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Properties of High-Temperature Gases

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Overview

As the title indicates, this chapter is devoted to gases at elevated temperatures. The temperature dependence of thermodynamic properties is explained using models for estimation of the inherent energy of molecules (or atoms in the case of mono-atomic gases), i.e. in this chapter we step down from the continuum levels of observation where we study fluid particles to study individual molecules and atoms.

Chapter Roadmap

Sections

16.1 Introduction

At high temperature and/or Mach the calorically perfect gas model does not hold and therefore better models are needed. The introduction section gives a few examples that where the calorically perfect gas assumption is used at conditions where it is not applicable in order to highlight the importance of using proper gas models.

16.2 Microscopic Description of Gases

The fundamental energy types of molecules are described; translational energy, rotational energy, vibrational energy, and electronic energy. Each of these energy types is quantized, which means that it can exist in certain discrete values. The lowest possible energy of each type is defined as the zero-point energy. The zero-point energy or the ground state is the energy at zero Kelvin. The rotational zero-point energy is exactly zero whereas the other energy types has finite zero-level values. The electronic energy has the larges zero-point value of the defined energy types. This is explained by the fact that the electrons would fall into the nucleus and the atoms would collapse.

Each possible energy level can occur in different states. It turns out that the energy states are quantized and only occurs in distinguishable discrete values. These are called the degenerate states.

The concept macrostate is introduced and explained and the relation between the most probable macrostate and the population of degenerate states is explained.

16.3 Counting the Number of Microstates for a given Macrostate

This section discusses micro- and macrostates and gives an introduction to the following section.

16.4 The Most Probable Macrostate

The most probable macrostate is the macrostate with the largest number of microstates.

16.5 The Limiting Case: Boltzmann Distribution

The Boltzmann distribution:

$$N_j^*=N\frac{g_j {\mathrm{e}}^{-\varepsilon_j/kT}}{Q}$$

where \(Q=f(T,V)\) is the state sum defined as

$$Q\equiv\sum_j g_j {\mathrm{e}}^{-\varepsilon_j/kT}$$

\(g_j\) is the number of degenerate states, \(\varepsilon_j=\varepsilon'_j-\varepsilon_o\), and \(k\) is the Boltzmann constant

For molecules or atoms of a given species, quantum mechanics says that a set of well-defined energy levels \(\varepsilon_j\) exists, over which the molecules or atoms can be distributed at any given instant, and that each energy level has a certain number of energy states, \(g_j\).

For a system of \(N\) molecules or atoms at a given \(T\) and \(V\), \(N_j^*\) are the number of molecules or atoms in each energy level \(\varepsilon_j\) when the system is in thermodynamic equilibrium.

16.6 Evaluation of Thermodynamic Properties in Terms of the Partition Function

The state sum \(Q\) defined in the previous section is related to the internal energy of the fluid.

$$E=NkT^2\left(\frac{\partial \ln Q}{\partial T}\right)_V$$

$$e=\frac{E}{M}=\frac{NkT^2}{Nm}\left(\frac{\partial \ln Q}{\partial T}\right)_V=\left\{\frac{k}{m}=R\right\}=RT^2\left(\frac{\partial \ln Q}{\partial T}\right)_V$$

16.7 Evaluation of the Partition Function in Terms of T and V

Expressions for each of the energy types are introduced.

$$\varepsilon'_{trans}=\frac{h^2}{8m}\left(\frac{n_1^2}{a_1^2}+\frac{n_2^2}{a_2^2}+\frac{n_3^2}{a_3^2}\right)$$

• \(n_1-n_3\) quantum numbers (1,2,3,...)
• \(a_1-a_3\) linear dimensions that describes the size of the system
• \(h\) Planck's constant
• \(m\) mass of the individual molecule

$$\varepsilon'_{rot}=\frac{h^2}{8\pi^2I}J(J+1)$$

• \(J\) rotational quantum number (0,1,2,...)
• \(I\) moment of inertia (tabulated for common molecules)
• \(h\) Planck's constant

$$\varepsilon'_{vib}=h\nu\left(n+\frac{1}{2}\right)$$

• \(n\) vibrational quantum number (0,1,2,...)
• \(\nu\) fundamental vibrational frequency (tabulated for common molecules)
• \(h\) Planck's constant

16.8 Practical Evaluation of Thermodynamic Properties for a Single Species

The relations introduced in previous sections are used to obtain the ratio of specific heat for mono-atomic and molecule-based gases. For calorically perfect gases, vibrational energy is unpopulated and this results in constant values of \(C_p\) and \(C_v\). When vibrational energy gets populated we switch to thermally perfect gas and analysis of the obtained expressions shows why \(C_p\) and \(C_v\) varies with temperature as temperature is increased.

Translational energy:

$$Q_{trans}=\left(\frac{2\pi mkT}{h^2}\right)^{3/2}V$$

$$\ln Q_{trans}=\frac{3}{2}\ln T +\frac{3}{2}\ln\frac{2\pi mk}{h^2}+\ln V$$

$$\left(\frac{\partial \ln Q_{trans}}{\partial T}\right)_V=\frac{3}{2}\frac{1}{T}$$

$$e_{trans}=RT^2\frac{3}{2T}=\frac{3}{2}RT$$

Rotational energy:

$$Q_{rot}=\frac{8\pi^2 IkT}{h^2}$$

$$\ln Q_{rot}=\ln T + \ln \frac{8\pi^2 Ik}{h^2}$$

$$\left(\frac{\partial \ln Q_{rot}}{\partial T}\right)_V=\frac{1}{T}$$

$$e_{rot}=RT^2\frac{1}{T}=RT$$

Vibrational energy:

$$Q_{vib}=\frac{1}{1-{\mathrm{e}}^{-h\nu/kT}}$$

$$\ln Q_{vib}=-\ln(1-{\mathrm{e}}^{-h\nu/kT})$$

$$\left(\frac{\partial \ln Q_{vib}}{\partial T}\right)_V=\frac{h\nu/kT^2}{\mathrm{e}^{h\nu/kT}-1}$$

$$e_{vib}=\frac{h\nu/kT}{\mathrm{e}^{h\nu/kT}-1}RT$$

$$\lim_{T\rightarrow \infty} \frac{h\nu/kT}{\mathrm{e}^{h\nu/kT}-1}=1\Rightarrow e_{vib}\le RT$$

Internal energy:

$$e=e_{trans}+e_{rot}+e_{vib}+e_{el}$$

$$e=\frac{3}{2}RT+RT+\frac{h\nu/kT}{\mathrm{e}^{h\nu/kT-1}}RT+e_{el}$$

Specific heat:

$$C_v\equiv\left(\frac{\partial e}{\partial T}\right)_V$$

Molecules with only translational and rotational energy

$$e=\frac{3}{2}RT+RT=\frac{5}{2}RT\Rightarrow C_v=\frac{5}{2}R$$ $$C_p=C_v+R=\frac{7}{2}R$$ $$\gamma=\frac{C_p}{C_v}=\frac{7}{5}=1.4$$

Mono-atomic gases with only translational and rotational energy

$$e=\frac{3}{2}RT\Rightarrow C_v=\frac{3}{2}R$$ $$C_p=C_v+R=\frac{5}{2}R$$ $$\gamma=\frac{C_p}{C_v}=\frac{5}{3}=1\frac{2}{3}\simeq 1.67$$

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Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.