an automobile tire contains a certain volume of air at 30 psig and 70 degree fahrenheit. The barometric pressure is 29.50 in.Hg. If due to running conditions the temperature of the air in the tire rises to 160 fahrenheit, what will be the gage pressure? assume that the air is an ideal gas and that the tire does not stretch.
37.5 psig
To solve this problem, we can use the ideal gas law, which states that:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 = initial pressure (psig)
V1 = initial volume (unknown)
T1 = initial temperature (Fahrenheit)
P2 = final pressure (unknown)
V2 = final volume (unknown)
T2 = final temperature (Fahrenheit)
First, we need to convert all temperatures to the absolute scale using the equation:
T(°F) + 460 = T(°R)
Given:
P1 = 30 psig
T1 = 70°F + 460 = 530°R
T2 = 160°F + 460 = 620°R
Let's solve this step-by-step:
Step 1: Calculate initial volume (V1)
Since the volume is constant, V1 and V2 are the same, so we can eliminate them from the equation:
P1 / T1 = P2 / T2
Step 2: Calculate the final pressure (P2)
Rearrange the equation to solve for P2:
P2 = P1 * (T2 / T1)
Step 3: Substitute the given values into the equation and solve:
P2 = 30 psig * (620°R / 530°R)
P2 = 35.85 psig (rounded to two decimal places)
Therefore, the gauge pressure when the temperature is 160°F is approximately 35.85 psig.
To find the gauge pressure inside the tire at 160 degrees Fahrenheit, we first need to convert the initial temperature of the air inside the tire (70 degrees Fahrenheit) to absolute temperature.
1. Convert 70 degrees Fahrenheit to Rankine:
Rankine = Fahrenheit + 459.67
Rankine = 70 + 459.67 = 529.67 degrees Rankine
Next, we need to convert the initial pressure from psig (pounds per square inch gauge) to absolute pressure.
2. Convert 30 psig to absolute pressure:
Absolute pressure = Gauge pressure + Atmospheric pressure
Atmospheric pressure is given as 29.50 in.Hg, which can be converted to psi.
- Convert inches of mercury (in.Hg) to pounds per square inch (psi):
1 in.Hg = 0.491154 psi (approximately)
So, 29.50 in.Hg = 29.50 * 0.491154 psi = 14.503 psi (approximately)
Absolute pressure = 30 psig + 14.503 psi = 44.503 psi
Now we have the initial absolute pressure and temperature. To find the gauge pressure at 160 degrees Fahrenheit, we can use the ideal gas law and the concept of absolute temperature.
3. Apply the ideal gas law:
P1 * V1 / T1 = P2 * V2 / T2
Where:
P1 = Initial absolute pressure
V1 = Initial volume of air in the tire (assumed constant)
T1 = Initial absolute temperature
P2 = Final absolute pressure (unknown - gauge pressure)
V2 = Final volume of air in the tire (assumed constant)
T2 = Final absolute temperature
Since the volume of air in the tire is assumed constant, we can simplify the equation to:
P1 / T1 = P2 / T2
Rearranging the equation and substituting the given values:
P2 = (P1 * T2) / T1
P1 = 44.503 psi (initial absolute pressure)
T1 = 529.67 degrees Rankine (initial absolute temperature)
T2 = 160 + 459.67 = 619.67 degrees Rankine (final absolute temperature)
P2 = (44.503 * 619.67) / 529.67
P2 ≈ 52.274 psi
Finally, to find the gauge pressure, subtract the atmospheric pressure from the absolute pressure:
Gauge pressure = P2 - Atmospheric pressure
Atmospheric pressure = 14.503 psi
Gauge pressure ≈ 52.274 psi - 14.503 psi
Gauge pressure ≈ 37.771 psi
Therefore, the gauge pressure inside the tire at 160 degrees Fahrenheit would be approximately 37.771 psi.