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Understanding the Two Specific Heats of a Gas

In thermodynamics, the specific heat of a substance is a critical property that quantifies how much heat energy is required to raise the temperature of a unit mass of the substance by one degree Celsius (or one Kelvin). Gases exhibit two distinct specific heat values depending on the conditions under which heat transfer occurs: at constant volume and at constant pressure. These two specific heats are commonly denoted as Cv (specific heat at constant volume) and Cp (specific heat at constant pressure). This assignment delves into the definitions, calculations, and significance of Cv and Cp in the context of gas behavior.

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Specific Heat at Constant Volume (Cv)

Definition

Specific heat at constant volume (Cv) is defined as the amount of heat required to raise the temperature of a unit mass of gas by one degree Celsius while keeping the volume constant. Under these conditions, no work is done by the gas on the surroundings.

Mathematical Expression

The mathematical expression for Cv can be represented as:

Cv = (∂Q/∂T)V

Where:

• Cv = Specific heat at constant volume
• Q = Heat added to the system
• T = Temperature
• The subscript V indicates constant volume.

Characteristics

• When heat is added to the gas at constant volume, the internal energy of the gas increases, leading to an increase in temperature.
• The value of Cv depends on the type of gas (monoatomic, diatomic, polyatomic) and can be derived from the degrees of freedom of the gas molecules.

Example Calculation

For a monoatomic ideal gas, the specific heat at constant volume is given by:

Cv = (3/2) R

Where R is the universal gas constant (R ≈ 8.314 J/(mol K)). Therefore, for a monoatomic ideal gas, Cv ≈ 12.47 J/(mol K).

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Specific Heat at Constant Pressure (Cp)

Definition

Specific heat at constant pressure (Cp) is defined as the amount of heat required to raise the temperature of a unit mass of gas by one degree Celsius while maintaining constant pressure. In this case, some of the heat energy is used to do work against the atmospheric pressure as the gas expands.

Mathematical Expression

The mathematical expression for Cp can be represented as:

Cp = (∂Q/∂T)P

Where:

• Cp = Specific heat at constant pressure
• The subscript P indicates constant pressure.

Characteristics

• At constant pressure, the heat added to the gas increases both its internal energy and the work done by the gas as it expands.
• Cp is always greater than Cv because it accounts for the additional work done by the gas.

Example Calculation

For an ideal gas, the specific heat at constant pressure is given by:

Cp = (5/2) R

For a monoatomic ideal gas, Cp ≈ 20.79 J/(mol K).

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Relationship Between Cp and Cv

The relationship between Cp and Cv for an ideal gas is given by the equation:

Cp - Cv = R

This equation highlights that the difference between the specific heats at constant pressure and constant volume is equal to the universal gas constant R.

Derivation of the Relationship

1. First Law of Thermodynamics: The first law states that the change in internal energy (dU) is equal to the heat added to the system (dQ) minus the work done by the system (dW).

   dU = dQ - dW

2. For Constant Volume: No work is done, so:

   dU = dQv ⇒ dQv = Cv dT

3. For Constant Pressure: The work done by the gas is dW = PdV. Therefore, the first law becomes:

   dU = dQp - PdV

4. Using the ideal gas law PV = nRT and substituting, it can be shown that:

   dQp = Cp dT

5. Ultimately, combining these equations leads to the relation:

   Cp - Cv = R.

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Significance of Specific Heats in Thermodynamics

1. Understanding Heat Transfer: The specific heats of gases are crucial for understanding how energy is transferred during heating and cooling processes in engines, refrigerators, and other thermal systems.
2. Thermodynamic Cycles: In thermodynamic cycles, such as the Carnot cycle or Otto cycle, the specific heats play a significant role in calculating efficiencies and work output.
3. Material Properties: Knowing the specific heats helps in characterizing materials and their behavior under various temperature conditions.
4. Chemical Reactions: In chemical thermodynamics, specific heats are vital for calculating reaction enthalpies and understanding reaction kinetics.

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Conclusion

The specific heats at constant volume (Cv) and constant pressure (Cp) are fundamental properties of gases that influence their behavior during heating and cooling processes. Understanding these specific heats and their relationship is essential for various applications in thermodynamics, engineering, and material science. By quantifying the energy required for temperature changes, Cv and Cp provide valuable insights into the thermodynamic processes governing gas behavior, making them crucial for both theoretical and practical applications.

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References

1. Cengel, Y. A., & Boles, M. A. (2015). Thermodynamics: An Engineering Approach. McGraw-Hill Education.
2. Sonntag, C. E., & Borgnakke, C. (2013). Introduction to Thermal Systems Engineering: Thermodynamics, Fluid Mechanics, and Heat Transfer. Wiley.
3. Moran, M. J., & Shapiro, H. N. (2006). Fundamentals of Engineering Thermodynamics. Wiley.

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