Translational energy heat capacity

All calculations will be done for the standard pressure of 1 bar and, unless otherwise noted, at T = 298.15 K for one mole of gas. Table 8.1 lists the calculated molecular partition function, thermal energy (energy in excess of the ground-state energy), heat capacity, and entropy. The individual contributions from translation, rotation, each of the six vibrational modes, and from the first excited electronic energy level are included. 

Fig. 3-11 shows that, foi watei, entropy and heat capacity ai e summations in which two terms dominate, the translational energy of motion of molecules treated as ideal gas paiticles. and rotational, energy of spin about axes having nonzero rnorncuts of inertia terms (see Prublerris). 

The total partition function may be approximated to the product of the partition function for each contribution to the heat capacity, that from the translational energy for atomic species, and translation plus rotation plus vibration for the diatomic and more complex species. Defining the partition function, PF, tlrrough the equation... 

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 value of this standard molar Gibbs energy, p°(T), found in data compilations, is obtained by integration from 0 K of the heat capacity determined by the translational, rotational, vibrational and electronic energy levels of the gas. These are determined experimentally by spectroscopic methods [14], However, contrary to what we shall see for condensed phases, the effect of pressure often exceeds the effect of temperature. Hence for gases most attention is given to the equations of state. 

The specific energy from Eq. 10.13 is reported in J/g for convenience in analyzing the process. If a change in a process causes the specific energy to increase by 50 J/g, the troubleshooter can translate this to an adiabatic temperature increase of 20 °C because many unfilled resins have heat capacities near 2.5 J/(g °C). This temperature change calculation, however, is an approximation because extruders typically do not operate adiabatically. 

Let us consider a system of M ideal monatomic gas molecules in a cubic box kept at a constant temperature T. For a very dilute gas, where the molecules do not interact with one another, the quantum mechanical solution is a number of electronic wave functions with three quantum numbers nx, riy, and for the translational energies in three dimensions. The energy of a molecule for a set of quanmm numbers, the observed average energy, and the heat capacity at constant volume are given by... 

For a monatomic gas, where the heat capacity involves only translational energy, V is independent of sound oscillation frequency (except at ultra-high frequencies, where a classical visco-thermal dispersion sets in). For a relaxing polyatomic gas this is no longer so. At sound frequencies, where the period of the oscillation becomes comparable with the relaxation time for one of the forms of internal energy, the internal temperature lags behind the translational temperature throughout the compression-rarefaction cycle, and the effective values of CT and V in equation (3) become frequency dependent. This phenomenon occurs at medium ultrasonic frequencies, and is known as ultrasonic dispersion. It is accompanied by... 

The thermophysical properties necessary for the growth of tetrahedral bonded films could be estimated with a thermal statistical model. These properties include the thermodynamic sensible properties, such as chemical potential /t, Gibbs free energy G, enthalpy H, heat capacity Cp, and entropy S. Such a model could use statistical thermodynamic expressions allowing for translational, rotational, and vibrational motions of the atom. 

The combined translational, rotational and vibrational contributions to the molar heat capacity, heat content, free energy and entropy for 1,3,4-thiadiazoles are available between 50 and 2000 K. They are derived from the principal moments of inertia and the vapor-phase fundamental vibration frequencies (68SA(A)36l). 

Clearly, a monatomic gas has no rotational or vibrational energy but does have a translational energy of RT per mole. The constant-volume heat capacity of a monatomic perfect gas is thus... 

When an ideal gas is heated in a rigid container in which no change in volume occurs, there can be no PV work (AV = 0). Under these conditions all the energy that flows into the gas is used to increase the translational energies of the gas molecules. Thus Cv, the molar heat capacity of an ideal gas at constant volume, is fR, the result anticipated in the above discussion ... 

Macroscopic observables, such as pressure P or heat capacity at constant volume c y. The allowed quantum states for the translational energy are determined by placing file molecule m a box , i.e. the poleiilial is zero inside the box but infimte outside. ... 

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