A tire is filled with air at 17 degreees C to a gauge pressure of 220 kPa. If the tire reaches a temperature of 39 degrees C, what fraction of the original air must be removed if the original pressure of 220 kPa is to be maintained? Please help step by step!
The absolute temperature increases by a ratio factor (273 +39)/(273+17) = 312/290 = 1.076
To keep BOTH the pressure and the tire volume the same, the number of moles must decrease by a factor 1/1.076 = 0.9295
The fraction of air than must be removed is 1 - 0.9295 = 7.1%
To solve this problem, we'll need to use the ideal gas law, which states that:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles of gas
R = ideal gas constant
T = temperature
Step 1: Determine the initial volume of the tire.
Since the problem doesn't provide the volume, we'll assume it remains constant throughout the process. Therefore, to maintain the original pressure, the original volume needs to be maintained as well.
Step 2: Convert temperatures to Kelvin.
To use the ideal gas law, we need to express temperatures in Kelvin by adding 273.15 to the Celsius temperature.
Initial temperature (T1) = 17°C + 273.15 = 290.15 K
Final temperature (T2) = 39°C + 273.15 = 312.15 K
Step 3: Convert pressure units to Pascal.
Since the ideal gas law uses pressure in Pascal, we need to convert kPa to Pascal. The conversion factor is 1 kPa = 1000 Pa.
Initial pressure (P1) = 220 kPa × 1000 Pa/kPa = 220,000 Pa
Step 4: Apply the ideal gas law to calculate the number of moles of gas.
Since the volume remains constant, we can rewrite the ideal gas law as:
P/n = RT
To simplify calculations, we'll assume that the initial and final pressures are proportional to the number of moles. Therefore:
P1/n1 = P2/n2
Substituting the known values:
P1/n1 = P2/n2
220,000 Pa / n1 = 220,000 Pa / n2
Since the pressures are equal, we can solve for n2:
n2 = (n1 * P2) / P1
Step 5: Calculate the fraction of air that needs to be removed.
Since the initial pressure needs to be maintained, we can calculate the fraction of air that needs to be removed as:
Fraction removed (f) = (n1 - n2) / n1
Substituting the known values:
f = (n1 - (n1 * P2) / P1) / n1
Step 6: Calculate the final answer.
Substitute the known values into the equation and calculate the fraction of air that needs to be removed.
f = (n1 - (n1 * P2) / P1) / n1
f = (1 - (1 * 312.15 K) / 290.15 K) / 1
f = (1 - 1.076) / 1
f = -0.076
The fraction of air that needs to be removed is -0.076, which implies that no air needs to be removed.