The pressure and density of a diatomic gas \((\gamma=7 / 5)\) change from \(\left(P_{1}, d_{1}\right)\) to \(\left(P_{2},\right.\), \(d_{2}\) ) adiabatically. If \(d_{2} / d_{1}=32\), then what is the value of \(P_{2} / P_{1}\) ? (a) 32 (b) 64 (c) 128 (d) 256

An adiabatic process is a thermodynamic change happening within a system where no heat is exchanged with the surroundings. This means any change in energy for a system undergoing an adiabatic process is due to work done by or on the system. For diatomic gases—those consisting of molecules with two atoms—the behavior during an adiabatic process is particularly interesting due to their energy absorption and release capabilities.

During such a process, the internal energy of a diatomic gas changes solely because of the work done, which affects the temperature, pressure, and volume of the gas. In mathematics, this is manifested through the pressure-density relationship and is described by the equation \( Pd^{-\gamma} = \text{constant} \) for a given quantity of gas. Here, \(\gamma\) is the heat capacity ratio, which is unique to the type of gas. For diatomic gases, such as oxygen or nitrogen, \(\gamma\) typically has a value of 7/5. By understanding how diatomic gases behave adiabatically, we can predict how parameters like pressure and volume will evolve without computing heat transfer, which is highly significant in areas such as aerodynamics and astrophysics.

The pressure-density relationship is a cornerstone in understanding adiabatic processes. It shows how the pressure of a gas changes in response to a change in density when no heat is transferred to or from the gas. This relationship can be formulated using the equation for an adiabatic process, \( Pd^{-\gamma} = \text{constant} \) where \( P \) is the pressure, \( d \) is the density, and \( \gamma \) is the heat capacity ratio. For an ideal diatomic gas going through an adiabatic change, if the density increases, the pressure will also increase.

The math behind this relationship allows us to solve problems involving changes in gas conditions without dealing with the complexities of heat exchange. This proves especially useful in practical applications like understanding how the atmosphere behaves when air rises and expands or compresses as it falls. By applying the formula correctly and considering the initial and final states of the gas, we can find the resulting pressures after adiabatic processes, which is fundamental in thermodynamics and fluid dynamics.

The heat capacity ratio, often denoted by \(\gamma\), is the ratio of the specific heat at constant pressure (\(C_p\)) to the specific heat at constant volume (\(C_v\)), that is, \(\gamma = C_p/C_v\). It is a crucial parameter in thermo-fluid dynamics as it influences the thermodynamic properties of a substance undergoing an adiabatic process.

For diatomic gases, the degree of freedom is more than in monatomic gases, leading to a \(\gamma\) value typically around 1.4 or 7/5. This specific ratio is essential when we deal with an adiabatic process because it determines how pressure and temperature will change when volume changes. A higher \(\gamma\) means the gas will react more dramatically to volume changes in terms of pressure and temperature. This property is utilized in a variety of engineering applications such as the design of nozzles and engines, as well as in understanding natural phenomena like sound speed propagation in the atmosphere.