Internal energy translational contribution

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

The contribution of translational and rotational motion to the internal energy can be estimated from the temperature. 

Because RT = 2.48 kj-mol-1 at 25°C. the translational motion of gas molecules contributes X 2.48 kJ-moC1 = 3.72 kj-mol 1 to the internal energy of the sample at 25°C. Apart from the energy arising from the internal structures of the atoms themselves, this is the only motional contribution to the internal energy of a monatomic gas, such as argon or any other noble gas. 

Internal energy is stored as molecular kinetic and potential energy. The equipartition theorem can be used to estimate the translational and rotational contributions to the internal energy of an ideal gas. 

The molar heat capacities of gases composed of molecules (as distinct from atoms) are Higher than those of monatomic gases because the molecules can store energy as rotational kinetic energy as well as translational kinetic energy. We saw in Section 6.7 that the rotational motion of linear molecules contributes another RT to the molar internal energy ... 

The internal energy is, as indicated above, connected to the number of degrees of freedom of the molecule that is the number of squared terms in the Hamiltonian function or the number of independent coordinates needed to describe the motion of the system. Each degree of freedom contributes jRT to the molar internal energy in the classical limit, e.g. at sufficiently high temperatures. A monoatomic gas has three translational degrees of freedom and hence, as shown above, Um =3/2RT andCy m =3/2R. 

The thermal internal energy function calculated at 298.15 K [E — 0] is also listed in Table 8.1. The translational and rotational contributions are found using Eqs. 8.80 and 8.82, respectively. The vibrational contributions (Eq. 8.84) are much less, as expected. Mode 2 makes a significant contribution to the total internal energy at this temperature. Vibrational modes 5 and 6 also make smaller, but nonnegligible, contributions. The electronic contribution was calculated directly from Eq. 8.76. Through application of Eq. 8.118, the total enthalpy is [H — Ho] - 11146.71 J/mole. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

A molecule can move through space along any of three dimensions, so it has three translational degrees of freedom. It follows from the equipartition theorem that the average translational energy of a molecule in a sample at a temperature T is 3 X kT = kT. The molar contribution to the internal energy is therefore NA times this value, or... 

Since the translational energy is proportional to the square of the velocity and the energy associated with internal degrees of freedom is approximately independent of the velocity of the molecule (see Section D.2), the translational and internal energies should be transported differently, leading to different constants of proportionality between X and pic for each contribution. Equation (44) should be valid for the translational contribution, but the internal part will be transported more like momentum, whence X = may be approximately true for the internal contribution. Adding the two contributions therefore yields... 

As shown in Chapter 17, the rotations and vibrations of diatomic or polyatomic molecules make additional contributions to the energy. In a monatomic gas, these other contributions are not present thus, changes in the total internal energy AU measured in thermodynamics can be equated to changes in the translational kinetic energy of the atoms. If n moles of a monatomic gas is taken from a temperature Ti to a temperature Tx, the internal energy change is... 

The internal energy of a material is the sum of the total kinetic energy of its molecules and the potential energy of interactions between those molecules. Total kinetic energy includes the contributions from translational motion and other components of motion such as rotation. The potential energy includes energy stored in the form of resisting intermolecular attractions between molecules. Enthalpy... 

The contribution to the internal energy (measured relative to the zero-point energy) and to the heat capacity of the vibrational motion is also negligible (Table 9.4) whereas that from translational and rotational motion is classical. Thus... 

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