Abstract
It is of course desirable to combine thermodynamics with our knowledge of the structure of matter. In particular we want to calculate Thermodynamic quantities on the basis of microscopic interactions between atoms and molecules or even subatomic particles.
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Notes
- 1.
Thus far \(E_\nu \) corresponded to the energy of a system. And systems do contain large number of particles. Now there is only one! We assume that there is so little interaction that each particle in a large system may be studied individually. But we also require that there is just sufficient interaction between this particle an its surroundings for it to reach thermal equilibrium. The idea is that one can collect instantaneous but uncorrelated (!) copies of this one particle, which, after one has obtained very many copies, are combined into one system and that this system is a system at equilibrium in the thermodynamic sense.
- 2.
The assumption of large \(N\) already entered our formalism via the truncated expansion (5.3).
- 3.
We use the Stirling approximation including \(\sqrt{2 \pi N}\). Otherwise the result is \(\Delta S = 0\).
- 4.
In this case the angle \(\psi \) does not enter and the above equations relating the momenta to the angular velocities reduce to \(p_\varphi = \mathcal I _2 \omega _2 \sin \theta \) and \(p_\theta = \mathcal I _1 \omega _1\).
- 5.
Of course, all this is expected because of the equipartition theorem of statistical mechanics stating that every term in the sum in Eq. (5.85) contributes \(k_B/2\) to the heat capacity.
- 6.
Consider for instance:
$$\begin{aligned} \frac{1}{2} E_\nu ^2 \left( \frac{d^2}{dE^2} \ln \Omega (E) \right) _{E} = \frac{1}{2} E_\nu ^2 \left( \frac{d E}{d\beta } \right) ^{-1}_{E} = - \frac{1}{2} E_\nu ^2 \left( k_B T^2 C_V^{Syst} \right) ^{-1} . \end{aligned}$$We have however
$$\begin{aligned} E^2_\nu \left( k_B T^2 C_V^{Syst} \right) ^{-1} \propto \beta E_\nu N/N^{Syst}. \end{aligned}$$Therefore this term is negligible compared to the leading one .
- 7.
via \( \sum \nolimits _{n=0}^{\infty } q^n = \left( 1 - q \right) ^{-1} \) for \( q< 1\).
- 8.
Wolfgang Pauli, Nobel prize in physics for his discovery of the exclusion principle, 1945.
- 9.
\(\zeta [s] = \sum \nolimits _{k=1}^\infty k^{-s}\) is the Riemann Zeta-function (Abramowitz and Stegun 1972).
- 10.
Somebody may object that \(\epsilon _0=0\) is not really possible due to the zero-point energy. But note that for a particle trapped in a cubic box one finds \(\beta \epsilon _0 \sim (\Lambda _T/L)^2\). Here \(V=L^3\) is the box volume. The thermal wavelength, \(\Lambda _T\), is on the order of Å, so that for every macroscopic \(L\) we find that \(\beta \epsilon _0\) is vanishingly small. This also is the reason why we consider the limit \(z=1\) rather than \(\exp [\beta \epsilon _0]\).
- 11.
The superfluid behavior exhibited by the helium isotope \({}^4He\) below \(2.1768K\), the so-called lambda point, is a manifestation of Bose condensation. The mass density at this temperature is about \(145 kg/m^3\). If we insert this number into Eq. (5.151) using \(C= (gV/(4 \pi ^2)) (2m_{He}/\hbar ^2)^{3/2}\), where \(g=2 s+1\) and \(s\) is the boson’s spin (in this case \(s=0\)), we obtain a transition temperature of about \(3.1K\), which, despite the ideality assumption, is rather close to the above value (an in depth discussion is given in R. P. Feynman (1972) Statistical Mechanics, Addison Wesley; Nobel prize in physics for his contributions to quantum electrodynamics, 1965). Notice that we have neglected the second term in Eq. (5.151), because for fixed \(z\) just slightly less than \(1\) the factor \(1/V\) dominates and this term is vanishingly small.
- 12.
In the case of \({}^4 He\), i.e. using \(T=3.2K\), we obtain \(\Delta H \approx 34 J/mol\). This is about three times less than the experimental value.
- 13.
Walther Hermann Nernst, Nobel prize in chemistry for his contributions to thermodynamics, 1920.
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Hentschke, R. (2014). Microscopic Interactions. In: Thermodynamics. Undergraduate Lecture Notes in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36711-3_5
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