Isothermal Compressibility Given Volumetric Coefficient of Thermal Expansion and Cv Calculator

Formula Used:

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

Isentropic Compressibility (K_S):

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

Isothermal Compressibility (K_T):

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1. What is Isothermal Compressibility?

Isothermal compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure change at constant temperature. It quantifies how much a material compresses under pressure when temperature is held constant.

2. How Does the Calculator Work?

The calculator uses the thermodynamic relationship:

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

Where:

\( K_T \) — Isothermal compressibility (m²/N)
\( K_S \) — Isentropic 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 isothermal compressibility to isentropic compressibility through thermal expansion properties and heat capacities.

3. Importance of Isothermal Compressibility Calculation

Details: Isothermal compressibility is crucial in understanding material behavior under pressure, designing pressure vessels, studying fluid dynamics, and in various engineering applications where pressure-volume relationships are important.

4. Using the Calculator

Tips: Enter all values in appropriate units. Ensure positive values for all inputs. The calculator provides results in square meters per newton (m²/N).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between isothermal and isentropic compressibility?
A: Isothermal compressibility measures volume change at constant temperature, while isentropic compressibility measures it at constant entropy (adiabatic and reversible process).

Q2: Why is the universal gas constant included in the formula?
A: The universal gas constant appears because the formula involves molar specific heat capacity, which relates to the gas constant through thermodynamic relationships.

Q3: What are typical values for isothermal compressibility?
A: Values vary widely by material. For liquids: 10⁻¹⁰ to 10⁻⁹ m²/N, for gases: 10⁻⁵ to 10⁻⁴ m²/N under standard conditions.

Q4: Can this calculator be used for both liquids and gases?
A: Yes, the formula applies to both liquids and gases, though the values and significance of terms may differ between phases.

Q5: How does temperature affect isothermal compressibility?
A: Generally, compressibility increases with temperature for liquids but decreases for ideal gases (inverse relationship with pressure).