A gas undergoes an adiabatic compression during which its volume drops to half its original value. If the gas pressure increases by a factor of \(2.55,\) what's its specific-heat ratio \(\gamma ?\)

Understanding the specific-heat ratio, often denoted as \(\gamma\), is a key concept in the study of thermodynamics. It represents a property of a gas that is crucial in understanding how it responds to different processes, such as adiabatic compression.

The specific-heat ratio is the ratio of the specific heat at constant pressure (\(C_p\)) to the specific heat at constant volume (\(C_v\)). These specific heats are measures of the amount of heat required to raise the temperature of a substance by a given amount under constant pressure or volume respectively.

\(\gamma = \frac{C_p}{C_v}\)

For ideal gases, the specific-heat ratio is greater than one and varies depending on the gas. Monatomic gases like helium typically have a \(\gamma\) of about 1.67, diatomic gases such as nitrogen have about 1.4, and for air, which is a mixture, it is approximately 1.4 under normal conditions. The concept plays a fundamental role in several thermodynamic equations, especially ones concerning adiabatic processes, which is reflected in our textbook exercise.

Thermodynamics is a branch of physics concerned with heat, work, temperature, and their relation to energy, radiation, and physical properties of matter. One of the fascinating aspects of thermodynamics is its ability to predict the behavior of a system through its laws without knowing the detailed properties of the microscopic particles within the system.

The four laws of thermodynamics describe the principles of energy conservation, the increase in entropy, and the temperatures and pressures at which systems reach equilibrium. These laws provide the foundation for various equations used to solve thermodynamic problems, including the adiabatic process we encounter in the exercise.

With its central theme of energy transformation, thermodynamics plays an indispensable role in fields ranging from mechanical engineering to chemical reactions, and even in the understanding of stellar structures and atmospheric phenomena.

An adiabatic process is one where no heat is exchanged between a system and its surroundings. In other words, all the work done on the system or by the system takes place at the expense or benefit of internal energy, without any heat transfer. This kind of process is an idealization because, in reality, there's always some heat loss or gain. However, it perfectly exemplifies scenarios where the process happens so quickly that there is insufficient time for such heat exchange.

In an adiabatic compression, the work done on the gas increases its internal energy, leading to a rise in temperature, even though no heat is added to the system. The relationship between pressure and volume in an adiabatic process for an ideal gas is given by the equation:\, where \(P_1\) and \(V_1\) are the initial pressure and volume, \(P_2\) and \(V_2\) are the final pressure and volume, and \(\gamma\) is the specific-heat ratio.

The exercise provided illustrates this principle, showing how to determine \(\gamma\) by relating the change in volume and pressure during adiabatic compression. Understanding the adiabatic process is not only crucial for solving this exercise but also for a wide range of practical applications such as refrigeration, internal-combustion engines, and the dynamics of atmospheric systems.