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The Ideal Gas Law is a fundamental equation in chemistry and physics that describes the behavior of an ideal gas. It's given by the equation \( PV = nRT \), where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume it occupies.
• \( n \) is the number of moles of the gas.
• \( R \) is the ideal gas constant (8.314 J/(mol·K)).
• \( T \) is the temperature in Kelvin.

This equation shows how pressure, volume, and temperature are interrelated for a given amount of gas. It's useful for solving various problems, such as calculating the number of moles in a gas at a particular temperature and pressure or finding the volume occupied by a gas when conditions change. In our basketball example, the Ideal Gas Law helped determine the number of moles of air in the ball, critical for later calculating the change in internal energy during compression.

Thermodynamics is the branch of physics concerned with heat, temperature, and their relation to energy and work. It encompasses principles that describe how energy is transferred and transformed. A key concept in thermodynamics is the adiabatic process, where no heat is transferred into or out of the system.
In the context of a bouncing basketball, when you compress it quickly, the process is considered adiabatic. During such a process:

• The internal energy of the system changes, but it does so without heat exchange with the environment.
• For an ideal gas, this results in a change in temperature and pressure directly linked to changes in volume.

Overall, understanding thermodynamics allows us to predict how temperature will increase when a gas, like the air in the basketball, is compressed quickly without allowing it to exchange heat with its surroundings.

The change in internal energy \( \Delta U \) of a system is a core concept in thermodynamics, especially involving ideal gases. It's calculated using the equation \( \Delta U = nC_v(T_2 - T_1) \), where:

• \( n \) is the number of moles of gas.
• \( C_v \) is the molar heat capacity at constant volume.
• \( T_2 \) and \( T_1 \) are the final and initial temperatures, respectively.

In this basketball scenario, the internal energy change is calculated based on the difference in temperature due to compression. Since the compression is adiabatic, the increase in pressure leads to a rise in temperature. By using the earlier derived number of moles and knowing \( C_v \) for nitrogen, which is typically \( \frac{5}{2}R \), the change in internal energy was determined. In our problem, this resulted in an increase in internal energy, indicating the system did work while energy was transformed within.