain the same. Therefore, we can cancel out "n" from the equation:
T2 = (2.95 ✕ 10^5 Pa) / (8.314 J/(mol⋅K))
Calculating this expression will give us the final temperature (T2) in Kelvin.