On the Motion of Particles Suspended in Liquids at Rest, as Required by the Molecular-Kinetic Theory of Heat

§ 2. The osmotic pressure from the standpoint of the molecular-kinetic theory of heat.^[1)]Show the German source

If $p_1 p_2 \ldots p_l$ are variables of state of a physical system that completely determine its instantaneous state (for example, the coordinates and velocity components of all the atoms of the system), and if the complete system of the equations of change of these variables of state is given in the formShow the German source $$\frac{\partial p_\nu}{\partial t} = \varphi_\nu(p_1 \ldots p_l) \quad (\nu = 1, 2 \ldots l)$$$p_{\nu }$State variable$t$Elapsed time${\phi }_{\nu }$State variable rate$p_1$State variable$p_l$State variableExplain this equationShow the German source where $\sum \frac{\partial \varphi_\nu}{\partial p_\nu} = 0$, then the entropy of the system is given by the expression:Show the German source $$S = \frac{\bar{E}}{T} + 2 \varkappa \lg \int e^{-\frac{E}{2 \varkappa T}} d p_1 \ldots d p_l.$$$S$Entropy$\bar{E}$System energy$T$Absolute temperature$\varkappa$Boltzmann constant$E$Microstate energy$p_1$State variable$p_l$State variableExplain this equationShow the German source Here $T$ denotes the absolute temperature, $\bar{E}$ the energy of the system, and $E$ the energy as a function of the $p_\nu$.Show the German source The integral is to be extended over all combinations of values of the $p_\nu$ that are compatible with the conditions of the problem.Show the German source $\varkappa$ is connected with the constant $N$ mentioned above by the relation $2 \varkappa N = R$.Show the German source For the free energy $F$ we therefore obtain:Show the German source $$F = -\frac{R}{N} T \lg \int e^{-\frac{E N}{R T}} d p_1 \ldots d p_l = -\frac{R T}{N} \lg B.$$$F$Free energy$R$Gas constant$N$Avogadro's number$T$Absolute temperature$E$Microstate energy$p_1$State variable$p_l$State variable$B$Configuration integralExplain this equationShow the German source

We now imagine a liquid enclosed in the volume $V$; in the partial volume $V^*$ of $V$ let there be $n$ dissolved molecules or suspended bodies, which are held in the volume $V^*$ by a semipermeable wall; the limits of integration of the integral $B$ that occurs in the expressions for $S$ and $F$ are thereby affected.Show the German source Let the total volume of the dissolved molecules or suspended bodies be small compared with $V^*$.Show the German source Let this system be completely described, in the sense of the theory mentioned, by the variables of state $p_1 \ldots p_l$.Show the German source

Even if the molecular picture were fixed down to every detail, the evaluation of the integral $B$ would still present such difficulties that an exact calculation of $F$ could hardly be contemplated.Show the German source Here, however, we need only know how $F$ depends on the size of the volume $V^*$ in which all the dissolved molecules or suspended bodies (called “particles” for short in what follows) are contained.Show the German source

We call $x_1, y_1, z_1$ the rectangular coordinates of the center of gravity of the first particle, $x_2, y_2, z_2$ those of the second, etc., $x_n, y_n, z_n$ those of the last particle, and we assign to the centers of gravity of the particles the infinitely small parallelepiped-shaped regions $d x_1 d y_1 d z_1, d x_2 d y_2 d z_2 \ldots d x_n d y_n d z_n$, all of which are to lie in $V^*$.Show the German source Let the value be sought of the integral occurring in the expression for $F$, with the restriction that the centers of gravity of the particles lie in the regions just assigned to them.Show the German source This integral can in any case be brought into the formShow the German source $$d B = d x_1 d y_1 \ldots d z_n \cdot J$$$B$Configuration integral$x_1$Latent position (one dimension)$y_1$Latent position (one dimension)$z_n$Latent position (one dimension)$J$Configuration integral factorExplain this equationShow the German source where $J$ is independent of $d x_1 d y_1$ etc., and of $V^*$, that is, of the position of the semipermeable wall.Show the German source But $J$ is also independent of the particular choice of the positions of the regions for the centers of gravity, and of the value of $V^*$, as is now to be shown.Show the German source For let a second system of infinitely small regions for the centers of gravity of the particles be given, denoted by $d x'_1 d y'_1 d z'_2, d x'_2 d y'_2 d z'_2 \ldots d x'_n d y'_n d z'_n$, regions that are to differ from those originally given only in their position and not in their size, and that are likewise all to be contained in $V^*$; then, analogously,Show the German source $$d B' = d x'_1 d y'_1 \ldots d z'_n \cdot J',$$$B^{\prime }$Configuration integral$x_1^{\prime }$Latent position (one dimension)$y_1^{\prime }$Latent position (one dimension)$z_n^{\prime }$Latent position (one dimension)$J^{\prime }$Configuration integral factorExplain this equationShow the German source whereShow the German source $$d x_1 d y_1 \ldots d z_n = d x'_1 d y'_1 \ldots d z'_n.$$$x_1$Latent position (one dimension)$y_1$Latent position (one dimension)$z_n$Latent position (one dimension)$x_1^{\prime }$Latent position (one dimension)$y_1^{\prime }$Latent position (one dimension)$z_n^{\prime }$Latent position (one dimension)Explain this equationShow the German source We therefore have:Show the German source $$\frac{d B}{d B'} = \frac{J}{J'}.$$$B$Configuration integral$B^{\prime }$Configuration integral$J$Configuration integral factor$J^{\prime }$Configuration integral factorExplain this equationShow the German source

From the molecular theory of heat given in the papers cited, however, it can easily be inferred^[1)] that $d B/B$ (respectively $d B'/B$) is equal to the probability that at an arbitrarily chosen instant the centers of gravity of the particles lie in the regions $(d x_1 \ldots d z_n)$ (respectively in the regions $(d x'_1 \ldots d z'_n)$).Show the German source Now if the motions of the individual particles are independent of one another (to a sufficient approximation), if the liquid is homogeneous, and if no forces act on the particles, then for regions of equal size the probabilities belonging to the two systems of regions must be equal to each other, so that:Show the German source $$\frac{d B}{B} = \frac{d B'}{B}.$$$B$Configuration integral$B^{\prime }$Configuration integralExplain this equationShow the German source But from this equation and the one found last it follows thatShow the German source $$J = J'.$$$J$Configuration integral factor$J^{\prime }$Configuration integral factorExplain this equationShow the German source

It is thus proved that $J$ depends neither on $V^*$ nor on $x_1, y_1 \ldots z_n$.Show the German source By integration one obtainsShow the German source $$B = \int J d x_1 \ldots d z_n = J V^{*n}$$$B$Configuration integral$J$Configuration integral factor$x_1$Latent position (one dimension)$z_n$Latent position (one dimension)$V$Volume$n$Particle countExplain this equationShow the German source and from thisShow the German source $$F = -\frac{R T}{N} \{ \lg J + n \lg V^* \}$$$F$Free energy$R$Gas constant$T$Absolute temperature$N$Avogadro's number$J$Configuration integral factor$n$Particle count$V^{\ast }$VolumeExplain this equationShow the German source andShow the German source $$p = -\frac{\partial F}{\partial V^*} = \frac{R T}{V^*} \frac{n}{N} = \frac{R T}{N} \nu.$$$p$Osmotic pressure$F$Free energy$V^{\ast }$Volume$R$Gas constant$T$Absolute temperature$n$Particle count$N$Avogadro's number$\nu$Number densityExplain this equationShow the German source

This consideration shows that the existence of osmotic pressure is a consequence of the molecular-kinetic theory of heat, and that according to this theory dissolved molecules and suspended bodies present in equal numbers behave in exactly the same way with respect to osmotic pressure at great dilution.Show the German source

Bern, May 1905.Show the German source

(Received 11 May 1905.)Show the German source