# 5512 calorically imperfect for calorically imperfect

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5.5.1.2 Calorically imperfect For calorically imperfect ideal gases (CIIG), e.g. O 2 at moderate to high temperatures (300 K<T < 6000 K ): CC BY-NC-ND. 2011, J. M. Powers.

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120 CHAPTER 5. THE FIRST LAW OF THERMODYNAMICS u = u ( T ), c v = c v ( T ), h = h ( T ), c P = c P ( T ). For such temperatures, our assumption of constant c v is not as valid. But for ideal gases, we can still take c v = c v ( T ), so du dT = c v ( T ) . (5.115) We can integrate via separation of variables to get du = c v ( T ) dT, (5.116) integraldisplay 2 1 du = integraldisplay 2 1 c v ( T ) dT, (5.117) u 2 u 1 = integraldisplay 2 1 c v ( T ) dT. (5.118) More generally, we could say u ( T ) = u o + integraldisplay T T o c v ( ˆ T ) d ˆ T, (5.119) valid for all ideal gases . Here ˆ T is a dummy variable of integration. Similarly, we could show h ( T ) = h o + integraldisplay T T o c P ( ˆ T ) d ˆ T, (5.120) valid for all ideal gases . Now c v , c P and R all have units of kJ/kg/K . Let us consider the ratio c v R = c v M RM = c v M R M = c v R . (5.121) The ratio is now in terms of molar specific properties with c v and R having units of kJ/kmole/K . Note that R is the universal gas constant. A plot of c v / R versus T for a variety of simple molecules is given in Fig. 5.11. We note some remarkable facts: For monatomic gases, such as Ar , O , and H , c v / R = 3 / 2 for a wide variety of temper- atures. CC BY-NC-ND. 2011, J. M. Powers.
5.5. CALORIC EQUATIONS OF STATE. 121 0 1000 2000 3000 4000 5000 6000 0 1 2 3 4 5 6 7 T (K) c v / R Ar, H O CO 2 H 2 O O 2 H 2 5/2 3/2 Figure 5.11: c v / R as a function of T for several molecules. For diatomic gases, such as O 2 and H 2 for T < 600 K , c v / R 5 / 2, and for T > 600 K , c v / R 7 / 2 For larger molecules such as CO 2 or H 2 O , c v / R is larger still. What we are seeing actually reflects some fundamental physics. We first note that sta- tistical thermodynamics proves Temperature is a measure of the average translational kinetic energy of a set of molecules. Now we consider some features of Fig. 5.11. Monatomic molecules, such as Ar , O or H have three fundamental modes of kinetic energy: translation in the x , y , and z directions. Each mode contributes 1 / 2 to c v / R , which sums to 3 / 2. For diatomic molecules, we summarize the behavior in the sketch given in Fig. 5.12. At very low temperatures, diatomic molecules, such as H 2 or O 2 , act like monatomic molecules. At low temperatures, diatomic molecules begin to rotate, and the rotational en- ergy becomes an important component. In fact when energy is added to diatomic CC BY-NC-ND. 2011, J. M. Powers.

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122 CHAPTER 5. THE FIRST LAW OF THERMODYNAMICS T(K) c v / R 1 3 600 2000 trans trans+rot trans+rot+vib k ~ 1.4 variable k model diatomic ideal gas 7/2 5/2 3/2 Figure 5.12: c v / R as a function of T for a model diatomic gas. (Note a real gas would liquefy in the very low temperature region of the plot! So this model is really for a non-existent gas that has no liquid-inducing intermolecular forces.) molecules, some is partitioned to translation and some is partitioned to rota- tion. There are two non-trivial axes of rotation, each adding 1 / 2 to c v / R , giving c v / R 5 / 2.
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