Isentropic Compressibility Given Volumetric Coefficient Of Thermal Expansion And Cv Calculator

Isentropic Compressibility Formula:

\[ K_S = K_T - \frac{(\alpha^2) \cdot T}{\rho \cdot (C_v + [R])} \]

Isothermal Compressibility (K_T):

m²/N

Volumetric Coefficient of Thermal Expansion (α):

1/K

Temperature (T):

Density (ρ):

kg/m³

Molar Specific Heat Capacity at Constant Volume (C_v):

J/K·mol

Isentropic Compressibility (K_S):

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1. What Is Isentropic Compressibility?

Isentropic compressibility is the measure of the relative volume change of a fluid or solid as a response to a pressure change at constant entropy. It represents how much a substance compresses under pressure when no heat is exchanged with the surroundings.

2. How Does The Calculator Work?

The calculator uses the thermodynamic formula:

\[ K_S = K_T - \frac{(\alpha^2) \cdot T}{\rho \cdot (C_v + [R])} \]

Where:

\( K_S \) — Isentropic compressibility (m²/N)
\( K_T \) — Isothermal compressibility (m²/N)
\( \alpha \) — Volumetric coefficient of thermal expansion (1/K)
\( T \) — Temperature (K)
\( \rho \) — Density (kg/m³)
\( C_v \) — Molar specific heat capacity at constant volume (J/K·mol)
\( [R] \) — Universal gas constant (8.314 J/K·mol)

Explanation: This formula relates isentropic compressibility to isothermal compressibility through thermal expansion and heat capacity properties, accounting for temperature and density effects.

3. Importance Of Isentropic Compressibility

Details: Isentropic compressibility is crucial in thermodynamics and fluid dynamics for understanding sound propagation speed, shock waves, and compressible flow behavior in various engineering applications.

4. Using The Calculator

Tips: Enter all values in appropriate SI units. Ensure temperature is in Kelvin, density in kg/m³, and heat capacity in J/K·mol. All input values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between isentropic and isothermal compressibility?
A: Isentropic compressibility occurs at constant entropy (adiabatic process), while isothermal compressibility occurs at constant temperature.

Q2: Why is the universal gas constant included in the formula?
A: The gas constant appears because the formula relates molar properties and follows from thermodynamic relationships between different types of compressibility.

Q3: What are typical values for isentropic compressibility?
A: Values vary widely by material. For liquids, it's typically around 10⁻¹⁰ m²/N, while for gases it's much higher, around 10⁻⁵ m²/N at atmospheric pressure.

Q4: How does temperature affect isentropic compressibility?
A: Generally, compressibility increases with temperature as molecular motion increases and intermolecular forces decrease.

Q5: Can this calculator be used for both gases and liquids?
A: Yes, the formula applies to both phases, though the input values will differ significantly between gases and liquids.