A car tire has the following dimensions outer diameter 21 in

A car tire has the following dimensions: outer diameter 21 inches, inner diameter 15 inches, width 6 inches.

a) Using geometry, determine the volume of the tire in cubic feet.

b) The tire has been inflated to 28 psig. This is done when the tire is cold, so the temperature is 45°F. Recalling that the molecular weight of air is 29, determine the mass of the air inside the tire in lb_m.

c) If the tire is driven for a while, it will heat up to 120°F. Due to the fact that materials expand when they are hotter, the volume of the tire also increases slightly, by 3%. Of course the mass cannot change since no new air was added. What is the new gauge tire pressure?

Solution

a) Volume of tire = pi (21^2 - 15^2) x 6

V = 4071.504 inch^3

inch = 0.0833 m

V = 4071.504 x 0.08333^3 m^3 = 2.36 ft^3

b) P = 28 psig = 28 x 6894.76 pa = 193053 Pa

V = 2.36 ft^3 = 0.0668 m^3

T = 45 deg F = 280.4 K

using PV = n RT

193053 x 0.0688 = n x 8.314 x 280.4

n = 5.53 mol

mass of gas = 5.53 x 29 = 160.4 g .........Ans

c) V = 1.03 x 0.0668 = 0.0687 m^3

T = 120 deg F = 322 K

n = 5.53

PV = n RT

P (0.0687) = 5.53 x 8.314 x 322

P = 215425.83 Pa

P (in psig) = 31.24 psig