A bicycle tire is inflated to a gauge pressure of 2.5 atm...

A bicycle tire is inflated to a gauge pressure of $2.5$ atm when the temperature is $15^{\circ} \mathrm{C}$. While a man is riding the bicycle, the temperature of the tire increases to $45^{\circ} \mathrm{C}$. Assuming the volume of the tire does not change, what is the gauge pressure in the tire at the higher temperature?

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Key Concepts

Gauge Pressure vs. Absolute Pressure

Gauge pressure is the pressure measured relative to the ambient atmospheric pressure, whereas absolute pressure is measured relative to a perfect vacuum. It is important to convert gauge pressure to absolute pressure when applying the ideal gas law, and subsequently convert back to gauge pressure if needed, ensuring accurate calculations in thermodynamic processes.

Temperature Conversion

Temperature conversion is essential because gas law calculations must be performed using an absolute temperature scale such as Kelvin. Converting from Celsius to Kelvin (by adding 273.15) ensures that the relationship between temperature and pressure is correctly applied.

Ideal Gas Law

The ideal gas law (PV = nRT) relates the pressure, volume, and temperature of a gas. In this context, it underpins the relationship between temperature and pressure when the volume and the number of moles remain constant, meaning that the pressure of the gas will change directly proportional to its absolute temperature.

Constant Volume Process

In a constant volume process, the pressure of a gas is directly proportional to its absolute temperature (as long as the number of gas molecules remains unchanged). This relationship is often expressed as P/T = constant, allowing for determination of how pressure changes with temperature.

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