The Entropy Equation for Solids and Gases and the Universal Quantum of Energy.

(Simon Ratnowsky. Die Entropiegleichung fester Körper und Gase und das universelle Energiequantum. Berichte d. deutsch. physik. Gesellschaft, 1916, p. 263).

The initial hypotheses of the derivation are the following:

a) Every system of material elements possesses a “zero energy,” present in the body also at absolute zero temperature. With a change in the energy of the body, produced from outside, there is associated a change in the system’s proper zero energy.

b) The system is assumed to be canonical (in Gibbs’s sense), i.e. the number of elements situated in the volume \(dh\) of the statistical \(2n\)-dimensional space:

\[ \rho(q,p)dh = N e^{-\frac{\psi-\varepsilon}{\theta}}dh \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (1) \]

\(q\) and \(p\) are generalized coordinates, \(N\) is the total number of elements, \(\psi\) is the statistical free energy, \(\theta\) is proportional to the absolute temperature, \(\varepsilon\) is the total energy.

c) The total energy of the system \(\varepsilon\) is chosen as a quadratic function of the coordinates \(p\) and \(q\):

\[ \varepsilon = \frac{f}{2}(q_1^2+q_2^2+\cdots+q_n^2) +\frac{1}{2m}(p_1^2+p_2^2+\cdots+p_n^2) \ . \ . \ (2) \]

d) The magnitude of the proper energy falling to one degree of freedom of the system cannot exceed a definite limit characteristic of the given system.

e) The number of degrees of freedom of a system of \(N\) elements,

\[ n=3N \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (3) \]

Calculating by the usual methods first the complete free energy of the system \(\psi\) on the basis of hypotheses b) and c), Ratnowsky then introduces hypotheses a), d), and e), finds an expression for the free thermal energy \(\psi_1\), and obtains the equations of entropy and energy

\[ S=3Nk\left[ \frac{\frac{\varepsilon_0}{kT}}{e^{-\frac{\varepsilon_0}{kT}}-1} -\log\left(1-e^{-\frac{\varepsilon_0}{kT}}\right) \right] \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (4) \]

\[ U=3N\frac{\varepsilon_0}{e^{-\frac{\varepsilon_0}{kT}}-1} \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (5) \]

where \(k=1347\cdot 10^{-16}\), \(\varepsilon_0\) is the limiting energy of one degree of freedom. Further, Ratnowsky shows that from hypotheses a) and d) it necessarily follows that

\[ \varepsilon_0=h\nu \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (6) \]

If we denote

\[ \frac{1}{\theta}\left[\frac{f}{2}q^2+\frac{1}{2m}p^2\right]=x \]

then, on the basis of all the hypotheses made, the free proper energy \(\psi_0\) is determined from the relation:

\[ e^{-\frac{\psi_0}{\theta}} = \left[ 6{.}2\pi \sqrt{\frac{m}{f}} \int_0^{x_0} e^{-x}\,dx \right]^{3N} \ldots\ldots\ldots\ldots\ldots \tag{7} \]

where \(x_0=\dfrac{\varepsilon_0}{kT}\), \(\theta=kT\); but, generally speaking,

\[ 2\pi\sqrt{\frac{m}{f}}=\frac{1}{\nu} \ldots\ldots\ldots\ldots\ldots \tag{8} \]

where \(\nu\) is a certain frequency of oscillation. Substituting in (7), we find:

\[ \psi_0=-3NkT\log\frac{kT}{\nu}\left(1-e^{-\frac{\varepsilon_0}{kT}}\right) \]

for the “intrinsic entropy” (Eigenentropie) we find:

\[ S_0=-\frac{d\psi_0}{dT} = -3Nk \left[ \frac{\frac{\varepsilon_0}{kT}}{e^{\frac{\varepsilon_0}{kT}}-1} -\log\left(1-e^{-\frac{\varepsilon_0}{kT}}\right) \right] -3Nk\log\frac{\nu}{kT}+3Nk \tag{9} \]

Classical theory gives the following expression for the entropy of a system of \(N\) elements of a solid body:

\[ S=3R\log T+S' \ldots\ldots\ldots \tag{10} \]

where \(R=kN\), \(S'\) is a constant. Equating, for the case

\[ kT \gg \varepsilon_0 \]

\(S'\) to the intrinsic entropy (9), we find:

\[ \frac{\varepsilon_0}{\nu}=h \ldots\ldots\ldots\ldots \tag{11} \]

where \(h\) is a universal constant. In that case formula (5) completely coincides with the known formula of Planck—Einstein.

It remains an open question to what extent the simultaneous application of hypotheses \(a)\) and \(d)\), on the one hand, and \(b)\), \(c)\), on the other, is compatible. The universality of \(h\) in formula (11) is also doubtful under restriction \(e)\).

S. Vavilov.