Early one cool \(\left(60.0^{\circ} \mathrm{F}\right)\) morning you start on a bike ride with the atmospheric pressure at \(14.7 \mathrm{lb}\) in. \(^{-2}\) and the tire gauge pressure at \(50.0 \mathrm{lb}\) in. \(^{-2}\). (Gauge pressure is the amount that the pressure exceeds atmospheric pressure.) By late afternoon, the air had warmed up considerably, and this plus the heat generated by tire friction sent the temperature inside the tire to \(104^{\circ} \mathrm{F}\). What will the tire gauge now read, assuming that the volume of the air in the tire and the atmospheric pressure have not changed?

When discussing the behavior of gases, the concept of pressure is fundamental. Gas pressure is defined as the force exerted by gas molecules as they collide with the surfaces of their container. This pressure is the result of the gas molecules moving and bumping into each other and the container walls. The faster they move, the more collisions occur and the higher the pressure.

In the context of our problem, the pressure inside the bicycle tire can be thought of as the cumulative push that the gas molecules exert on the tire's inner surface. The gauge pressure, which is the pressure read by the tire gauge, measures the pressure inside the tire above the atmospheric pressure, which is the pressure exerted by the air in our atmosphere. Understanding this difference is key for solving problems related to pressure changes in gases due to temperature variations or volume changes.

To apply the ideal gas law correctly, we must express temperature using an absolute scale: the Kelvin scale. This scale starts at absolute zero, which is the theoretical lowest temperature possible, where molecular motion virtually stops. Unlike Celsius or Fahrenheit, Kelvin does not use degrees, and thus we simply refer to it as kelvins.

To convert from Fahrenheit to Kelvin, as required in our example, we use the formula: \( K = (F - 32) \times \frac{5}{9} + 273.15 \). One must be careful to ensure that the subtraction and multiplication by the ratio \(\frac{5}{9}\) (which converts Fahrenheit to Celsius) are conducted before adding 273.15, converting Celsius to Kelvin. Proper temperature conversion is crucial, as the Kelvin scale allows us to utilize the ideal gas law, wherein the relationship between pressure, volume, and temperature determines the behavior of an ideal gas.

The Kelvin scale is indispensable in scientific work, especially regarding gas laws, as it provides a linear relationship between energy and temperature. In an absolute scale like Kelvin, zero reflects the point at which there is an absence of thermal energy - absolute zero. Unlike Celsius and Fahrenheit, Kelvin allows for the comparison and computation of thermal energy in a proportionate manor, critical for relating temperature changes to changes in gas pressure or volume.

The use of Kelvin is particularly important when applying the ideal gas law, which helps us discover unknown variables given the other conditions remain constant. For instance, in the bicycle tire scenario, we need to know the temperature in Kelvins to determine how much the gas pressure inside the tire increases with the rise in temperature caused by the day's heat and friction from use.