Abstract
To brown something faster, one turns up the heat. This paper shows that taking away water works better—and that the effect of temperature is itself not constant. No new theorem or law is claimed.
First, raising the temperature by $10$ K multiplies the rate by $\mathbf{2.3193}$ (Section 2).
Second, applying that rule twice should give $5.3792$, but the truth is $\mathbf{5.1539}$, short by $\mathbf{4.3726}$ per cent (Section 2).
Third, the second ten kelvin buys $\mathbf{2.2222}$, less than the first’s $2.3193$ (Section 2).
Fourth, this is the core of the paper. Dropping the water activity from $1.0$ to $0.65$ multiplies the rate by $\mathbf{11.9334}$ (Section 3).
Fifth, matching that with heat alone would take $\mathbf{29.47}$ K (Section 3).
Sixth, the separator is raising the speed against removing the obstruction (Section 4).
Keywords: water activity, Maillard reaction, browning, Arrhenius equation, food preservation.
Scope (what is not claimed)
No new theorem or law is claimed—the Arrhenius equation, water activity and the moisture dependence of browning are all standard. The reaction mechanism is not treated—which intermediates lead to which pigments is not the subject. No activation energy is asserted—$100$ kJ/mol is representative and runs from $80$ to $160$ kJ/mol across systems; it is not the value of any particular food. The moisture dependence is not asserted as measurement—the peak is placed near $0.65$ to match a known guide; the shape is not derived here. No cooking method is proposed—how one ought to cook is not the subject. Flavour and safety are not discussed—neither the aroma of the products nor the toxicity of anything formed on heating is the subject. Relation to earlier papers: Paper 442 showed that white is made by an index difference rather than a colour—there too the cause lay outside the conspicuous quantity. Paper 300 showed that common root and distinct root can be decided—“faster by heat” and “faster by less water” are distinct roots by that criterion. Paper 434 showed that what makes things cheaper is the count, not the calendar—there too the axis to move was not the one assumed. Paper 416 showed that an equal product need not give an equal result—here too a convenient rule of thumb does not agree with the exact expression. What is added is giving the ten-kelvin factor as $2.3193$, showing that applying it twice does not square, short by $4.3726$ per cent, confirming by inversion that the second step is weaker, tabulating the rate at six water activities, converting the effect of drying into $29.47$ K of heating, and placing the separator at “speed or obstruction”.
1. Notation and the Question
Setting: The rate of a reaction follows the Arrhenius form in temperature.
$$k(T)\propto\exp\!\left(-\frac{E_a}{RT}\right)$$Take $E_a=100$ kJ/mol and $R=8.314$ J/(mol K). The rate of browning is also known to peak with water activity $a_w$ near $0.6$ to $0.7$.
Observation 1.1 To brown something faster, one turns up the heat. Raise the flame—the most obvious move.
Observation 1.2 But wet meat will not brown however hot the pan. Until the water has evaporated the surface stays at $100$ degrees—something other than temperature is acting.
Question: Temperature or water, and by how much.
2. “Double per Ten Kelvin” Cannot Be Applied Twice
Computation 2.1: Take the ratio of rates from $100$ to $110$ degrees.
It is $\mathbf{2.3193}$. Close to the familiar “double per ten kelvin”—consistent with the known guide.
Computation 2.2: Now go from $100$ to $120$ degrees, a rise of $20$ K.
| Quantity | Value |
|---|---|
| The actual ratio | $\mathbf{5.1539}$ |
| “$2.3193$ per ten kelvin” applied twice | $5.3792$ |
| Shortfall | $\mathbf{4.3726}$ % |
Observation 2.3 They do not agree. The truth is $4.3726$ per cent smaller—because the Arrhenius form is exponential in $1/T$, not in $T$.
Computation 2.4: Take the second step alone: $110$ to $120$ degrees.
It is $\mathbf{2.2222}$, less than the first step’s $2.3193$.
Observation 2.5 So it is not “a fixed factor per ten kelvin”. The higher one goes, the less it buys—the same ten kelvin gives a different factor depending on where one starts.
Observation 2.6 This is not an approximation breaking down but two different expressions. $Q_{10}$ is linear in $T$; Arrhenius is exponential in $1/T$—close over a narrow range, and apart once compounded.
3. Taking Water Away Multiplies by 11.9334
Computation 3.1: Vary the water activity and take the relative rate.
| Water activity | Relative rate |
|---|---|
| $0.20$ | $0.0152$ |
| $0.40$ | $0.2749$ |
| $0.60$ | $0.9497$ |
| $0.70$ | $\mathbf{0.9497}$ |
| $0.80$ | $0.6282$ |
| $1.00$ | $\mathbf{0.0796}$ |
Observation 3.2 It peaks. The maximum is between $0.6$ and $0.7$—too much water is slow, and so is too little.
Observation 3.3 It falls on both sides, but unevenly. $0.0796$ on the wet side, $0.0152$ on the dry—because the peak sits off centre.
Computation 3.4: Take the factor gained by going from $1.0$ to $0.65$.
It is $\mathbf{11.9334}$.
Observation 3.5 More than five times the $2.3193$ of a ten-kelvin rise. From a modest reduction of water—more than the flame can do.
Computation 3.6: How much heating would match it?
$\mathbf{29.47}$ K. From $100$ degrees that is $129.5$—unreachable while water remains on the surface.
4. The Separator
The separator: raising the speed, or removing the obstruction.
Observation 4.1 Temperature makes the molecules move faster. The chance of crossing the barrier rises—it acts on the reaction itself.
Observation 4.2 Water acts in two directions. Too much dilutes the reactants and holds the temperature down by evaporating—too little leaves the molecules unable to move.
Observation 4.3 So water has an optimum and temperature does not. Temperature is faster the higher it goes—the shapes differ.
Observation 4.4 These are distinct roots by the criterion of Paper 300. Both are called “speeding it up”—one acts monotonically, one with a peak.
Observation 4.5 And removing water is also raising the temperature. Once evaporation stops, the surface heats—the two are not independent; moving one moves the other.
Observation 4.6 To be honest—$100$ kJ/mol and a peak at $0.65$ are both representative. They move with the system—what does not move is the difference in shape: water has an optimum and temperature does not.
Observation 4.7 One last thing—“double per ten kelvin” is easy to remember and must not be compounded. Two steps are already off by $4.3726$ per cent, three by more—a convenient rule is for use inside its range.
5. Roll Call and Closing
- $2.3193$ per ten kelvin (Computation 2.1): matching the known guide.
- $5.1539$ over twenty (Computation 2.2): not the square.
- The square is $5.3792$ (Computation 2.2): short by $4.3726$ %.
- The second step is $2.2222$ (Computation 2.4): weaker than the first.
- The peak is at $0.6$--$0.7$ (Computation 3.1): falling on both sides.
- Wet $0.0796$, dry $0.0152$ (Observation 3.3): asymmetric.
- $1.0\to0.65$ gives $11.9334$ (Computation 3.4): the core.
- Over five times a ten-kelvin rise (Observation 3.5): more than the flame.
- Matching it takes $29.47$ K (Computation 3.6): unreachable while wet.
- The separator is speed or obstruction (Observation 4.4): distinct roots by Paper 300.
Closing: To brown something faster one turns up the heat, which is the most obvious move—but wet meat will not brown however hot the pan, since until the water has evaporated the surface stays at $100$ degrees. Something other than temperature is acting. First, the effect of temperature is itself not constant—going from $100$ to $110$ degrees multiplies the rate by $\mathbf{2.3193}$, close to the familiar “double per ten kelvin”. But applying that rule twice should give $5.3792$, while the truth over twenty kelvin is $\mathbf{5.1539}$, short by $\mathbf{4.3726}$ per cent. Taking the second step alone gives $\mathbf{2.2222}$, less than the first step’s $2.3193$—because the Arrhenius form is exponential in $1/T$, not in $T$. This is not an approximation breaking down but two different expressions: $Q_{10}$ is linear in $T$ and Arrhenius exponential in $1/T$, close over a narrow range and apart once compounded. And water acts far more strongly—dropping the water activity from $1.0$ to $0.65$ multiplies the rate by $\mathbf{11.9334}$, more than five times what a ten-kelvin rise buys. Matching that with heat alone would take $\mathbf{29.47}$ K, which from $100$ degrees means $129.5$—unreachable while water remains on the surface. The rate peaks with water activity between $0.6$ and $0.7$, falling on both sides but unevenly, $0.0796$ on the wet side against $0.0152$ on the dry, because the peak sits off centre. One thing separates them—raising the speed, or removing the obstruction. Temperature makes the molecules move faster and acts on the reaction itself; water acts in two directions, too much diluting the reactants and holding the temperature down by evaporating, too little leaving the molecules unable to move. So water has an optimum and temperature does not—the shapes differ. They are distinct roots by the criterion of Paper 300, one acting monotonically and one with a peak. And removing water is also raising the temperature, since once evaporation stops the surface heats: the two are not independent. To be honest—$100$ kJ/mol and a peak at $0.65$ are both representative and move with the system. What does not move is the difference in shape. One last thing—“double per ten kelvin” is easy to remember and must not be compounded. Two steps are already off by $\mathbf{4.3726}$ per cent and three by more. A convenient rule is for use inside its range.
References
- Maillard, L. C. (1912). Action des acides aminés sur les sucres. Comptes Rendus.
- Arrhenius, S. (1889). Über die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch Säuren. Zeitschrift für Physikalische Chemie.
- Labuza, T. P. (1980). The Effect of Water Activity on Reaction Kinetics of Food Deterioration. Food Technology.
- Hodge, J. E. (1953). Dehydrated Foods: Chemistry of Browning Reactions in Model Systems. Journal of Agricultural and Food 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). The Product Is the Same and the Result Is Not [Paper 416]. Zenodo. 10.5281/zenodo.22699831
- Yamagishi, Y. (2026). What Makes It Cheaper Is the Count, Not the Calendar [Paper 434]. Zenodo. 10.5281/zenodo.22708735
- Yamagishi, Y. (2026). White Is Made by a Difference, Not by a Colour [Paper 442]. Zenodo. 10.5281/zenodo.22732404
Reference
Yamagishi, Y. (2026). Taking Water Away Beats Turning the Heat Up [Paper 443]. YOI: https://www.yamagishiyuuki.com/yoi/443
DOI: 10.5281/zenodo.22732419
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 443. This paper is a preprint and has not been peer reviewed.