How many moles of gas must be forced into a 4.8-L tire to give it a gauge pressure of \(32.4\) psi at \(25^{\circ} \mathrm{C}\) ? The gauge pressure is relative to atmospheric pressure. Assume that atmospheric pressure is \(14.7\) psi so that the total pressure in the tire is \(47.1 \mathrm{psi}\).

When we talk about the pressure of a gas inside a tire, we often refer to it as 'gauge pressure.' This is the pressure of the tire relative to the surrounding atmospheric pressure, not the total pressure inside the tire. Imagine gauge pressure as the extra push you give to the air already compressed by Earth's atmosphere.

In this example, a gauge pressure of 32.4 psi means the pressure exerted by the gas is 32.4 psi more than the atmospheric pressure. However, for calculations, we need to consider the total pressure which is gauge pressure plus atmospheric pressure. So if the atmospheric pressure is 14.7 psi, the total pressure in the tire would be the sum of the two, which in our case is 47.1 psi. The concept of gauge pressure becomes significant because it impacts how much air we need to add or release to achieve a certain level of firmness in the tire.

The Systeme Internationale (SI) is used as a standard measurement system globally, ensuring that scientists and engineers speak the same language in terms of calculations and data.

In the context of pressure, the SI unit is the Pascal (Pa). Converting from pounds per square inch (psi) to Pascals can be done using the conversion factor where 1 psi is approximately equivalent to 6894.76 Pa. So, to convert our total tire pressure from psi (47.1) to Pascals, we simply multiply it by 6894.76.

Similarly, volume in liters must be converted to cubic meters (m³), and temperature from Celsius to Kelvin, the SI unit for absolute temperature—which is essential for calculations involving gases. Remembering these conversions is vital for precision in science and engineering fields.

Understanding absolute temperature is key when working with the ideal gas law. The Kelvin scale, which measures absolute temperature, is essential because it begins at absolute zero—the point where all molecular motion stops.

In practical terms, you can think of absolute temperature as being 'Celsius plus 273.15.' This conversion factors in the difference between where the Celsius scale starts (the freezing point of water) and absolute zero. For our tire, a given temperature of 25°C is not just 25 in our formula; it has to be converted to Kelvin by adding 273.15, making it 298.15 K. Such conversions are vital in gas calculations to ensure that the temperature variable reflects the true kinetic energy of the gas particles.

Moles are a foundational concept in chemistry, representing a specific quantity of particles (atoms, molecules, etc.). Calculating moles is an essential skill, particularly when dealing with gases and the ideal gas law.

The ideal gas law relates pressure (P), volume (V), temperature (T), and the number of moles (n) of a gas to each other. The formula is represented as: \( PV = nRT \). With it, if you know the pressure, volume, and temperature, you can solve for 'n', the moles of gas.

In the tire example, after converting all the necessary measurements to SI units, you'd rearrange the formula to solve for n: \( n = \frac{PV}{RT} \). Plugging the pressure (in Pascals), volume (in cubic meters), and temperature (in Kelvin) into this equation, along with the universal gas constant 'R', provides the number of moles of gas needed to fill the tire to the desired pressure.