At higher temperatures diatomic molecules begin to

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At higher temperatures, diatomic molecules begin to vibrate as well, and this energy becomes an important component. There are two vibrational modes, and each adds another 1 / 2 to c v / R , giving c v / R 7 / 2 at high temperature. at higher temperatures still, the diatomic molecules begin to dissociate, e.g. O 2 O + O . at even higher temperatures, its electrons are stripped, and it becomes an ionized plasma. This is important in engineering applications ranging from welding to atmospheric re-entry vehicles. For triatomic molecules such as H 2 O or CO 2 , there are more modes of motion which can absorb energy, so the specific heat is higher still. Feynman 9 summarizes the argument that this preference for one type of energy over an- other (translation, rotational, vibrational) depending on temperature is surprising to those not versed in quantum mechanics and violates standard assumptions of classical statistical mechanics. In fact, he notes that Maxwell had a hint of the problem as early as 1859, and 9 R. P. Feynman, R. B. Leighton, and M. Sands, 1963, The Feynman Lectures on Physics , Volume 1, Addison-Wesley, Reading, Massachusetts, pp. 40-7–40-10. CC BY-NC-ND. 2011, J. M. Powers.
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5.5. CALORIC EQUATIONS OF STATE. 123 stated this concern more directly in 1869. Feynman argues that the reason for the energy partition observed in diatomic gases is a “failure of classical physics” and instead is a pure effect of quantum mechanics; that is to say k = c P ( T ) /c v ( T ) = k ( T ) is a non-classical result! Employment of the theories of quantum and statistical mechanics allows an accounting for the observation that there is a preference of molecules to exist in lower energy states, and at those states, the discrete quantization is important. High energy vibrational states are less likely than translational states at low temperature. At higher temperature, there is a higher probability that all states are populated, and one recovers results consistent with classical physics. Let us also recall that c P ( T ) c v ( T ) = R ; thus, c P ( T ) c v ( T ) = R. Let us summarize for monatomic gases, c v = 3 2 R, (5.122) c P = c v + R = 5 2 R, (5.123) c P c v = k = 5 2 R 3 2 R = 5 3 = 1 . 6667 . (5.124) for diatomic gases at moderate temperature, 50 K<T < 600 K , c v = 5 2 R, (5.125) c P = c v + R = 7 2 R, (5.126) c P c v = k = 7 2 R 5 2 R = 7 5 = 1 . 4 . (5.127) To summarize, usually the most problematic case is whether or not specific heats vary with temperature in ideal gases. For low temperatures, the specific heat is well modeled as a constant; here the internal energy change is strictly proportional to the temperature change. For moderate to high temperatures, a temperature-variation of the specific heat is observed. Changes in internal energy are no longer strictly proportional to changes in temperature. The behavior is analogous to solid mechanics. At low strain ǫ , stress σ is proportional to strain, and the constant of proportionality is the modulus of elasticity E . For high strains, the linearity is lost; we could say the elastic modulus becomes a function of strain. We give a sketch in Fig. 5.13 of the comparison to solid mechanics
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