Quote of the Day
Courage is what it takes to stand up and speak; courage is also what it takes to sit down and listen.
— Sir Winston Churchill
Introduction
I woke up this morning to a rather brisk temperature of 23 °F (31 °C). This is cold. At work, we had a short hallway discussion on the impact of temperature on tire pressure. I mentioned the following "rule of thumb" for tire pressure versus temperature (one of many sources):
Tire pressure decreases by 1 psi for each 10 °F decrease temperature.
This rule of thumb is easily derived and I thought I would put out a quick post on the subject. According to this rule, my tire pressure has dropped 6 psi since I filled my tires last week. The rule as stated assumes a nominal tire pressure of ~40 psi. You will see the rule changed to 2 psi for 10 °F of temperature change for a commercial truck tire, which are often inflated to 80 psi or more. As you will see below, the nominal tire pressure affects the amount of pressure variation with temperature.
Background
My derivation will be based on the ideal gas law. I will also make the following assumptions:
 The ideal gas law is valid for air under pressure.
The ideal gas law works well for lowpressure gases and gas mixtures, but it becomes less accurate as pressures increase. It turns out that at typical tire pressures, the error is less than 1% (see Appendix A).
 The volume of the tire changes insignificantly with pressure.
Tires are pretty stiff (e.g. they may have Kevlar or steel cords embedded in them). I know that severe underinflation is obvious because the tire material is sagging. However, tires near full inflation vary little in volume as pressure changes.
Analysis
General Result
Given the assumptions above, we can derive an expression that states that the percentage change of air pressure in a tire is inversely proportional to the tire air temperature (Equation 1).
Eq. 1 
where
 T is the absolute temperature of the tire air.
 p(T) is the tire air pressure as a function of temperature.
 is the percentage tire pressure change with temperature.
We can derive Equation 1 as shown in Figure 1.
Specific Example
Figure 2 shows how we can apply Equation 1 to a common tire pressure scenario to give us the rule of thumb. The tire pressure scenario is:
 Tire air temperature of 20 °C (68 °F).
 Sea level air pressure of 14. 7 psi.
 Nominal tire pressure of 40 psi (gauge pressure)
I should note that Mathcad automatically converts the temperatures to an absolute scale (e.g. Kelvin). So when you see an expression like "(20)°F", understand that this temperature is being converted into Kelvin.
We see in the example that the change in tire pressure with temperature is about 1 psi for every 10 °F change in temperature for a 40 psi tire pressure at 68 °F. In Figure 2, I look at two different air temperatures. You will notice that for a given tire pressure, the pressure change with temperature increases with lower reference temperature.
Figure 3 shows a graph of how the tire pressure (40 psi @ 68 °C) varies with temperature.
Conclusion
Straightforward application of the ideal gas law. Pretty simple post.
Appendix A: Compressibility Factor for Air
The compressibility factor for air is shown in Table 1 (Source). The compressibility factor is the ratio of the molar volume of a gas to the molar volume of an ideal gas at the same temperature and pressure. The compressibility factor can be used as a correction factor for the ideal gas law when applied to real gases.
As you can see from the table, the compressibility factor deviates from 1 by a fraction of a one percent even at 60 psi (~5 bar absolute). This means that using the ideal gas law will result in an error of less than 1%.
Pressure, 

Temp, 
1 
5 
10 
20 
40 
60 
80 
75 
0.0052 
0.0260 
0.0519 
0.1036 
0.2063 
0.3082 
0.4094 
80 
0.0250 
0.0499 
0.0995 
0.1981 
0.2958 
0.3927 

90 
0.9764 
0.0236 
0.0453 
0.0940 
0.1866 
0.2781 
0.3686 
100 
0.9797 
0.8872 
0.0453 
0.0900 
0.1782 
0.2635 
0.3498 
120 
0.9880 
0.9373 
0.8860 
0.6730 
0.1778 
0.2557 
0.3371 
140 
0.9927 
0.9614 
0.9205 
0.8297 
0.5856 
0.3313 
0.3737 
160 
0.9951 
0.9748 
0.9489 
0.8954 
0.7803 
0.6603 
0.5696 
180 
0.9967 
0.9832 
0.9660 
0.9314 
0.8625 
0.7977 
0.7432 
200 
0.9978 
0.9886 
0.9767 
0.9539 
0.9100 
0.8701 
0.8374 
250 
0.9992 
0.9957 
0.9911 
0.9822 
0.9671 
0.9549 
0.9463 
300 
0.9999 
0.9987 
0.9974 
0.9950 
0.9917 
0.9901 
0.9903 
350 
1.0000 
1.0002 
1.0004 
1.0014 
1.0038 
1.0075 
1.0121 
400 
1.0002 
1.0012 
1.0025 
1.0046 
1.0100 
1.0159 
1.0229 
450 
1.0003 
1.0016 
1.0034 
1.0063 
1.0133 
1.0210 
1.0287 
500 
1.0003 
1.0020 
1.0034 
1.0074 
1.0151 
1.0234 
1.0323 
600 
1.0004 
1.0022 
1.0039 
1.0081 
1.0164 
1.0253 
1.0340 
800 
1.0004 
1.0020 
1.0038 
1.0077 
1.0157 
1.0240 
1.0321 
1000 
1.0004 
1.0018 
1.0037 
1.0068 
1.0142 
1.0215 
1.0290 
I made a spreadsheet to play with the temperature and pressure, and it calculates it asuming no volumechange when other pressure , wich it will be for an already presurised tire.
the website I filled in is my public map of skydrive that belongs to my hotmail adress, with lots of tirepressure maps and files. In the motorhome tire pressure calculator map is the pressurecalculation with temp spreadsheet wich you first have to download to your computer by RIGHTCLICKING and then choose DOWNLOAD. Other means go wrong like leftclicking.
After download open in Excell or Open office Calc on your computer .
Would like further contact with you about the pressure calculation. can reach me at my hotmail.com adress with the username jadatis ( to prefent spamm like this) and I can sent you the copy of page 14 of standards manual of ETRTO in wich formula to calculate pressure for a sertain load, wich all the tyre makers use .
Nice work.
Mathscinotes
have you tried nitrogen filled tires on your vehicles yet? look for the green valve stem caps on vehicles.
I have heard people mention nitrogen for tires, but I have never seen it. I will ask some of our local motorheads about it.
mathscinotes
Dry atmospheric air already contains 78% nitrogen so you would only gain another 22% if you inflated your tires with pure nitrogen, a very expensive way to inflate standard tires with no noticeable benefit.
Professional race cars use it in tires specifically made for racing and are typically inflated to somewhere around 8psi...a horse of a different color from regular tires.
Nitrogen filling does not influence the pressure/temperature relation.
for every gas or combination of gasses that dont get liquid in the temperature range the tire is used expands the same with temperature chanching.
So exeption to that can be water that can be liquid and gas shaped in the tire in the temperatures that can exist . Pure nitrogen contains no water by the proces of making it.
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could you pls clarify the steps after taking natural log of all sides in ideal gas law