Molecular investigation of the Volume Dependence of Entropy

5. Molecular Theoretical investigation of the Volume Dependence of the Entropy of Gases and Dilute Solutions

In calculating Entropy in molecular theory, “probability” is often used in a meaning that isn’t covered by the definition in probability theory.

Especially the “cases of equal probability” are often set by hypothesis, where the applied theoretical representation is sufficiently definite to deduce probabilities without fixing them by hypothesis.

I will show in a separate work that in considerations of thermal processes one obtains a complete result with the so-called “statistical probability”. This way I hope to remove a logical difficulty that is in the way of fully implementing Boltzmann’s principle.

Here however only its general formulation and application in quite specific cases will be given.

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Every increase of entropy can be described as a transition to a more probable state. Then the entropy S1 of a system is a function of the probability W1 of its instantaneous state. In the case of two systems S1 and S2, one can state:

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If one considers these systems as a single system with entropy S and probability W, then:

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and

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The latter equation expresses that the states of the two systems are independent.

From these equations it follows:

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and hence finally

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The quantity C is also a universal constant.

It follows from kinetic gas theory, where the constants R and N have the same meaning as above.

Denoting the entropy at a particular starting state as S0, and the relative probability of a state with entropy S as W we have in general:

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We now consider the following special case.

Let a number (n) of movable points (for example molecules) be present in a volume v0, these points will be the subject of our considerations. Other than these, arbitrarily many other movable points can be present.

As to the law that describes how the considered points move around in the space the only assumption is that no part of the space (and no direction) is favored over others. The number of the (first-mentioned) points that we are considering is so small that mutual interactions are negligible.

The system considered, which can be for example an ideal gas or a diluted solution, has a certain entropy. We take a part of the volume v0 with a size of v and we think of all n movable points displaced to that volume v, with otherwise no change of the system.

Clearly this state has another entropy (S), and here we want to determine that entropy difference with the help of Boltzmann’s principle.

We ask: how large is the probability of the last-mentioned state relative to the original state? Or, what is the probability that at some point in time all n independently moving points in a volume v0 have by chance ended up in the volume v?

For this probability, which is a “statistical probability” one obtains the value:

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one derives from this, applying Boltzmann’s principle:

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It’s noteworthy that for this derivation, from which the Boyle-Gay-Lussac law and the identical law of osmotic pressure can be easily derived thermodynamically [6], there is no need to make any assumption regarding the way the molecules move.

Interpretation of the Volume Dependence of the Entropy of Monochromatic Radiation using Boltzmann’s Principle In paragraph 4 we found for the dependence of Entropy of the monochromatic radiation on volume the expression:

This formula can be recast as follows:

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Comparing this with the general formula that expresses Boltzmann’s principle

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we arrive at the following conclusion:

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If monochromatic radiation of frequency ν and energy E is enclosed (by reflecting walls) in the volume v0, then the probability that at an arbitrary point in time all of the radiation energy located in a part v of the volume v0 is:

Subsequently we conclude:

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In terms of heat theory monochromatic radiation of low density (within the realm of validity of Wien’s radiation formula) behaves as if it consisted of independent energy quanta of the magnitude Rβν/N.

We also want to compare the average magnitude of the energy quanta of the “black body radiation” with the mean average energy of the center-of-mass-motion of a molecule at the same temperature. The latter is 3/2(R/N)T, and for the average energy of the Energy quanta Wien’s formula gives:

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The fact that monochromatic radiation (of sufficiently low density) behaves as regards to dependency of entropy on volume like a discontinuous medium that consists of energy quanta of magnitude Rβν/N suggests we should investigate whether the laws of generation and transformation of light are what they must be if light consisted of such energy quanta. In the following we will address that question.

Stokes’ Rule

Let monochromatic light be transformed by photoluminence into light of another frequency.

Let it be assumed that according to the result just obtained the generating as well as the generated light consists of energy quanta of magnitude (R/N)βν, where ν is the corresponding frequency.

The transformation process can then be interpreted as follows.

Each generating energy quantum of frequency ν1 is absorbed and generates—at least with sufficiently small density of the generating energy quanta—by itself a light quantum of frequency ν2; possibly other light quanta of frequency ν3, ν4 etc. as well as other form of energy (e.g heat) can be generated simultaneously.

Through which intermedia processes the final result comes about is immaterial. If the photoluminescing substance isn’t a continuous source of energy it follows from the energy principle that the energy of the generated energy quanta are not larger than the generating light quanta; therefore the following relation must hold:

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or

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As is well known this is Stokes’ rule.

In my conception, the quantity of light emitted under low illumination must be proportional to the strength of the incident light.

This is because each incident energy quantum will cause an elementary process of the postulated kind, independently of the action of other incident energy quanta.

There will be no lower limit for the intensity of incident light necessary to excite the fluorescent effect.

This means that deviations from Stokes’s Rule are conceivable in the following cases:

1. When the number of simultaneously interacting energy quanta per unit volume is so large that an energy quantum of emitted light can receive its energy from several incident energy quanta

2. When the incident (or emitted) light is not of such a composition that it corresponds to blackbody radiation within the range of validity of Wien’s Law. For example, when the incident light is produced by a body of such high temperature that for the wavelengths under consideration Wien’s Law is no longer valid.

According my conception, the possibility is not excluded that a “non-Wien radiation” of very low density can exhibit an energy behavior different from that of a blackbody radiation within the range of validity of Wien’s Law.