The constant volume molar heat capacity (Cv,m) of a non-linear polyatomic gas is 3(N-1)R, where 'N' represents the number of atoms in the gas molecule and 'R' is the ideal gas constant. This value is derived from the principle of equipartition of energy, which distributes energy equally among a molecule's available degrees of freedom.
Understanding Constant Volume Molar Heat Capacity (Cv,m)
The constant volume molar heat capacity (Cv,m) is a fundamental thermodynamic property that quantifies the amount of heat energy required to raise the temperature of one mole of a substance by one Kelvin, while keeping its volume constant. For gases, Cv,m is primarily determined by how the gas molecules can store energy, which relates directly to their molecular structure and the available degrees of freedom.
The Equipartition Principle and Degrees of Freedom
The equipartition principle is a classical statistical mechanics theorem that states that each quadratic term in the expression for the energy of a system contributes (1/2)kT to the average energy per molecule, where 'k' is the Boltzmann constant and 'T' is the absolute temperature. For molar quantities, this translates to (1/2)R for each degree of freedom in the heat capacity.
For a molecule composed of N atoms, the total number of degrees of freedom is 3N. These degrees of freedom can be categorized into translational, rotational, and vibrational motions.
- Translational Degrees of Freedom: These refer to the movement of the entire molecule through space along the x, y, and z axes. Every molecule, regardless of its shape, has 3 translational degrees of freedom. Each contributes (1/2)R to Cv,m.
- Rotational Degrees of Freedom: These describe the rotation of the molecule around its center of mass.
- For a non-linear polyatomic gas, molecules can rotate around three independent axes, thus having 3 rotational degrees of freedom. Each contributes (1/2)R to Cv,m.
- (In contrast, linear molecules only have 2 rotational degrees of freedom).
- Vibrational Degrees of Freedom: These represent the oscillatory motions of atoms within the molecule relative to each other, like stretching or bending bonds. The number of vibrational degrees of freedom is calculated by subtracting the translational and rotational degrees from the total:
- Total degrees of freedom = 3N
- Translational degrees of freedom = 3
- Rotational degrees of freedom (for non-linear) = 3
- Vibrational degrees of freedom = 3N - 3 - 3 = 3N - 6.
Each vibrational degree of freedom contributes R to Cv,m because vibrational energy involves both kinetic and potential energy components, each contributing (1/2)R.
Derivation of Cv,m for Non-Linear Polyatomic Gas
Combining these contributions, the constant volume molar heat capacity for a non-linear polyatomic gas is:
- Translational Contribution: 3 × (1/2)R = (3/2)R
- Rotational Contribution: 3 × (1/2)R = (3/2)R
- Vibrational Contribution: (3N - 6) × R
Summing these up:
Cv,m = (3/2)R + (3/2)R + (3N - 6)R
Cv,m = 3R + (3N - 6)R
Cv,m = (3 + 3N - 6)R
Cv,m = (3N - 3)R = 3(N-1)R
This formula provides the expected value for Cv,m based on the classical equipartition principle at temperatures where all modes are active.
Summary of Degrees of Freedom Contributions
The following table summarizes how each type of motion contributes to the constant volume molar heat capacity for a non-linear polyatomic gas:
| Type of Motion | Number of Degrees of Freedom | Contribution to Cv,m (per mode) | Total Contribution to Cv,m |
|---|---|---|---|
| Translational | 3 | (1/2)R | (3/2)R |
| Rotational | 3 | (1/2)R | (3/2)R |
| Vibrational | 3N - 6 | R | (3N - 6)R |
| Total (Cv,m) | 3N | - | 3(N-1)R |
Examples of Non-Linear Polyatomic Gases
Let's apply the formula to common non-linear polyatomic gases:
- Water (H₂O): N = 3 atoms (2 Hydrogen, 1 Oxygen)
- Cv,m = 3(3-1)R = 3(2)R = 6R
- Methane (CH₄): N = 5 atoms (1 Carbon, 4 Hydrogen)
- Cv,m = 3(5-1)R = 3(4)R = 12R
- Ammonia (NH₃): N = 4 atoms (1 Nitrogen, 3 Hydrogen)
- Cv,m = 3(4-1)R = 3(3)R = 9R
These values are theoretical maximums based on the equipartition principle, assuming all vibrational modes are fully excited.
Factors Influencing Cv,m
While the 3(N-1)R formula provides a strong theoretical basis, the actual Cv,m can vary, primarily with temperature. At very low temperatures, quantum effects become significant, and vibrational degrees of freedom may not be fully excited (they are "frozen out"). As temperature increases, more vibrational modes become active, leading to an increase in the measured Cv,m towards the classically predicted value.
Understanding Cv,m is crucial for various thermodynamic calculations, including predicting temperature changes, heat transfer, and energy conversion processes in systems involving polyatomic gases.
