For a polyatomic gas, the specific heat capacities depend on the universal gas constant ($R$) and the number of active vibrational degrees of freedom ($f$). Specifically, the specific heat at constant volume ($C_V$) is given by $(3 + f)R$, and the specific heat at constant pressure ($C_P$) is given by $(4 + f)R$.
Understanding Specific Heat Capacity
Specific heat capacity is a measure of the amount of heat energy required to raise the temperature of a substance by a certain amount. For gases, there are two primary specific heat capacities:
- Specific Heat at Constant Volume ($C_V$): This refers to the heat required to raise the temperature of a unit mass or mole of a gas by one degree Celsius (or Kelvin) while its volume is kept constant. In this process, no work is done by or on the gas.
- Specific Heat at Constant Pressure ($C_P$): This refers to the heat required to raise the temperature of a unit mass or mole of a gas by one degree Celsius (or Kelvin) while its pressure is kept constant. In this case, the gas expands and does work.
Formulas for Polyatomic Gases
The specific heat capacities for polyatomic gases are determined by their internal structure, which allows for various forms of energy storage (translational, rotational, and vibrational).
- Specific Heat at Constant Volume ($C_V$):
$C_V = (3 + f)R$ - Specific Heat at Constant Pressure ($C_P$):
$C_P = (4 + f)R$
Where:
- $R$ is the Universal Gas Constant, approximately $8.314 \text{ J mol}^{-1} \text{K}^{-1}$. It relates energy to temperature and the amount of substance.
- $f$ represents the number of active vibrational degrees of freedom of the polyatomic molecule. Unlike translational and rotational degrees of freedom, vibrational modes typically become active and contribute to specific heat only at higher temperatures.
Degrees of Freedom Explained
The total energy of a gas molecule is distributed among its various degrees of freedom. For a polyatomic molecule, these include:
- Translational Degrees of Freedom (3): These correspond to the molecule's movement along the x, y, and z axes. All molecules (monoatomic, diatomic, polyatomic) have 3 translational degrees of freedom, each contributing $(1/2)R$ to $C_V$.
- Rotational Degrees of Freedom:
- Linear Polyatomic Molecules (e.g., CO₂, N₂O): Have 2 rotational degrees of freedom (rotation about two axes perpendicular to the molecular axis). Each contributes $(1/2)R$ to $C_V$.
- Non-linear Polyatomic Molecules (e.g., H₂O, CH₄): Have 3 rotational degrees of freedom (rotation about three perpendicular axes). Each contributes $(1/2)R$ to $C_V$.
- Vibrational Degrees of Freedom ($f$): These correspond to the internal oscillations of the atoms within the molecule. Each vibrational mode involves both kinetic and potential energy, contributing $R$ to $C_V$ when fully excited (at high temperatures). The number of vibrational modes depends on the molecular structure:
- For a non-linear molecule with $N$ atoms, there are $3N-6$ vibrational modes.
- For a linear molecule with $N$ atoms, there are $3N-5$ vibrational modes.
The term 'f' in the given formulas specifically accounts for the contribution of these vibrational modes to the specific heat. At very low temperatures, vibrational modes may be "frozen out," meaning they do not contribute to the specific heat. As temperature increases, more vibrational modes become active, increasing the value of $f$.
Practical Applications
Understanding the specific heat of polyatomic gases is crucial in various fields, including:
- Thermodynamics: Calculating heat transfer, work done, and changes in internal energy in chemical reactions and industrial processes.
- Engineering: Designing engines, turbines, and refrigeration systems where gas properties at different temperatures and pressures are critical.
- Atmospheric Science: Modeling atmospheric processes, as components like CO₂ (a polyatomic gas) play a significant role.
Relationship Between $C_P$ and $C_V$
For ideal gases, the relationship between $C_P$ and $C_V$ is always $C_P = C_V + R$. This is known as Mayer's relation.
If $C_V = (3+f)R$, then $C_P = (3+f)R + R = (4+f)R$, which perfectly aligns with the given formulas.
Summary Table
| Property | Formula for Polyatomic Gas | Notes |
|---|---|---|
| Specific Heat at Constant Volume ($C_V$) | $(3 + f)R$ | $f$ is the number of active vibrational degrees of freedom. |
| Specific Heat at Constant Pressure ($C_P$) | $(4 + f)R$ | $R$ is the Universal Gas Constant ($8.314 \text{ J mol}^{-1} \text{K}^{-1}$). |
| Mayer's Relation | $C_P - C_V = R$ | Applies to all ideal gases. |
This detailed understanding allows for accurate calculations and predictions of gas behavior under varying conditions.
