brownian motion

The equation explained

K \nu - \frac{\partial p}{\partial x} = 0.

In wordsThe force on one particle times the number of particles in unit volume, less how steeply the osmotic pressure changes along the column, is zero.OverviewThe same equilibrium written with the osmotic pressure of the first two sections in it: the force on the particles of unit volume is balanced by the fall of the osmotic pressure across that volume.Full explanationSections 1 and 2 gave the osmotic pressure of the suspended particles as

p = \frac{R T}{N} \nu

. The second term of equation (1) is exactly the derivative of that pressure with respect to

x

, since

R T / N

does not vary along the column, so equation (1) can be written in this form. The sentence under it says what the form is for: the force

K

is balanced by osmotic pressure forces. That is the point at which the argument of the first two sections enters the theory of diffusion. The equilibrium is the familiar one of a fluid in a field of force, with the osmotic pressure of the suspended substance playing the part of the pressure, and it holds only where that pressure law does, which is at great dilution.Every step

p = \frac{R T}{N} \nu

The osmotic pressure of the suspended particles, from sections 1 and 2.

\frac{\partial p}{\partial x} = \frac{R T}{N} \frac{\partial \nu}{\partial x}

Its derivative along the column. The gas constant, the temperature and the number in a gram-molecule are the same everywhere, so only the number density is differentiated.

- K \nu + \frac{\partial p}{\partial x} = 0

Equation (1) with that derivative in place of its second term.

K \nu - \frac{\partial p}{\partial x} = 0

Multiplied by minus one, which is how the paper prints it: the force on the particles of unit volume against the fall of the osmotic pressure.AlsoOverviewFull explanationEvery step

Read it in the paper