12-91 Consider an infinitesimal reversible adiabatic...

12-91 Consider an infinitesimal reversible adiabatic compression or expansion process. By taking $s=s(P, U)$ and using the Maxwell relations, show that for this process $P U^{k}=$ constant, where $k$ is the isentropic expansion exponent defined as $$ k=-\frac{U}{P}\left(\frac{\partial P}{\partial U}\right)_{s} $$ Also, show that the isentropic expansion exponent $k$ reduces to the specific heat ratio $c_{p} / c_{v}$ for an ideal gas.

12-91 Consider an infinitesimal reversible adiabatic compression or expansion process. By taking $s=s(P, U)$ and using the Maxwell relations, show that for this process $P U^{k}=$ constant, where $k$ is the isentropic expansion exponent defined as
$$
k=-\frac{U}{P}\left(\frac{\partial P}{\partial U}\right)_{s}
$$
Also, show that the isentropic expansion exponent $k$ reduces to the specific heat ratio $c_{p} / c_{v}$ for an ideal gas.

Key Concepts

Infinitesimal Process

An infinitesimal process considers extremely small changes in the state variables, allowing the use of differential calculus to obtain precise relationships between them. This approach is particularly useful in deriving how state variables vary during adiabatic processes, leading to expressions like PU^k being constant.

Isentropic Expansion Exponent

The isentropic expansion exponent is a parameter that describes the power-law relationship between pressure and internal energy during an isentropic process. Defined as k = - (U/P) (?P/?U)_s, it quantifies how a differential change in internal energy affects the pressure when the entropy is held constant, thereby characterizing the dynamics of the adiabatic process.

Specific Heat Ratio

The specific heat ratio, expressed as c_p/c_v, is the ratio of the heat capacity at constant pressure to that at constant volume for a substance, and it is especially significant in the study of ideal gases. In an ideal gas, this ratio directly relates to the thermodynamic behavior during adiabatic processes, and the isentropic expansion exponent is shown to reduce to this specific heat ratio, highlighting the deep connection between ideal gas behavior and isentropic transformations.

Maxwell Relations

Maxwell relations arise from the equality of mixed second derivatives of thermodynamic potentials and play a critical role in connecting different thermodynamic variables. By using these relationships, it is possible to convert derivatives with respect to one variable into derivatives with respect to another, which is essential in establishing connections between pressure, internal energy, and entropy in the derivation.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged between the system and its surroundings. In the context of reversible adiabatic or isentropic processes, the lack of heat transfer implies that the process occurs with constant entropy, which significantly simplifies the analysis of the relationships between thermodynamic variables.

Isentropic Process

An isentropic process is characterized by a constant entropy throughout the transformation. This property is key in deriving invariants, such as the constant product of a certain power of pressure and internal energy. It provides a natural framework for applying differential calculus to the state functions, leading to important thermodynamic relationships.

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