A bicycle tire whose volume is 4.1 x 10-4 m3 has a temperature of 296 K and an absolute pressure of 4.69 x 105 Pa. A cyclist brings the pressure up to 6.75 x 105 Pa without changing the temperature or volume. How many moles of air must be pumped into the tire?
To solve this problem, we can use the ideal gas law equation, which states:
PV = nRT
Where:
P is the absolute pressure
V is the volume
n is the number of moles
R is the ideal gas constant (8.314 J/mol·K)
T is the temperature
We are given:
Initial pressure, P1 = 4.69 x 10^5 Pa
Final pressure, P2 = 6.75 x 10^5 Pa
Volume, V = 4.1 x 10^-4 m^3
Temperature, T = 296 K
Using the ideal gas law, we can rearrange the equation to solve for the number of moles (n):
n = PV / RT
First, we need to convert the pressure from pascals (Pa) to atmospheres (atm) since the ideal gas constant (R) is typically expressed in those units. 1 atm = 101325 Pa.
P1 = 4.69 x 10^5 Pa = 4.61 atm
P2 = 6.75 x 10^5 Pa = 6.66 atm
Now, we can plug in the values into the equation:
n = (P2 * V) / (R * T)
n = (6.66 atm * 4.1 x 10^-4 m^3) / (8.314 J/mol·K * 296 K)
Simplifying the equation:
n = 0.0187 mol
Therefore, you would need to pump approximately 0.0187 moles of air into the tire to increase the pressure from 4.69 x 10^5 Pa to 6.75 x 10^5 Pa, keeping the temperature and volume constant.