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Among the more confusing subjects of thermodynamics are the many different ways • Nmol: the number of moles of the gas in the volume V , −1 −1 in which the ideal gas equation can be written. • R = NAk8.3145 J mol K : the universal gas constant The one I prefer for astronomy is To summarise, each of these equations has its own uses, and which one you want to use, really depends on the circumstances of the problem you are solving. For your P = nkT future life as physicists, try to remember one of them, and then understand how you where get from this one to the others, instead of memorising all four ones. This approach − will need less memory and lead to a better understanding of what is really going on • P : Pressure (measured in N m 2) − behind the scenes. • n: particle density (i.e., number of particles per cubic meter, unit: m 3) − − • k = 1.38066 × 10 23 J K 1: Boltzmann constant • T : Temperature (measured in Kelvins) This equation has the advantage that it counts all particles individually (thus using n). If you know the mass of the gas particles, mgas then another way of writing the ideal gas equation is nm 1 P = gas kT = ρkT mgas mgas illustrating that for an ideal gas, P ∝ ρ, where ρ is the mass density. Another way to write the ideal gas equation is in terms of the total number of gas molecules, N = nV , where V is the volume. The ideal gas equation then is N P V P = kT ⇐⇒ P V = NkT ⇐⇒ = Nk V T This version has the problem, however, that the number of gar molecules is typically rather large (there are 6 × 1023 molecules in a volume of 22.4 liters of gas, this number of particles is called one mole). Because working with smaller numbers is generally thought a good idea, chemists prefer to work with moles. Per definition, 23 the unit of particle number here is the Avogadro number NA = 6.0221 × 10 . So, if you want to work with moles, then the above equation becomes N P V = AkT = NmolRT NA where 10 3–11 ”) auli P alues v their the empty is of QM one discrete in e (almost) v in. is least ha ks at I kic , only space phenomena can lude inciple mions”), ., phase pr er i.e (“ inter anger ”, (“F erent. str ed lem QM diff ), z xclusion the e be prob of mv a quantiz , electrons ⇒ “ .g., ust y = e m not , as 2). One mv are / : is , 1 are x ), : − z such dense , this Sun , mv 2 , ( y umbers / , n 1 umbers the x inciple n gas ( ticles + umbers momentum, pr mechanics n of ) s par ( becomes spin: these it ALE or XAN CO D I R R I typical
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Main absolute Magnitude 0.0 −7.5 −5.0 −2.5 +2.5 +5.0 +7.5 ost −10.0 P 3.4 PN Core Helium Flash RGB Ejection He Stars: 3.6 C+O H AGB 3.8 1 Msun He Mass log T 4.0 Horizontal Branch Sun Solar 4.2 Low metallicity stars only the 4.4 of
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that: regimes: strongly M e v o tw (1.5 Abo cycle of K . 6 10 CNO × energy Sun: xistence Stars of to 18 from e e w of ation sho ) similar 17 amounts atures ) Massiv T 5 gener T ∝ ucture ulations equal ∝ ( temper str ( e Sim al energy v olution: : ha Ev centr produce . since or f pp-chain CNO-cycle stars : : core Sequence center Stellar e hull core ation: ation: e e ards Main hull CNO-cycle Sun ectiv w e v to . gener gener the sequence sequence the and adiativ adiativ r r con ed ectiv of on v main main peak ALEX energy inner con energy inner outer O AN IC D R R
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Maeder & Meynet, 1989 Kippenhahn et al. (1965): Evolution of the internal structure of a 5 M star. Kippenhahn et al. (1965)
Evolution of a 5 M star in the HRD. Evolution of the structure of a 7 M star to the carbon burning phase (Maeder & Meynet, 1989). Internal structure of a 7 M star which just starts its carbon burning phase.
Maeder & Meynet, 1989 ; 34
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