For a reversible adiabatic ideal gas expansion \(\frac{\mathrm{a} P}{P}\) is equal to : (a) \(\gamma \frac{d V}{V}\) (b) \(-\gamma \frac{d V}{V}\) (c) \(\left(\frac{\gamma}{\gamma-1}\right) \frac{d V}{V}\) (d) \(\frac{d V}{V}\)

Poisson's Law is critical to understanding adiabatic processes in physical chemistry, particularly those involving ideal gases. Named after French mathematician Siméon Denis Poisson, it describes how pressure (\(P\)) and volume (\(V\)) of an ideal gas change during an adiabatic process. This relationship is defined by the formula \( PV^\gamma = \text{constant} \), where \( \gamma \) represents the heat capacity ratio (\(C_p/C_v\)) of the gas.

The importance of Poisson's Law lies in its ability to help us predict the outcome of an adiabatic expansion or compression without any external heat exchange. Understanding this formula allows students to grasp why an adiabatic process leads to changes in a gas's temperature and pressure without the need for heat flow. It is particularly useful in thermodynamics problems, where calculating the resulting temperature and pressure of an expanding gas is necessary.

Reversible adiabatic expansion is a process where a gas expands without exchanging heat with its surroundings, and it is done in such a way that the process could be reversed by an infinitesimal change. Crucially, 'reversible' means that every intermediate state of the expansion is an equilibrium state, which implies that the process can be traced back exactly in the opposite direction.

To analyze such an expansion, we apply Poisson's Law, and through a series of mathematical steps, we can describe the relationship between the changes in pressure and volume. In the given exercise, after isolating \(dP/P\), we find that \(\frac{dP}{P} = -\gamma \frac{dV}{V}\), which corresponds to a reversible adiabatic expansion. This negative sign indicates that as the volume (\(V\)) increases, the pressure (\(P\)) decreases, which aligns with our physical intuition about gases expanding and contracting.

The heat capacity ratio, \( \gamma \), is a dimensionless number which plays a pivotal role in the thermodynamics of gases. It is the ratio of the heat capacity at constant pressure (\(C_p\)) to the heat capacity at constant volume (\(C_v\)). Since \(C_p\) is always greater than \(C_v\), the \(\gamma\) value is greater than one for an ideal monatomic gas (where it is approximately 1.67), diatomic gas (approximately 1.4), and polyatomic gases (with values closer to 1.3).

The concept of heat capacity ratio is central to understanding processes involving heat and work done by gases. It is linked to the gas's degree of freedom—translational, rotational, and vibrational—which dictates how energy is divided in the system. A higher value of \( \gamma \) indicates a gas with lower heat capacity in comparison to its ability to do work. This ratio also factors into speed of sound in a gas and adiabatic processes, where it impacts how rapidly conditions within the gas change when no heat is transferred.