AsToo w

(Cv)vib = K (classical limit) - asT-»0(w-»«•),

(Cv)vib = 0 (mode inactive) Cv = (^v)trtiu + (Cv)rot + (Cy)vib = 3/2 R + 3/2 R + 0 R = 3R Cp = Cv + R = 4R = 33 J/mole-K

Vibration each mode contributes R to Cv > >

FIGURE 2.6 Contributions to the specific heat of water vapor.

The atomic gases (Ar, Ne, He) possess only translational kinetic energy but no internal eneigy modes. Consequently, an atom moving in three-dimensional space possesses 3/2kT of translational energy, where k is Boltzmann's constant. Per mole of gas, Cv = 3/2/?, and, by Equation (2.12), CP = 5/2R. CP for the atomic gases is independent of temperature because translation is the only mode of eneigy storage.

The diatomic gases 02 and N2 (or air in Figure 2.5) exhibit an apparent low-temperature limit of CP = 7/2/?. This is because rotation of these dumbbell-shaped molecules provides an additional energy-storage mode that adds one unit of R to Cy, and hence to CP

The curve for H2 in Figure 2.5 shows that the rotational mode first appears at low temperature but does not fully contribute until -400 K. The continued increase in CP of the diatomic gases at higher temperatures is due to the contributions of the various vibrational modes of the molecules.

The triatomics C02 and H20 have higher values of CP because their more complex structures give these molecules a greater number of rotational and vibrational modes than the diatomic gases.

Example: Figure 2.6 illustrates the contributions of the different modes of energy storage to the specific heat of water vapor. In addition to three degrees of rotational freedom, the water vapor curve in Figure 2.5 includes three degrees of rotation, shown schematically in Figure 2.6. All three modes persist to near 0 K because transformation to liquid or solid states is not considered in the plot. At low temperatures, the vibrational modes are not active, so CF includes 3/2R for translation, 3/2R for rotation and a final R to convert Cv to CP The total is AR, or 33 J/mole-K. At T > -250 K, energy storage by vibration begins. Each of the three modes of vibration is quantized. Depending on the frequency of the mode, each grows in differently with temperature.

In the high-temperature (classical) limit the quantum sum in Figure 2.6 approaches 3R, at which point CP should be 7R, or 58 J/mole-K. This appears to be the high-temperature limit of the water vapor curve in Figure 2.5, but the slope is not zero. This means that other modes of eneigy storage are growing in.

The high-temperature contributions to CP include ionization, illustrated for the diatomic molecule A2:

The fraction of the A2 molecules that are ionized absorbs energy (enthalpy) according to:

AH^ffd where I = ionization energy and/ = fraction ionized, obtained from the equilibrium of the above reaction (see Chapter 9). The specific heat contribution from this source is:

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Solar Panel Basics

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