English translation · §5 · independent configurations

Translation

Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt

§ 5. Molecular-theoretical investigation of the dependence of the entropy of gases and dilute solutions on the volume.

In calculating entropy by molecular-theoretical methods, the word “probability” is often used in a sense that does not coincide with the definition of probability as it is given in the calculus of probabilities. In particular, the “cases of equal probability” are often fixed hypothetically in cases where the theoretical pictures used are definite enough to give a deduction in place of that hypothetical stipulation. I shall show in a separate paper that in considerations of thermal processes one manages completely with the so-called “statistical probability”, and I hope thereby to remove a logical difficulty that still stands in the way of carrying through Boltzmann’s principle. Here, however, only its general formulation and its application to quite special cases will be given.

If it makes sense to speak of the probability of a state of a system, and if, further, every increase in entropy can be conceived as a transition to a more probable state, then the entropy $S_1$ of a system is a function of the probability $W_1$ of its momentary state. If, therefore, two systems $S_1$ and $S_2$ are given that do not interact with each other, one can set: $\begin{aligned} S_1 &= \varphi_1(W_1), \\ S_2 &= \varphi_2(W_2). \end{aligned}$ EntropyEntropyProbability of the configurationEntropyEntropyProbability of the configurationExplain this equation If one regards these two systems as a single system of entropy $S$ and probability $W$ , then we have: $S = S_1 + S_2 = \varphi(W)$ EntropyEntropyEntropyEntropyProbability of the configurationExplain this equation and $W = W_1 \cdot W_2.$ Probability of the configurationProbability of the configurationProbability of the configurationExplain this equation The latter relation says that the states of the two systems are events independent of each other.

From these equations it follows: $\varphi(W_1 \cdot W_2) = \varphi_1(W_1) + \varphi_2(W_2)$ EntropyProbability of the configurationProbability of the configurationEntropyEntropyExplain this equation and from this, finally, $\begin{aligned} \varphi_1(W_1) &= C \lg(W_1) + \text{konst.}, \\ \varphi_2(W_2) &= C \lg(W_2) + \text{konst.}, \\ \varphi(W) &= C \lg(W) + \text{konst.} \end{aligned}$ EntropyProbability of the configurationUniversal entropy constantEntropyProbability of the configurationEntropyProbability of the configurationExplain this equation The quantity $C$ is therefore a universal constant; as follows from the kinetic theory of gases, it has the value $R/N$ , where the constants $R$ and $N$ are to be given the same meaning as above. If $S_0$ denotes the entropy in a certain initial state of a system under consideration and $W$ the relative probability of a state of entropy $S$ , we thus obtain in general: $S - S_0 = \frac{R}{N} \lg W.$ EntropyEntropyGas constantAvogadro's numberProbability of the configurationExplain this equation

We first treat the following special case. In a volume $v_0$ let there be a number $(n)$ of movable points (e.g., molecules) to which our consideration is to refer. Besides these, arbitrarily many other movable points of any kind may be present in the space. About the law by which the points under consideration move in the space, nothing is presupposed except that, with respect to this motion, no part of the space (and no direction) is distinguished from the others. Further, let the number of the (first-mentioned) movable points under consideration be so small that any action of the points on one another can be disregarded.

The system under consideration, which may be, e.g., an ideal gas or a dilute solution, has a certain entropy $S_0$ . We imagine a part of the volume $v_0$ of size $v$ , and all $n$ movable points transferred into the volume $v$ , without anything else in the system being changed. To this state there evidently belongs a different value of the entropy $(S)$ , and we now want to determine the difference in entropy with the help of Boltzmann’s principle.

We ask: How large is the probability of the last-mentioned state relative to the original one? Or: How large is the probability that at a randomly chosen instant of time all $n$ points movable independently of one another in a given volume $v_0$ are (by chance) in the volume $v$ ?

For this probability, which is a “statistical probability”, one evidently obtains the value: $W = \left( \frac{v}{v_0} \right)^n ;$ Probability of the configurationVolumeVolumeNumber of independent pointsExplain this equation from this one obtains, by applying Boltzmann’s principle: $S - S_0 = R \left( \frac{n}{N} \right) \lg \left( \frac{v}{v_0} \right).$ EntropyEntropyGas constantNumber of independent pointsAvogadro's numberVolumeVolumeExplain this equation

It is remarkable that in deriving this equation, from which the Boyle-Gay-Lussac law and the law of the same form for osmotic pressure can easily be derived thermodynamically^[1)]; one need make no assumption about the law according to which the molecules move.

- d(E - T S) = p d v = T d S = R \frac{n}{N} \frac{d v}{v} ;

System energyTemperature of the radiationEntropyGas pressureVolumeGas constantNumber of independent pointsAvogadro's numberExplain this equation

p v = R \frac{n}{N} T.

Gas pressureVolumeGas constantNumber of independent pointsAvogadro's numberTemperature of the radiationExplain this equation

Bern, 17 March 1905.

(Received 18 March 1905.)

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On a Heuristic Point of View Concerning the Generation and Transformation of Light

§ 5. Molecular-theoretical investigation of the dependence of the entropy of gases and dilute solutions on the volume.

§ 5. Molecular-theoretical investigation of the dependence of the entropy of gases and dilute solutions on the volume.Show the German source

§ 5. Molecular-theoretical investigation of the dependence of the entropy of gases and dilute solutions on the volume.