Like many people, I check my car tires about once a month, since correct pressure is necessary for good car handling, a smooth ride, and good gas mileage. When I checked my front tires recently with my Topeak digital gauge (image below), one was at the correct pressure (30 psi), but the other was higher, at about 33 psi. What puzzled me was that they had both been at 30 psi the previous time. I know tires can lose pressure, but I had never heard of a case where the pressure increased on its own.

This inexpensive digital-pressure gauge is a pleasure to use: It reads up to 160 psi/11 bar (useful for bike tires and suspensions) to three significant figures; can be switched among psi, Bar, or kg/cm² readings; and handles Presta and Schrader valves — a big improvement over the old “pencil-type” mechanical tire-pressure gauges.

I gave this some thought and saw only two possibilities at first: Someone was playing a practical joke on me (very unlikely); or my previous reading for just that one tire alone was in error (also unlikely, as all tires were measured twice, and at the same time).

Then I looked at the car and saw that the tire that read high was in full sunlight, while the other was in the shadow, and a black tire certainly does heat up from solar radiation. Mystery solved — or maybe not. I pulled out my custom-made “back of the envelope” pad and did a quick calculation using the ideal gas law:

…where P is pressure, V is volume, T is the absolute temperature, and K is a constant, which depends on the amount and type of gas (here, the value is irrelevant).

My hand-made “back of the envelope” pad reminds me that doing quick, rough estimates is often a good first step to understanding the parameters of a problem.

Thus if the pressure I measured was about 10% higher than the original, and the volume was constant, then the temperature of the air in the tire also should have gone up about 10%. The “cold” ambient temperature was about 77°F (25°C) or about 300K (remember, this is a rough estimate we're doing), so the delta rise would be 30K (30°C), or about 55°F.

Then I worried that perhaps a change in the tire's volume would affect my estimate, but I realized it was a non-issue for two reasons. First, a car tire is not an easily expandable balloon; it is a rubber enclosure restrained by steel-wire belts. Therefore, its volume stays fairly constant, especially for modest variations around a nominal value (this is a type of assumption we often use in many simplified models).

Second, even if the tire did expand slightly due to the increase in internal pressure, that would actually cause a decrease in the resultant pressure — again, the gas law. (I recall seeing a complex differential equation embodying the relationship among a tire's construction, pressure, and volume, for more advanced modeling.)

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