Question

Imagine a car tire that contains 5.1 moles of air when at a gauge pressure of 2.1×10^5N/m2 (the pressure above atmospheric pressure) and a temperature of 27 degrees C. The temperature increases to 37 degrees C, the volume decreases to 0.8 times the original volume, and the gauge pressure decreases to 1.6×10^5N/m2.

How many moles of air are left in the tire?

Answer 1

To solve this problem it is necessary to use the concepts given through the ideal gas equation.

For this it is defined that

`PV = nRT`

Where,

P = Pressure

V = Volume

R = Gas ideal constant

T = Temperature

n = number of moles.

In this problem we have two states in which the previous equation can be applied, so

`1) P_1V_1 = n_1RT_1`

`2) P_2V_2 = n_2RT_2`

From the first state we can calculate the Volume

`V_1 = (n_1RT_1)/(P_1)`

Replacing

`V_1 = (5.1*8.314*300.15)/(3.1*10^5)`

`V_1 = 0.041m^3`

From the state two we can calculate now the number of the moles, considering that there is a change of 0.8 from Volume 1, then

`n_2 = (P_2(0.8*V_2))/(RT_2)`

`n_2 = (2.6*10^5(0.8*0.041))/(8.314*310.15)`

`n_2 = 3.3moles`

Therefore there are 3.3moles of air left in the tire.