The vibrational modes of molecular nitrogen are 'frozen out' at room temperature but become active at temperatures above ≈1500 K. The temperature in the combustion chamber of a jet engine can reach 2000 K, so an engineering analysis of combustion requires knowing the thermal properties of materials at these temperatures. What is the expected specific heat ratio γ for nitrogen at 2000 K?

Verified step by step guidance

1. Identify the degrees of freedom for nitrogen (N2) at high temperatures. At 2000 K, both translational and vibrational modes are active. Nitrogen, being a diatomic molecule, has 5 degrees of freedom at room temperature (3 translational and 2 rotational) and 2 additional vibrational degrees of freedom become active at high temperatures, making a total of 7 degrees of freedom.
2. Use the formula for the specific heat capacities at constant volume (Cv) and at constant pressure (Cp) for a diatomic gas with active vibrational modes. The formula for Cv is Cv = (f/2)R, where f is the degrees of freedom and R is the gas constant.
3. Calculate Cp using the relation Cp = Cv + R.
4. Determine the specific heat ratio γ, which is the ratio of Cp to Cv. Use the formula γ = Cp/Cv.
5. Apply these calculations specifically for nitrogen at 2000 K, considering all 7 degrees of freedom to find the expected specific heat ratio γ.

Key Concepts

Specific Heat Ratio (γ)

The specific heat ratio, denoted as γ (gamma), is the ratio of the specific heat at constant pressure (Cp) to the specific heat at constant volume (Cv). It is a crucial parameter in thermodynamics, particularly in understanding the behavior of gases during processes such as combustion. For diatomic gases like nitrogen, γ typically approaches 1.4 at room temperature but can change with temperature due to the activation of vibrational modes.

Vibrational Modes

Vibrational modes refer to the different ways in which the atoms in a molecule can vibrate. For diatomic molecules like nitrogen (N2), these modes include stretching and bending vibrations. At lower temperatures, many of these vibrational modes are 'frozen out' and do not contribute to the heat capacity, but as temperature increases, more modes become active, affecting the specific heat and, consequently, the specific heat ratio.

Thermal Properties of Materials

Thermal properties of materials, such as specific heat, thermal conductivity, and thermal expansion, describe how materials respond to changes in temperature. In the context of combustion and high-temperature environments like jet engines, understanding these properties is essential for predicting material behavior, efficiency, and performance. At elevated temperatures, the specific heat can vary significantly, influencing the energy balance in combustion processes.