= Avogadro constant from sedimentation and diffusion
{c}
{title2=$N_A^{\rm sed}=RTb/(m_bg),\quad N_A^{\rm diff}=RT/(6\pi\eta aD)$}
For two observed layers $z_2>z_1$, define the positive logarithmic count <slope>
$$
b=\frac{\ln[\overline C(z_1)/\overline C(z_2)]}{z_2-z_1}=\frac1H.
$$
Since $k_B=R/N_A$, where $R$ is the <molar gas constant>, the <sedimentation scale height> gives
$$
N_A^{\rm sed}=\frac{RTb}{m_bg}=\frac{RT}{m_bgH}.
$$
Several heights check whether one <slope> fits the whole profile. A mixture of grain radii generally gives a sum of <exponential functions>, with the larger grains depleted faster aloft; one apparent radius need not describe every height.
An independent readout uses horizontal displacements at lag $\tau$, avoiding the vertical gravitational <drift>. Define the one-coordinate <mean squared displacement> $M_x(\tau)=\langle[x(t+\tau)-x(t)]^2\rangle$. In a stationary <liquid> without horizontal <advection>, <translational diffusion> predicts
$$
M_x(\tau)=2D\tau,\qquad
\langle(\Delta x)^2+(\Delta y)^2\rangle=4D\tau.
$$
A camera sees two horizontal coordinates, so its squared radial displacement has twice the one-coordinate mean. Neither observable is path length or instantaneous <velocity>. The bulk <Stokes–Einstein relation>, with <dynamic viscosity> $\eta$, then gives
$$
D=\frac{k_BT}{6\pi\eta a},\qquad
N_A^{\rm diff}=\frac{RT}{6\pi\eta aD}.
$$
The 1926 lecture compares displacement-based and equilibrium determinations of the <Avogadro constant>.<Nobel Lectures, Physics 1922–1941/Discontinuous Structure of Matter by Jean Baptiste Perrin>[Nobel Lectures, Physics 1922–1941; Discontinuous structure of matter, pp. 152–154]{ref}
<Rotational diffusion> supplies a further observable when an identifiable grain orientation can be followed; centre positions alone cannot measure it.
For the chosen model above and $\eta=1.00\times10^{-3}\,\mathrm{Pa\,s}$, the bulk prediction is $D\simeq0.8731\,\mathrm{\mu m^2\,s^{-1}}$. At $\tau=10\,\mathrm s$, $M_x\simeq17.46\,\mathrm{\mu m^2}$ and the planar value is $34.92\,\mathrm{\mu m^2}$. Reconstructing $N_A$ from these stipulated model outputs returns $6.02214076\times10^{23}\,\mathrm{mol^{-1}}$ by construction; this arithmetic consistency check is not experimental confirmation. Historical force comes from comparing separately measured profiles and displacements using independently checked material parameters.
The <diffusive flux> and gravitational particle flux provide another useful connection. With constant drag coefficient $\zeta$, zero vertical net flux reads
$$
j_z=-D\frac{dn}{dz}-\frac{m_bg}{\zeta}n=0,\qquad
H=\frac{D\zeta}{m_bg}=\frac{D}{v_s},\qquad
v_s=\frac{m_bg}{\zeta},
$$
where $v_s$ is the downward <settling velocity> magnitude. With $\zeta=6\pi\eta a$, the chosen model gives $v_s\simeq0.02725\,\mathrm{\mu m\,s^{-1}}$. The <equilibrium> profile forgets how rapidly it was established, while displacement measurements retain that rate information.
\Image[/perrin-brownian-displacement-diffusion.png]
{title=Independent horizontal displacement readout for the chosen Brownian model}
{description=Three simulated planar tracks illustrate sampled endpoints. The other panel compares exact one-coordinate and planar <mean squared displacement> laws with separate simulated ensembles at each lag. These model observations do not reuse the height counts and are not historical measurements. All use $D\simeq0.8731\,\mathrm{\mu m^2\,s^{-1}}$.}
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