A diatomic ideal gas is compressed adiabatically to \(\frac{1}{32}\) of its initial volume. In the initial temperature of the gas is \(T_{i}\) (in Kelvin) and the final temperature is \(a T_{i}\), the value of \(a\) is

The ideal gas law is a fundamental equation in the study of thermodynamics, describing the behavior of an ideal gas, which is a hypothetical gas composed of point particles that interact only through elastic collisions. It combines simple empirical laws such as Boyle's Law, Charles's Law, and Avogadro's Law into a single expression given by the formula:
\( PV = nRT \).
Here, \( P \) represents pressure, \( V \) is volume, \( n \) is the number of moles of the gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin. When examining thermodynamic processes, this equation is an essential starting point for understanding how variables such as pressure, volume, and temperature are interrelated in a gas.
For an adiabatic process, as mentioned in the exercise, the ideal gas law plays a critical role in correlating the initial and final states of temperature and volume. Because no heat is exchanged with the surroundings during an adiabatic process, the process does not follow the path dictated by the ideal gas law directly; instead, additional factors, such as the heat capacity ratio, come into play.

The heat capacity ratio, denoted by \( \gamma \) (Gamma), is a dimensionless parameter that describes the relationship between the specific heat at constant pressure \( C_p \) and the specific heat at constant volume \( C_v \) of a gas. It is expressed as
\( \gamma = \frac{C_p}{C_v} \).
The heat capacity ratio is particularly important in the analysis of adiabatic processes because it defines how the temperature and volume of an ideal gas change without the exchange of heat. For a diatomic gas, including gases like oxygen and nitrogen, which make up a significant portion of Earth's atmosphere, \( \gamma \) has a value of about 1.4. This value is constant for a given gas and affects the steepness of an adiabatic curve on a P-V diagram, which is steeper than an isothermal curve (a graph of a process occurring at constant temperature).
When an ideal gas undergoes an adiabatic compression or expansion, the product of the temperature and volume raised to the power \( \gamma - 1 \) remains constant (as shown in the step-by-step solution). This relationship is central to determining how the temperature varies when an ideal gas is compressed adiabatically, as seen in our textbook exercise.

In thermodynamics, processes describe how the properties of a system change and how these changes relate to energy transfer. There are several types of thermodynamic processes, each governed by specific laws and characteristics.

• Isobaric: Process at constant pressure.
• Isochoric: Process at constant volume, also known as isovolumetric.
• Isothermal: Process at constant temperature.
• Adiabatic: Process with no heat exchange with the surroundings.

In the context of our exercise, an adiabatic process is of particular interest because the gas is compressed without heat being added or removed from the system. While work is done on the gas to compress it, this energy does not translate into heat transfer but alters the internal energy, which in turn changes the gas's temperature. The ideal gas law and the heat capacity ratio come into play here to quantify these changes. Understanding the nuances of such thermodynamic processes helps us both predict and explain the natural behavior of gases under various conditions.