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Thermodynamics is a branch of physics that deals with the relationships and conversions between heat and other forms of energy. It plays a fundamental role in understanding how systems operate, including gases in an automobile tire. In this context, understanding how temperature and pressure of a gas change when conditions vary is essential.

Thermodynamics provides the underlying principles that explain the behavior of gases, particularly through various laws like the Ideal Gas Law. This law simplifies the complexities of thermodynamics by describing the state of an ideal gas and providing a foundation for solving real-world problems. It mainly concerns energy transfer and conversion, laying the groundwork for the study of pressure, volume, and temperature relationships in gases.

The pressure-temperature relationship of gases is crucial in thermodynamics. It outlines how changes in temperature can lead to changes in pressure, assuming constant volume. This relationship is part of Gay-Lussac's law, stating that the pressure of a fixed mass of gas is directly proportional to its absolute temperature, provided the volume remains constant.

• When temperature increases, the gas molecules move more vigorously, increasing the pressure inside a fixed volume, like a tire.
• Conversely, if temperature decreases, the pressure goes down as well.

This concept is mathematically represented through the Ideal Gas Law when it states that \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]In our problem, knowing the initial pressure and temperature allows us to calculate the final temperature, given a change in pressure. This understanding is fundamental in predicting how temperature impacts gas pressure, which was necessary to solve for the new temperature of the tire.

The Ideal Gas Law is one of the fundamental gas laws in thermodynamics, combining several individual laws into a single equation. It is expressed as \(PV = nRT\), where:

• \(P\) is the pressure of the gas.
• \(V\) is the volume.
• \(n\) is the number of moles.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.

While the full formula accounts for multiple variables, in many exercises, such as the one with the automobile tire, simplification occurs when certain quantities like volume and number of moles are constant.

In such cases, the simplified version used can be \[\frac{P_1}{T_1} = \frac{P_2}{T_2}\] This version shows the intrinsic link between the pressure and temperature of the gas when other factors don't change. Understanding this allows students not only to solve academic problems but also to see real-world applications, like why tire pressure increases on a hot day.

Temperature calculation in gases often revolves around changes when using the Ideal Gas Law. In this exercise, finding the final temperature after a change in pressure is crucial.

The formula above is rearranged to solve for the unknown temperature:\[T_2 = \frac{P_2 \times T_1}{P_1} \]Plugging the values from our problem:

• Initial pressure \( P_1 = 1.80 \text{ atm} \)
• Final pressure \( P_2 = 2.20 \text{ atm} \)
• Initial temperature \( T_1 = 300 \text{ K} \)

The calculation gives:\[T_2 = \frac{2.20 \times 300}{1.80} \approx 366.67 \text{ K}\]This result shows how the temperature inside the tire increased to approximately 366.67 K. It explains step by step, ensuring clarity, and connects academic understanding with practical scenarios like adjusting tire pressure on a hot day. Understanding each step and how numbers fit into the equation aids in mastering these thermodynamic calculations.