Adiabatic Index of Real Gas given Heat Capacity at Constant Pressure Calculator

Adiabatic Index Formula:

\[ k = \frac{C_p}{C_p - \left( \frac{v \cdot T \cdot \alpha^2}{K_T} \right)} \]

Heat Capacity Constant Pressure (Cp):

J/kg·K

Specific Volume (v):

m³/kg

Temperature (T):

Coefficient of Thermal Expansion (α):

1/K

Isothermal Compressibility (K_T):

m²/N

Adiabatic Index (k):

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1. What is the Adiabatic Index?

The Adiabatic Index (k) is the ratio of the heat capacity at constant pressure (C_p) to heat capacity at constant volume (C_v). For real gases, it accounts for thermal expansion and compressibility effects that differ from ideal gas behavior.

2. How Does the Calculator Work?

The calculator uses the Adiabatic Index formula:

\[ k = \frac{C_p}{C_p - \left( \frac{v \cdot T \cdot \alpha^2}{K_T} \right)} \]

Where:

\( C_p \) — Heat capacity at constant pressure (J/kg·K)
\( v \) — Specific volume (m³/kg)
\( T \) — Temperature (K)
\( \alpha \) — Coefficient of thermal expansion (1/K)
\( K_T \) — Isothermal compressibility (m²/N)

Explanation: The formula accounts for the deviation of real gases from ideal gas behavior by incorporating thermal expansion and compressibility effects.

3. Importance of Adiabatic Index Calculation

Details: Accurate Adiabatic Index calculation is crucial for understanding thermodynamic processes in real gases, including compression/expansion work, sound speed propagation, and stability analysis in fluid dynamics.

4. Using the Calculator

Tips: Enter all values in appropriate SI units. Ensure temperature is in Kelvin, specific volume in m³/kg, and all other parameters in their respective units. All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between adiabatic index for ideal vs real gases?
A: For ideal gases, k = C_p/C_v is constant. For real gases, it varies with temperature and pressure due to intermolecular forces.

Q2: What are typical values of adiabatic index?
A: For monatomic gases, k ≈ 1.67; for diatomic gases, k ≈ 1.4; for polyatomic gases, k can range from 1.1 to 1.33 depending on temperature and pressure.

Q3: How does temperature affect the adiabatic index?
A: As temperature increases, the adiabatic index generally decreases due to increased molecular vibrations and rotational degrees of freedom.

Q4: What is the significance of isothermal compressibility?
A: Isothermal compressibility measures how much a substance's volume changes with pressure at constant temperature, indicating how "compressible" the substance is.

Q5: When is this calculation most important?
A: This calculation is particularly important in high-pressure applications, refrigeration systems, and when working with gases near their critical points.