You inflate your car tires to a gauge pressure of 2.43 atmospheres at a temperature of 24 degrees C. After driving a couple of miles the temperature of the tire increases by 46 degrees C. What is the new gauge pressure to the nearest hundredth of an atmosphere? What fraction of the air would you have to let out of the tire in order for the pressure to return to its original value?
The ratio of final to initial ABSOLUTE temperature is 319/297 = 1.0741
Absolute pressure increases by the same ratio, so the final absolute pressure is (2.43+1)*1.0741 = 3.663
The final gauge pressure is 2.663 atm. Remember that absolute pressure is always 1 atm more than gauge pressure.
PV = nRT
P/(nT) = constant
To return to original pressure P, n*T must remains the same, so the amount of gas (n) must be multiplied by a factor 1/1.0741 = 0.931
That would require releasing 6.9% of the gas in the tire
To find the new gauge pressure of the car tires after the increase in temperature, we need to use the ideal gas law, which states:
PV = nRT
Where:
P is the pressure
V is the volume
n is the number of moles
R is the ideal gas constant
T is the temperature
First, we need to convert the initial temperature to Kelvin:
Initial temperature = 24 degrees C + 273.15 = 297.15 K
Next, we calculate the initial number of moles of air in the tires. Since we are only interested in the change in pressure, we can assume that the volume remains constant. Therefore, the initial number of moles can be calculated using the initial gauge pressure:
P1 = 2.43 atm (gauge pressure)
Atmospheric pressure = 1 atm
Absolute pressure = Gauge pressure + Atmospheric pressure
P1 absolute = 2.43 atm (gauge pressure) + 1 atm (atmospheric pressure) = 3.43 atm
Now, we can solve for the initial number of moles:
P1V = nRT
n = (P1V) / (RT)
Since volume (V), ideal gas constant (R), and initial temperature (T) are constant, we can ignore them for now. We are only interested in the change in pressure, so:
n = P1 / R
Substituting the values:
n = 3.43 atm / (0.0821 L*atm/(mol*K))
Now we can calculate the final absolute pressure using the final temperature:
Final temperature = 24 degrees C + 46 degrees C = 70 degrees C = 343.15 K
Using the ideal gas law again:
P2V = nRT
P2 = (nRT2) / V
Since the volume remains constant:
P2 = n * T2
Substituting the values:
P2 = (3.43 atm / (0.0821 L*atm/(mol*K))) * 343.15 K
Now we need to convert the final absolute pressure to gauge pressure:
P2 gauge = P2 - atmospheric pressure
Finally, we calculate the fraction of air that needs to be released to return the pressure to its original value:
Fraction of air released = (P2 - P1) / P2
Now you can perform the calculations to find the new gauge pressure and the fraction of air that needs to be released.