If in an adiabatic process, the pressure is increased by 2 /...

If in an adiabatic process, the pressure is increased by $2 / 3 \%$, then volume decreases by $\left(\right.$ Assume $\left.\frac{C_{P}}{C_{V}}=\frac{3}{2}\right)$ :
(a) $\frac{4}{9} \%$
(b) $\frac{2}{3} \%$
(c) $4 \%$
(d) $\frac{9}{4} \%$

Key Concepts

Differential Analysis of Small Changes

When analyzing small changes in pressure and volume during an adiabatic process, differential analysis is used to linearize the relationship between these variables. By applying logarithmic differentiation to Poisson's equation, one can derive an expression that relates the relative changes in pressure and volume. This method is essential for approximating the effects of small perturbations in state variables, making it easier to predict system behavior under near-equilibrium conditions.

Adiabatic Process

An adiabatic process is one in which no heat is transferred to or from the system, meaning that any change in the system's internal energy is solely due to work done. This concept is important in thermodynamics because it simplifies the analysis by removing heat exchanges from the energy balance equation, and it is commonly studied in both theoretical and applied settings such as engines and atmospheric phenomena.

Heat Capacity Ratio (Gamma)

The heat capacity ratio, commonly denoted by ? (gamma), is the ratio of the specific heat at constant pressure (C?) to the specific heat at constant volume (C?). This ratio is a crucial parameter in adiabatic processes because it governs the relationship between pressure and volume changes, as seen in Poisson's equation. Understanding gamma helps in determining how a system responds to changes in state variables during adiabatic compression or expansion.

Poisson's Equation for Adiabatic Process

Poisson's equation, expressed as P·V^? = constant, describes the behavior of an ideal gas undergoing an adiabatic process. It links the system's pressure and volume in a way that any change in one must be accompanied by a compensatory change in the other to maintain the constant value. This relation is vital for predicting the response of a gas when it is compressed or expanded without heat transfer.

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