Abstract
When a drug is called “poorly soluble”, two different quantities are being called by one name. This paper shows that one of them can be moved by a factor of a thousand while the other does not move at all——milling does not change the solubility. No new theorem or law is claimed.
First, at fixed mass the surface area is $A=6m/(\rho d)$, inversely proportional to particle size (Section 2).
Second, taking $100$ mg from $1$ mm to $1$ $\mu$m raises the area from $5.0\times10^{-4}$ to $\mathbf{5.0\times10^{-1}}$ m$^2$, exactly $\mathbf{1000}$ times (Section 2).
Third, and this is the core of the paper. The time to $90$ per cent falls from $\mathbf{49.34}$ hours to $\mathbf{2.96}$ minutes, and the solubility does not move at all (Section 3).
Fourth, the number of particles goes from $159$ to $\mathbf{1.591549\times10^{11}}$, exactly $\mathbf{1000}$ times per decade of size (Section 2).
Fifth, the speed is set by geometry and the solubility by thermodynamics; a mill touches only the former (Section 4).
Sixth, the separator is “how much dissolves at equilibrium” versus “how long it takes to get there” (Section 5).
Scope (what is not claimed)
No new theorem or law is claimed——the Noyes--Whitney equation, the relation between specific area and particle size, and the improvement of dissolution by milling are all known in pharmaceutics. No drug is designed——no formulation or dose is given. Only the independence of two quantities is examined, in one model. The organism is not treated——absorption, distribution, metabolism and excretion are not the subject. Only dissolution in a vessel is computed. The surface area is held constant——particles shrink as they dissolve, and this model ignores that. The absolute times are approximate; the ratios are the robust part. The Ostwald--Freundlich effect is not treated——for nanoparticles the solubility itself rises slightly, but at $1$ $\mu$m that correction is not the subject. Polymorphism is not treated——a change of crystal form does move the solubility; this paper mills within one crystal form. Relation to earlier papers: Paper 369 showed that pasteurisation is a dose, not a temperature——there too one phrase named two things. Paper 384 showed that free-space loss is not a loss——here “poor solubility” is not solubility. Paper 300 showed that same-root and different-root can be decided——solubility and dissolution rate are different-root by that criterion. Paper 362 showed that lifting the stones is not the bottleneck——here the solubility is not the bottleneck. What is added is putting numbers on area, count and time at four particle sizes, showing that the area changes by exactly $1000$, tracking the time to $90$ per cent from $49.34$ hours to $2.96$ minutes, confirming that the count changes by exactly $1000$ per decade of size, and placing the separator at “equilibrium amount or arrival time”.
1. Notation and the Question
Setting: divide a mass $m$ of a substance of density $\rho$ evenly into spheres of diameter $d$.
$$A=\frac{6m}{\rho d},\qquad N=\frac{6m}{\rho\pi d^{3}}$$Let dissolution follow the Noyes--Whitney equation.
$$\frac{dC}{dt}=\frac{DA}{hV}\left(C_{s}-C\right)$$$D$ is the diffusion coefficient, $h$ the boundary-layer thickness, $V$ the volume of medium, $C_{s}$ the solubility.
Observation 1.1 $C_{s}$ appears only inside the bracket. The prefactor of the rate is $DA/(hV)$——$C_{s}$ decides how far it goes and does not enter how fast.
Observation 1.2 $A$ moves with particle size. At the same mass, finer means more——$C_{s}$ does not move with particle size.
Observation 1.3 so the two can be moved independently. Milling touches only $A$——that independence is what the numbers show.
Question: when we say “poorly soluble”, what is it that is small?
2. The Area Grows a Thousandfold and the Count a Billionfold
Computation 2.1: surface area and particle count for $\rho=1200$ kg/m$^3$ and $m=100$ mg (computed in this paper).
| Size $d$ | Area $A$ [m$^2$] | Count $N$ |
|---|---|---|
| $1000$ $\mu$m | $5.000000\times10^{-4}$ | $1.591549\times10^{2}$ |
| $100$ $\mu$m | $5.000000\times10^{-3}$ | $1.591549\times10^{5}$ |
| $10$ $\mu$m | $5.000000\times10^{-2}$ | $1.591549\times10^{8}$ |
| $\mathbf{1}$ $\mu$m | $\mathbf{5.000000\times10^{-1}}$ | $\mathbf{1.591549\times10^{11}}$ |
Observation 2.2 the area changes by exactly $1000$ from top to bottom. From $5.0\times10^{-4}$ to $5.0\times10^{-1}$ m$^2$——$A\propto1/d$, so a thousandth of the size is a thousand times the area.
Observation 2.3 the count changes by exactly $1000$ per decade of size. From $159$ to $1.59\times10^{11}$——$N\propto1/d^{3}$.
Observation 2.4 $100$ mg of powder carries $0.5$ m$^2$ of surface. About eight sheets of A4——at one tenth of a gram.
Observation 2.5 every row is the same substance in the same amount. Nothing has changed chemically——only the way it is divided.
3. Forty-Nine Hours Becomes Three Minutes, and the Solubility Has Not Moved
Computation 3.1: dissolution for $D=7\times10^{-10}$ m$^2$/s, $h=30$ $\mu$m, $V=900$ mL (computed in this paper).
| Size $d$ | Time constant $\tau$ [s] | Time to $90$ % | Solubility $C_{s}$ |
|---|---|---|---|
| $1000$ $\mu$m | $77142.86$ | $\mathbf{49.34}$ hours | unchanged |
| $100$ $\mu$m | $7714.29$ | $4.93$ hours | unchanged |
| $10$ $\mu$m | $771.43$ | $29.60$ minutes | unchanged |
| $\mathbf{1}$ $\mu$m | $\mathbf{77.14}$ | $\mathbf{2.96}$ minutes | unchanged |
Observation 3.2 this is the core of the paper. The last column reads “unchanged” in all four rows——the time to $90$ per cent moves by a factor of a thousand, from $49.34$ hours to $2.96$ minutes, and the solubility does not move at all.
Observation 3.3 the final endpoint is also the same in all four rows. Wait long enough and the same amount dissolves——only the waiting differs.
Observation 3.4 $49.34$ hours is, in practice, “does not dissolve”. Longer than gastrointestinal transit——$2.96$ minutes is “dissolves at once”. The same drug becomes both, by way of a mill.
Observation 3.5 $\ln 10=2.302585$ is common to all four rows. $t_{90}=\tau\ln10$——the shape is identical and only the scale differs.
Observation 3.6 the shape is that of Paper 362. There lifting the stones was not the bottleneck——here the solubility is not the bottleneck. In both, the thing that carries the name is not the thing that acts.
4. A Mill Does Not Touch Thermodynamics
Observation 4.1 $C_{s}$ is set by substance, solvent and temperature. A thermodynamic equilibrium amount——particle size does not enter its definition.
Observation 4.2 $A$ is set by shape. A geometric quantity——what the substance is enters $A=6m/(\rho d)$ only as the density.
Observation 4.3 they are different-root by the criterion of Paper 300. One is set by thermodynamics, the other by geometry——one phrase, “poorly soluble”, names two things with different roots.
Observation 4.4 which one is at fault decides the tool. For solubility: make a salt, change the crystal form, use a solubiliser——for speed: mill. Misdiagnose and you pick a tool that cannot work.
Observation 4.5 at the extreme, however, they mix. For nanoparticles the Ostwald--Freundlich effect raises the solubility itself——small at $1$ $\mu$m, not negligible at $100$ nm. The separator breaks if the size is pushed down too far.
5. Separator: Equilibrium Amount, or Arrival Time
Observation 5.1 separator——does it fail to dissolve even given time, or does it dissolve given time but not in time.
| What is small | Set by | Moves on milling | Remedy |
|---|---|---|---|
| Solubility $C_{s}$ | thermodynamics | does not move | salt, polymorph, solubiliser |
| Dissolution rate | geometry ($A$) | moves $1000\times$ | milling |
Observation 5.2 the third column is the test. Does the dissolution curve move when the sample is milled——if it does not, the bottleneck is the solubility.
Observation 5.3 this separator can be run as an experiment. Mill the same sample and measure——the result falls into one of two cases.
Observation 5.4 the separator can be moved. Increase the medium until sink conditions hold and the ceiling set by solubility disappears——what remains is speed alone.
Observation 5.5 to be honest——both $49.34$ hours and $2.96$ minutes come from a model holding the area constant. Particles do shrink as they dissolve, so the late stage is slower——what does not move is that the ratio is exactly $1000$ and that $C_{s}$ is the same in all four rows.
Observation 5.6 one last thing——“poorly soluble” folds a statement about amount and a statement about time into one word. Folded, it hides which tool to reach for——one question unfolds it: what happens if you wait long enough.
6. Roll Call and Closing
- $A=6m/(\rho d)$ (Section 1): inversely proportional to size.
- $5.0\times10^{-4}$ to $5.0\times10^{-1}$ m$^2$ (Computation 2.1): exactly $1000$ times.
- $159$ to $1.591549\times10^{11}$ particles (Computation 2.1): $1000$ per decade of size.
- $100$ mg carries $0.5$ m$^2$ (Observation 2.4): about eight sheets of A4.
- $90$ % in $49.34$ hours or $2.96$ minutes (Computation 3.1): a factor of $1000$.
- Solubility unchanged in all four rows (Observation 3.2): it does not move at all.
- The endpoint is also the same (Observation 3.3): only the waiting differs.
- $t_{90}=\tau\ln10$ (Observation 3.5): $\ln10=2.302585$ throughout.
- Different-root (Observation 4.3): thermodynamics and geometry.
- They mix in the nanometre range (Observation 4.5): the separator breaks if pushed too far.
Closing: when a drug is called “poorly soluble”, two different quantities are being called by one name——how much dissolves, and how long it takes to get there. In the Noyes--Whitney equation the solubility $C_{s}$ appears only inside the bracket——$dC/dt=DA/(hV) (C_{s}-C)$, so $C_{s}$ decides how far it goes and does not enter how fast. The prefactor is $DA/(hV)$, and the only part of it that moves with particle size is $A$. At fixed mass the area is $A=6m/(\rho d)$, inversely proportional to size——taking $100$ mg of a substance of density $1200$ kg/m$^3$ from $1$ mm to $1$ $\mu$m raises the area from $5.0\times10^{-4}$ to $\mathbf{5.0\times10^{-1}}$ m$^2$, exactly $\mathbf{1000}$ times, and takes the count from $159$ to $\mathbf{1.591549\times10^{11}}$, exactly $1000$ per decade of size. One hundred milligrams of powder then carries about eight sheets of A4 in surface. Every row is the same substance in the same amount; only the way it is divided has changed. And the time to $90$ per cent falls from $\mathbf{49.34}$ hours to $\mathbf{2.96}$ minutes——a factor of a thousand. Yet the solubility does not move at all across the four sizes, and given enough time the final endpoint is identical. Only the waiting differs. Forty-nine hours is longer than gastrointestinal transit and so is, in practice, “does not dissolve”; three minutes is “dissolves at once”——the same drug becomes both, by way of a mill. A mill does not touch thermodynamics——$C_{s}$ is an equilibrium amount set by substance, solvent and temperature, and particle size does not enter its definition. $A$ is geometric, and what the substance is enters only as the density. They are different-root by the criterion of Paper 300, and one phrase names two things with different roots. As in Paper 362, where lifting the stones was not the bottleneck, the thing that carries the name is not the thing that acts. One thing does the separating——does it fail to dissolve even given time, or does it dissolve given time but not in time. The test is whether the dissolution curve moves when the sample is milled; if it does not, the bottleneck is the solubility. Which one is at fault decides the tool——salt, polymorph or solubiliser for solubility, milling for speed. Misdiagnose and you pick a tool that cannot work. To be honest——both $49.34$ hours and $2.96$ minutes come from a model holding the area constant, and particles do shrink as they dissolve, so the late stage is slower. What does not move is that the ratio is exactly $1000$ and that $C_{s}$ is the same in all four rows. The separator also breaks if the size is pushed down too far——in the nanometre range the Ostwald--Freundlich effect raises the solubility itself. One last thing——“poorly soluble” folds a statement about amount and a statement about time into one word. Folded, it hides which tool to reach for. One question unfolds it: what happens if you wait long enough.
References
- Noyes, A. A., Whitney, W. R. (1897). The Rate of Solution of Solid Substances in Their Own Solutions. Journal of the American Chemical Society.
- Dokoumetzidis, A., Macheras, P. (2006). A century of dissolution research: From Noyes and Whitney to the Biopharmaceutics Classification System. International Journal of Pharmaceutics.
- Amidon, G. L., Lennernäs, H., Shah, V. P., Crison, J. R. (1995). A Theoretical Basis for a Biopharmaceutic Drug Classification. Pharmaceutical Research.
- Aulton, M. E., Taylor, K. M. G. (eds.) (2017). Aulton’s Pharmaceutics (5th ed.). Edinburgh: Elsevier.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Lifting the Stones of a Pyramid Is Not the Bottleneck [Paper 362]. Zenodo. 10.5281/zenodo.22697498
- Yamagishi, Y. (2026). Pasteurisation Is a Pair of Temperature and Time, Not a Temperature [Paper 369]. Zenodo. 10.5281/zenodo.22697760
- Yamagishi, Y. (2026). “Free-Space Loss” Is Not a Loss: The Vacuum Absorbs Nothing [Paper 384]. Zenodo. 10.5281/zenodo.22699073
Reference
Yamagishi, Y. (2026). The Solubility Did Not Move at All, and the Dissolution Became a Thousand Times Faster [Paper 386]. YOI: https://www.yamagishiyuuki.com/yoi/386
DOI: 10.5281/zenodo.22699119
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 386. This paper is a preprint and has not been peer reviewed.