Molar Heat Capacity at Constant Volume given Volumetric Coefficient of Thermal Expansion Calculator

Formula Used:

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

Volumetric Coefficient of Thermal Expansion (α):

1/K

Temperature (T):

Isothermal Compressibility (K_T):

m²/N

Isentropic Compressibility (K_S):

m²/N

Density (ρ):

kg/m³

Molar Specific Heat Capacity at Constant Volume (C_v):

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1. What is Molar Specific Heat Capacity at Constant Volume?

Molar Specific Heat Capacity at Constant Volume (C_v) is the amount of heat required to raise the temperature of 1 mole of a substance by 1 degree Celsius at constant volume. It is a fundamental thermodynamic property that characterizes how a substance responds to heat input when its volume is fixed.

2. How Does the Calculator Work?

The calculator uses the thermodynamic relation:

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

Where:

\( \alpha \) — Volumetric coefficient of thermal expansion (1/K)
\( T \) — Temperature (K)
\( K_T \) — Isothermal compressibility (m²/N)
\( K_S \) — Isentropic compressibility (m²/N)
\( \rho \) — Density (kg/m³)
\( R \) — Universal gas constant (8.314 J/K·mol)

Explanation: This formula relates the molar heat capacity at constant volume to measurable thermodynamic properties through fundamental thermodynamic relationships.

3. Importance of C_v Calculation

Details: Accurate calculation of C_v is crucial for understanding thermal properties of materials, designing thermal systems, predicting phase transitions, and studying thermodynamic processes at constant volume.

4. Using the Calculator

Tips: Enter all values in appropriate units. Ensure that isothermal compressibility is greater than isentropic compressibility (K_T > K_S) to avoid division by zero. All input values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What is the difference between C_v and C_p?
A: C_v is heat capacity at constant volume, while C_p is at constant pressure. For ideal gases, C_p = C_v + R.

Q2: Why is C_v important in thermodynamics?
A: C_v helps determine internal energy changes and is essential for analyzing constant-volume processes in thermodynamic systems.

Q3: What are typical values of C_v for common substances?
A: For monatomic ideal gases, C_v = 3/2R ≈ 12.47 J/K·mol. For diatomic gases, C_v = 5/2R ≈ 20.79 J/K·mol at room temperature.

Q4: How does temperature affect C_v?
A: For ideal gases, C_v is constant. For real substances, C_v generally increases with temperature due to additional vibrational modes becoming active.

Q5: When is this formula particularly useful?
A: This formula is especially valuable when direct measurement of C_v is difficult, but other thermodynamic properties (α, K_T, K_S, ρ) can be measured more easily.