An automobile tire has a maximum rating of 38.0 psi (gauge...

An automobile tire has a maximum rating of 38.0 psi (gauge pressure). The tire is inflated (while cold) to a volume of 11.8 $\mathrm{L}$ and a gauge pressure of 36.0 psi at a temperature of $12.0^{\circ} \mathrm{C}$ . On a hot day, the tire warms to $65.0^{\circ} \mathrm{C},$ and its volume expands to 12.2 L. Does the pressure in the tire exceed its maximum rating? (Note: The gauge pressure is the difference between the total pressure and atmospheric pressure. In this case, assume that atmospheric pressure is 14.7 psi.)

An automobile tire has a maximum rating of 38.0 psi (gauge pressure). The tire is inflated (while cold) to a volume of 11.8 $\mathrm{L}$ and a gauge pressure of 36.0 psi at a temperature of $12.0^{\circ} \mathrm{C}$ . On a hot day, the tire warms to $65.0^{\circ} \mathrm{C},$ and its volume expands to
12.2 L. Does the pressure in the tire exceed its maximum rating? (Note: The gauge pressure is the difference between the total pressure and atmospheric pressure. In this case, assume that atmospheric pressure is 14.7 psi.)

Key Concepts

Ideal Gas Law

The Ideal Gas Law (PV = nRT) relates a gas's pressure, volume, and temperature to the number of moles and the universal gas constant. This law provides the basis for understanding how a gas’s pressure will change when its temperature and volume are altered, assuming that the quantity of gas remains constant.

Gauge Pressure versus Absolute Pressure

Gauge pressure is defined as the difference between the total (absolute) pressure and the atmospheric pressure. This concept is crucial when evaluating systems like tires, where the rating is given in gauge pressure, but the gas law calculations require the absolute pressure for accurate analysis.

Temperature Conversion in Gas Calculations

In gas law calculations, temperature must be expressed in an absolute scale (Kelvin) to ensure that it is directly proportional to the thermal energy of the system. Converting from Celsius to Kelvin is an essential step to correctly apply the Ideal Gas Law.

Effect of Volume Change on Gas Pressure

The volume of a gas is inversely proportional to its pressure at constant temperature, according to Boyle’s Law, which is a special case of the Ideal Gas Law. When a gas is heated and the container’s volume increases, as in the case of an expanding tire, both temperature and volume changes must be considered together to determine the resulting pressure change.

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