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When considering the behavior of gases, the ideal gas law is an essential relation that allows us to understand and predict how a given amount of gas will respond to changes in temperature, pressure, and volume. The law is commonly formulated as the equation \( PV = nRT \), where \( P \) stands for pressure, \( V \) for volume, \( n \) for the number of moles of gas, \( R \) for the universal gas constant, and \( T \) for temperature in Kelvin.

This equation assumes that the gas behaves ideally, meaning its molecules do not interact and occupy no volume. Although no gas is truly ideal, many gases under normal conditions can be approximated as such, and this aids greatly in solving physics problems. For instance, in the task of determining the temperature change in a basketball during an adiabatic compression, we employ the ideal gas law to infer the initial state of the gas before the process takes place.

Internal energy is a concept within the field of thermodynamics which refers to the total energy contained by a system - in this case, the gas within the basketball. This includes kinetic energy related to the translation, rotation, and vibration of the molecules. For ideal gases, internal energy is simply related to the temperature and the number of moles of gas present.

In the context of an adiabatic process, there is no heat exchange with the environment, so the change in internal energy \( \Delta U \) is determined by the work done on or by the system. The change in internal energy for an ideal diatomic gas, such as nitrogen (\( \mathrm{N}_{2} \)), undergoing adiabatic compression or expansion is given by the equation \( \Delta U = nC_v\Delta T \), where \( C_v \) is the specific heat at constant volume, and \( \Delta T \) is the change in temperature.

The heat capacity ratio, also known as the adiabatic index and represented as \( c \) or \( \gamma \), is a dimensionless quantity important in the analysis of adiabatic processes and is defined as the ratio of the specific heat capacity at constant pressure \( C_p \) to that at constant volume \( C_v \), i.e., \( c = C_p / C_v \). For diatomic gases like nitrogen, the value of \( c \) is typically around 1.4.

This ratio appears in the formulation of the adiabatic process which expresses the relationship between pressure and volume during an adiabatic change. The product \( PV^c \) remains constant if the process is adiabatic. When a basketball is bounced, the compression of the air inside is nearly adiabatic and so we can use this ratio to determine how the temperature changes with respect to volume change without considering heat exchange with the surroundings.