Abstract
A fixative in a perfume makes the scent last. It is easily taken that diluting the volatile component slows it in proportion to the dilution. What this paper shows is that even with Raoult’s law for an ideal solution alone, the slowing factor does not stop at the initial rate ratio (the inverse of the mole fraction) but grows as the volatile component evaporates and becomes more dilute. No new theorem or law is claimed.
First, the evaporation rate is proportional to mole fraction times vapour pressure——a volatile component dissolved at mole fraction $0.1$ in a non-volatile fixative starts evaporating at $\mathbf{0.100000}$ of the pure rate (Section 2).
Second, and this is the core. The slowing factor grows with the fraction evaporated——at mole fraction $0.1$, half takes $\mathbf{13.476649}$ times as long as for the pure substance, nine-tenths $\mathbf{24.025851}$ times and $99$ per cent $\mathbf{42.865184}$ times, all beyond the initial factor of $10$ (Section 2).
Third, the closed form agrees with the equation of motion——the time factor to evaporate a fraction $f$ is $1+(F/N_0)\ln(1/(1-f))/f$, agreeing with the numerically solved equation to a relative $1.3\times10^{-11}$ (Section 3).
Fourth, the form is the same for weaker dilution——at mole fraction $0.5$, nine-tenths takes $\mathbf{3.558428}$ times as long, and at $0.2$, $\mathbf{11.233712}$ times, both beyond the initial rate ratios of $2$ and $5$ (Section 2).
Fifth, a fixative also smooths the succession of notes——mixing equal moles of two components whose vapour pressures differ by $10^4$, the heavy component speeds up $\mathbf{2.000000}$-fold once the light one has gone, but only $\mathbf{1.111111}$-fold with eight times as much fixative as each component (Section 4).
Sixth, the separator is whether one looks at the initial rate or at the time to fade——the rate ratio is set by the mole fraction, the time ratio by the fraction evaporated (Section 5).
Keywords: fixative, Raoult’s law, vapour pressure, evaporation, perfume.
Scope (what is not claimed)
No new theorem or law is claimed——Raoult’s law, the proportionality of the evaporation rate to the vapour pressure, and the time obtained by integrating it are all standard. Only ideal solutions are treated——a real fixative holds back the volatile component through intermolecular attraction (activity coefficients), acting more or less strongly than here; this paper sets the activity coefficient to $1$. The evaporation rate coefficient is taken to be constant——gas-phase diffusion, wind and temperature changes are not included. The fixative is taken not to evaporate. The perception of scent is not treated——odour thresholds and olfactory adaptation are not the subject. No perfume is designed, and no substance is named. Relation to earlier papers: Paper 333 showed that the constancy of the ratio of osmotic pressure to freezing-point depression follows from an assumption, not from a measurement, and stated in its scope note that it does not set boiling-point elevation and vapour-pressure lowering side by side——this paper takes that seat and looks at how vapour-pressure lowering (Raoult’s law) slows the evaporation of the volatile component itself. Paper 392 showed that the boiling point is a property not of the substance alone but of the substance and the pressure together——here too, how fast the same substance evaporates depends on what surrounds it, and how much. Paper 300 showed that sharing a root is decidable——the initial rate ratio and the time ratio come from the same differential equation and share a root; they differ in the moment one looks at. What is added is setting out the time factors for three mole fractions and three evaporated fractions, checking the closed form against the equation of motion, comparing the succession of two components with and without fixative, and placing the separator on rate versus time.
1. Notation and the Question
Setting: let $N$ moles of a volatile component be dissolved in $F$ moles of a non-volatile fixative. In an ideal solution the component’s vapour pressure falls in proportion to its mole fraction $x=N/(N+F)$ (Raoult’s law), so with rate coefficient $k$ and pure vapour pressure $p$
$$\frac{dN}{dt}=-kp\,\frac{N}{N+F}.$$The time for a fraction $f$ of the initial $N_0$ to evaporate is $fN_0/(kp)$ for the pure substance and, in the fixative,
$$t=\frac{fN_0+F\ln\bigl(1/(1-f)\bigr)}{kp},\qquad \frac{t}{t_{\mathrm{pure}}}=1+\frac{F}{N_0}\cdot\frac{\ln\bigl(1/(1-f)\bigr)}{f}.$$Observation 1.1 it is easily taken that diluting the volatile component slows it in proportion to the dilution. Dilute it to mole fraction $0.1$ and the evaporation rate falls to a tenth, so the scent lasts ten times as long——whether that holds for the time the scent takes to fade is another question.
Question: by what factor does a fixative lengthen the time a scent takes to fade?
2. One Volatile Component
Computation 2.1: initial mole fraction and the time factor (pure substance $=1$) for each evaporated fraction.
| initial mole fraction | initial rate | half | nine-tenths | $99$ per cent |
|---|---|---|---|---|
| $0.5$ | $0.500000$ | $2.386294$ | $\mathbf{3.558428}$ | $5.651687$ |
| $0.2$ | $0.200000$ | $6.545177$ | $\mathbf{11.233712}$ | $19.606748$ |
| $0.1$ | $\mathbf{0.100000}$ | $\mathbf{13.476649}$ | $\mathbf{24.025851}$ | $\mathbf{42.865184}$ |
Observation 2.2 this is the core. At mole fraction $0.1$ the initial rate is a tenth, but half takes $13.476649$ times as long, nine-tenths $24.025851$ times and $99$ per cent $42.865184$ times——the more of the component evaporates, the lower the mole fraction of what remains, and the slower it goes. A pure substance evaporates at the same rate to the end; a diluted component evaporates less readily the fainter it becomes. The scent weakens but lingers.
Observation 2.3 the form is the same for weaker dilution. At mole fraction $0.5$ the initial rate ratio is $2$, but nine-tenths takes $3.558428$ times as long; at $0.2$, ratio $5$ against $11.233712$——the time factor is the initial rate ratio weighted by $\ln(1/(1-f))/f$, and as $f$ approaches $1$ this weight has no bound.
3. Checking with the Equation of Motion
Computation 3.1: as a route separate from the closed form, $dN/dt=-N/(N+F)$ ($kp=1$, $N_0=1$, $F=9$) was solved by Runge--Kutta and the time at which $N$ falls to $1-f$ was read. The times are $6.738325$ for half, $21.623266$ for nine-tenths and $42.436532$ for $99$ per cent, agreeing with the closed form to a relative $1.3\times10^{-11}$ or better.
Observation 3.2 “slower in proportion to the dilution” is right only for the first instant. The rate ratio concerns the first instant; the time ratio integrates all the dilution that follows——the same differential equation answers $10$ or $24$ depending on the moment one looks at.
4. Two Components
Computation 4.1: equal moles of a light and a heavy component whose vapour pressures differ by $10^4$ were mixed. Once the light component has evaporated, the heavy component’s mole fraction rises, without fixative, from $0.500000$ to $1.000000$, and its evaporation rate grows $\mathbf{2.000000}$-fold. With eight times as much fixative as each component, it rises only from $0.100000$ to $0.111111$, and the rate grows only $\mathbf{1.111111}$-fold.
Observation 4.2 a fixative also smooths the succession of notes. Without fixative, the heavy note strengthens as soon as the light note has gone——with plenty of fixative, every component’s mole fraction is set by the fixative, and the loss of one component barely changes the others.
5. The Separator
Observation 5.1 the separator is whether one looks at the initial rate or at the time to fade. The initial rate ratio is set by the mole fraction alone; the time ratio is set by the evaporated fraction $f$ and grows as $f$ approaches $1$——“how many times longer does a fixative make a scent last” has no single answer until one says how far faded counts as gone.
Observation 5.2 taking the seat left by Paper 333. It showed that the constancy of the ratio of osmotic pressure to freezing-point depression follows from an assumption, and set aside vapour-pressure lowering in its scope note——this paper applies that vapour-pressure lowering to the evaporation of the volatile component itself.
Observation 5.3 beside Paper 392. It showed that the boiling point belongs to the substance and the pressure together——here too, the evaporation rate is set not by the substance alone but by its proportion to the fixative around it.
Observation 5.4 one root, under Paper 300. The initial rate ratio and the time ratio come from the same differential equation——they differ only in the moment one looks at.
Observation 5.5 what is not said. Only ideal solutions are treated; real fixatives act differently through intermolecular attraction, and the perception of scent is not treated. No perfume is designed——what is shown is the form that follows from Raoult’s law alone: a diluted volatile component evaporates less readily the fainter it becomes.
6. Roll Call and Closing
- at mole fraction $0.1$ the initial rate is $\mathbf{0.100000}$ (Computation 2.1): slower in proportion to the dilution.
- half takes $\mathbf{13.476649}$ times as long (Computation 2.1): already beyond $10$.
- nine-tenths takes $\mathbf{24.025851}$ times (Observation 2.2): the core.
- $99$ per cent takes $\mathbf{42.865184}$ times (Computation 2.1): fainter means slower.
- $\mathbf{3.558428}$ at $0.5$ and $\mathbf{11.233712}$ at $0.2$ (Observation 2.3): the same form.
- equation of motion and closed form agree to $1.3\times10^{-11}$ (Computation 3.1): two routes.
- without fixative the heavy note strengthens $\mathbf{2.000000}$-fold (Computation 4.1): after the light note has gone.
- with fixative, only $\mathbf{1.111111}$-fold (Computation 4.1): the succession is smoothed.
- taking the seat left by Paper 333 (Observation 5.2): vapour-pressure lowering.
- the separator is initial rate or time to fade (Observation 5.1): one equation answers $10$ or $24$.
Closing: A fixative in a perfume makes the scent last, and it is easily taken that diluting the volatile component slows it in proportion to the dilution. That is right only for the first instant——dissolved at mole fraction $0.1$ in a non-volatile fixative, the component starts evaporating at $\mathbf{0.100000}$ of the pure rate, ten times slower. But the more of it evaporates, the lower its own mole fraction and the slower it goes, so half takes $\mathbf{13.476649}$ times as long, nine-tenths $\mathbf{24.025851}$ times and $99$ per cent $\mathbf{42.865184}$ times. At mole fractions of $0.5$ and $0.2$, nine-tenths takes $\mathbf{3.558428}$ and $\mathbf{11.233712}$ times, again beyond the initial rate ratios. The closed form agreed with the equation of motion to a relative $1.3\times10^{-11}$. A fixative also smooths the succession of notes——with equal moles of two components whose vapour pressures differ by $10^4$, the heavy note strengthens $\mathbf{2.000000}$-fold once the light one has gone, but only $\mathbf{1.111111}$-fold with fixative. The separator is whether one looks at the initial rate or at the time to fade——the same differential equation answers $10$ or $24$ depending on the moment; under Paper 300 the two share a root. Placed among the earlier papers——this takes the seat for vapour-pressure lowering that Paper 333 left in its scope note; as in Paper 392, the evaporation rate is not set by the substance alone. To be honest——only ideal solutions (activity coefficient $1$) are treated, with a constant rate coefficient and a non-volatile fixative. Real fixatives act differently through intermolecular attraction, and the perception of scent is not treated. No perfume is designed and no substance is named.
References
- Atkins, P., & de Paula, J. (2010). Physical Chemistry (9th ed.). Oxford University Press.
- Prausnitz, J. M., Lichtenthaler, R. N., & de Azevedo, E. G. (1999). Molecular Thermodynamics of Fluid-Phase Equilibria (3rd ed.). Prentice Hall.
- Sell, C. S. (2006). The Chemistry of Fragrances: From Perfumer to Consumer (2nd ed.). Royal Society of Chemistry.
- 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). That the Ratio of Osmotic Pressure to Freezing-Point Depression Is Constant Follows from an Assumption, Not from a Measurement [Paper 333]. Zenodo. 10.5281/zenodo.22681740
- Yamagishi, Y. (2026). The Boiling Point Is Not a Property of the Substance, but of the Substance and the Pressure Together [Paper 392]. Zenodo. 10.5281/zenodo.22699237
Reference
Yamagishi, Y. (2026). A Fixative Slows a Scent, but by a Factor That Is Not the Initial Rate Ratio and Grows as the Scent Fades [Paper 594]. YOI: https://www.yamagishiyuuki.com/yoi/594
DOI: 10.5281/zenodo.23014400
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 594. This paper is a preprint and has not been peer reviewed.