Question Need Help Please

jruiz3 · Junior Member Joined: 18 Sep 2006 Posts: 8

A bicycle tire is filled with air to a pressure of 100. psi at a temperature of 19 °C. Riding the bike on asphalt on a hot day increases the temperature of the tire to 54 °C. The volume of the tire increases by 4.1 %. What is the new pressure in the bicycle tire?

? psi

adrian · Posted: Fri Sep 29, 2006 7:45 am Post subject:

107.57 psi

Lawsen Lew · Member Joined: 11 Nov 2006 Posts: 15

It is not a very difficult problem as long as you obey the ideal gas laws. Tire inflation pressures are different in the winter than summers. Heavy equipment like cranes and lifting equipment on tires need to be aware of proper tire inflation pressures and regular tire condition checks to prevent equipment from toppling, a very dangerous situation. Pedaling efficiency is optimal with proper tire pressures.

The ideal gas law units must be used. That is liters for volume, atmosphere for pressure, and number of moles for the quantity of gases. This problem does not require the number of moles, because it would be complicated, because a tire is inflated with air. Air is a homogenous mixture of molecules, diatomic and other gases. You cannot see air, it is a gas. We will leave the number of moles of air in the tire out.

The ideal gas law will not work with other units, because gas research was based on mole, liters, atmospheres, and Kelvin.

The pressures have to be converted from psi to atm.
There is 14.6959 psi = 1 atm
I got this from my calculator, but one can look it up in a CRC or online somewhere.

P1=100 psi
100 psi (1 atm/14.6959 psi) = 6.8046 atm
P2 is unknown at this time and it will be given in atm

T1 = 19 C +273.15K = 292.15 K
T2 is after the tire was heated by friction on the pavement = 54 C + 273.15 K = 327.15 K

I am from North America, so I will spell L as liters, not litres. If you want, then write it as litres. The US and Burma, Myanmar are the few nations left on the planet that still uses the English units, feet, inch, yard, gallon, and acres. I can write and solve the best I can. I am not repsonsible, if you failed school or crashed your bicycle. These are your risks in life.

The volume will be in liters. This is a bit large for a bicycle tire. Liters are a requirement unit in the ideal gas laws. The problem wrote 4.1%, that is vague. I will have to assign an arbitrary number to solve this. I do not know who you are or your purpose in this problem. I will chose 100 liters, a nice simple number, even though a bicycle tire will not be able to hold that much air. It is for obeying the ideal gas law and mathematics than to be reality. Please, understand my intention. It increased by 4.1%. No, the tire is not 100 liters volume, but I have to assign a number and that is what makes this problem really tricky.

100 * 4.1%/100 + 100=104.1 L The tire increased in size, so I have to add 4.1 L with 100 L, Please understand, it is arbitrary, number for the math.

We could use a smaller number if this bothers you. I am going to use 100 to make the calculation easier. The reflex axiom of mathematics allows this. 100 = 100, a smaller number will do, too. 0.50 L of air will do, so, that will make it about 500 ml of air, 0.50=0.50.

0.50 * 4.1%/100 + 0.50 = 0.5205 L is V2
0.50 L is V1
The answer will still be the same as if I used 100 L for V1 and 104.1 L for V2, because the subsitution law and reflex axiom in mathematics.

Here is what we know, so far.
Volume initial, V1=100 L
Volume, V2 is 104.1 L
T1=292.15 K
T2=327.15 K
Pressure initial, P1=6.8046 atm
Pressure final, P2 is unknown

The ideal gas law is PV=nRT
R is the gas constant that requires us to use moles, liters, and atmospheres. We will not need the use the R constant or the n, number of moles. That will make the problem tricky.

R=0.08206 (L * atm)/(mol*K)

P1V1/T1=P2V2/T2

(6.8046 atm * 100 L)/292.15 K = (104.1 L * P2)/327.15K
2.32914 (atm L)/K = 0.318202 (L/K) P2

2.32914 (atm L)/K/0.318202 (L/K)= P2
7.3196 atm = P2

It is important to write the units down and make sure these cancel as you work it through.

The tire pressure when ridding this bicycle on the pavement with heat from friction and the sun, if it is day time is 7.3196 atm. The answer is best in psi, so it will match the tire pressure gauge or gage from Pep Boys, Kragens, NAPA, Northern Automotive Parts Association or bicycle shop.

7.3196 atm (14.6959 psi/1atm) = 107.57 psi. Most tire gages do not measure that many digits. It will be too costly to buy one. I will round the number to 108 psi, pounds per square inch.

This is a hard problem, because the volume is vaguely given. It depends upon the type of bicycle tires. A road or racing bicycle will have thin tires from Michelin tires and Mavic rims from France to reduce the coefficient of friction with the road, but risky for lost stability and you could fall or crash. A recumbent will have larger tires made in Asia. Off road bicycles will have knobby tires. A street bicycle, mountain bicycle or hybrid street-mountain bicycle will have larger width tires, thus larger volume of air, but a greater coefficient of friction. Wider tires mean more stability, safety, hydro planning in the wet pavement. Wider tires are less efficient and harder to peddle.
I assigned 100 liters with respecting the 4.1% increase in volume. The tire is elastic, stretch out to a limit. We could assign a more realistic number, 0.50 liters to be realistic and the answer would still be the same, because volume increase is 4.1% and reflex axiom in mathematics, and the ideal gas law with the kinetic molecular theory influencing gas behavior works the same independent when the volume of gas is changed in the same amount. Why not you try changing the volume to 0.50 liters and increase it by 4.1% on V2 side of the ideal gas equation?

I hope this will help you. I think I am not going to do anymore. I need to do my resumes.

I have decided to do this for you, even it might be too late. I have a good breakfast on my corner. I use an old T92 and I have newer HP-49G+ and a Lenovo notebook computer. I still use the old Macintosh G3 iBook 300 MHz. The new Mac Books have so many design flaws to be worked out. In a year, it will be a good computer, but I already have enough.

I have decided to prove that the volume does not influence this calculation. I do not know exactly why. It might be the kinetic energy theory. It is Charles Law that allows us to solve this problem. It was first explained by Jacques Charles in 1787. It cannot be Boyle's Law, because the temperature changed from 19 C (292.15 K) to 54 C (327.15 K).

Here are my calculations subsituting the 100 L of air with 0.50 L, more realistic of bicycle tire capacity.

0.50 L = V1
0.50 L * 4.1%/100 + 0.50 L = 0.5205 L

(6.8046 atm * 0.50 L)/292.15 K=(0.5205 L * P2)/327.15 K
1.15457299 * 10^-2 (atm * L)/K = 1.59101329 (L/K) * P2
1.16457299 * 10^-2 (atm * L)/K / 1.59101329 (L/K) =P2
7.31969 atm = P2

7.31969 atm is the exact same pressure with 104.1 L in the other calculation. This verifies Charles Law and kinetic energy theory of the ideal gas law.

7.3196 atm (14.6959 psi/1atm) = 107.57 psi. I will round the number to 108 psi, pounds per square inch.

There are automatic tire pressure monitoring computers and sensors. Some cars have this feature already. Large diesel rigs could have an aftermarket installed. Loads like large bulldozers and excavators on center of gravity trailers will be safer with a tire pressure monitor.

Surf this site:
http://www.pressureprosouth.com/files/Pressurepro/PPSproducts.html

adrian · Posted: Fri Nov 24, 2006 7:25 pm Post subject:

Well done!