A cold tire has a volume of \(0.03 \mathrm{~m}^{3}\) at \(-10^{\circ} \mathrm{C}\) and \(180 \mathrm{kPa}\) gauge. If the pressure and temperature increase to \(210 \mathrm{kPa}\) gauge and \(30^{\circ} \mathrm{C}\), find the final volume in \(\mathrm{m}^{3}\). a) \(0.0304\) c) \(0.0312\) b) \(0.0308\) d) \(0.0316\)

The study of gas laws is fundamental in engineering because it helps predict the behavior of gases under various temperature and pressure conditions. The combined gas law is particularly useful as it incorporates three of the basic gas laws: Boyle's Law (constant temperature), Charles's Law (constant pressure), and Gay-Lussac's Law (constant volume). This law is expressed by the formula: \[\begin{equation} P_1 V_1 / T_1 = P_2 V_2 / T_2 \text{,} \end{equation}\] where P represents pressure, V is volume, T is the temperature in Kelvin, and subscripts 1 and 2 refer to the initial and final states respectively.
In practical applications, engineers often deal with changes in the state of a gas due to processes such as heating, cooling, compression, and expansion in systems like HVAC, automotive engines, and aerodynamics. Understanding and applying the combined gas law allows for precise calculations of these changes, ensuring safety, efficiency, and functionality in the design and operation of equipment.

Temperature plays a crucial role in calculations involving gas laws. Engineers and scientists use the Kelvin scale for these types of calculations because it is an absolute temperature scale with its zero point at absolute zero, where all thermal motion ceases.
To convert Celsius to Kelvin, which is a critical step in gas law problems, you add 273.15 to the Celsius temperature. The equation for this conversion is simply: \[\begin{equation} T(K) = T(°C) + 273.15 \text{.} \end{equation}\] For example, to convert -10°C to Kelvin: \[\begin{equation} 263.15 K = -10°C + 273.15 \text{.} \end{equation}\] And for 30°C to Kelvin: \[\begin{equation} 303.15 K = 30°C + 273.15 \text{.} \end{equation}\] Precise temperature conversion is essential in ensuring the accuracy of gas volume and pressure calculations.

In gas law problems, it's crucial to differentiate between gauge pressure and absolute pressure. Gauge pressure measures pressure relative to the ambient atmospheric pressure, whereas absolute pressure includes the atmospheric pressure in its measurement.
The formula to convert gauge pressure to absolute pressure is: \[\begin{equation} P_{abs} = P_{gauge} + P_{atm} \text{,} \end{equation}\] where \[\begin{equation} P_{atm} \text{,} \end{equation}\] the atmospheric pressure, is approximately 101.325 kPa at sea level. For instance, a gauge pressure of 180 kPa equates to an absolute pressure of 281.325 kPa when you add the atmospheric pressure: \[\begin{equation} P_{abs} = 180 kPa + 101.325 kPa \text{.} \end{equation}\] Understanding this concept is essential for actual pressure measurements in engineering tasks.