The Armour Formula That's Secretly a Chemistry Equation

I have been doing chemical engineering for roughly a decade now, which is long enough for it to become a minor personality defect.

Now ordinary life keeps reminding of chemical engineering whether I ask it or not. Accounting statements looks like mass balance equarions. If the numbers do not close, I want to know which stream somebody forgot.

A few days ago, during the highly productive activity of doom-scrolling Instagram, I came across a reel about armour in a video game. It asked a surprisingly good question:

Should armour protect you equally well from one enormous hit and a hundred tiny ones?

A flat-percentage system would. If armour removes 30% of every hit, then 10,000 total damage loses 3,000 whether it arrives all at once or as a hundred hits of 100.

But some games do something stranger. Armour is excellent against small hits and progressively worse against large ones.

Before I turn a video-game mechanic into an adsorption problem, it is easier to get a feel for what that means.

Try the game mechanic first

One giant hit or lots of small ones?

Total incoming damage: 10,000 · Armour fixed at 18,000.

Split the damage into

100 hits

One big hit

1 × 10,000

2,647 blocked · 26%

7,353 gets through

Same damage, smaller hits

100 × 100

9,000 blocked · 90%

1,000 gets through

Same total damage. Same armour. Split into smaller hits, and the armour can block 3.4× as much.

That is the key behaviour. The armour does not care only about how much damage arrives. It also cares about how that damage is packaged into individual hits.

And this is not an equation I invented to make the chemistry analogy work.

A concrete example is Path of Exile 1. The community-maintained PoE Wiki documents armour reduction against a physical hit as

$$\mathrm{DR}=\min (0.9,\frac{A}{A+5D})$$

where A is armour and D is the incoming hit. The 0.9 is the game’s cap: armour never removes more than 90% of a hit. Armour rules differ between patches and in Path of Exile 2, so treat this as one documented version of the rule, not the final word. PoE Wiki ↗

So the 5 is the game’s number, not mine.

The important distinction: percentage blocked versus damage blocked

The equation above gives the fraction of a hit removed. As the hit gets larger, that fraction goes down.

But the quantity that reminded me of adsorption is different. It is the actual amount of damage prevented:

$$D_{\text{blocked}}=D·\min (0.9,\frac{A}{A+5D})$$

That amount goes up as the hit grows — but it cannot keep going up forever.

With 18,000 armour:

• a 100-damage hit has 90 damage blocked (the 90% cap);
• a 2,000-damage hit has about 1,286 blocked;
• a 100,000-damage hit still has only about 3,475 blocked.

The cap binds for any hit below A/45, which is 400 damage here, so the very-small-hit end of the curve is a straight line at 90% rather than pure Langmuir.

The armour is blocking more damage each time, but it is running into a ceiling.

For this formula, that ceiling is simply

$$\frac{A}{5}$$

so 18,000 armour can never block much more than 3,600 damage from one hit.

At that point the graph started looking like something I had seen far too many times in papers, lectures and group meetings.

Same curve, different nouns

Watch armour run into its ceiling

Armour fixed at 18,000 · per-hit ceiling 3,600

Game armour Adsorption

Incoming hit

2,000

2,000incoming damage

1,286damage blocked

3,600maximum block per hit

36% of the per-hit ceiling reached

Why adsorption does the same thing

A Langmuir adsorption isotherm describes a surface with a finite number of places for molecules to sit. If you have spent any meaningful amount of time around catalysis or surface science, this curve becomes difficult to unsee.

At low pressure, most sites are empty, so adding more gas makes adsorption rise quickly. As pressure increases, the sites fill. Eventually the surface starts running out of real estate and the amount adsorbed approaches a maximum.

Irving Langmuir wrote down the model more than a century ago. Langmuir, 1918 ↗

The usual form is

$$q=q_{\text{max}}\frac{KP}{1+KP}$$

where q is the amount adsorbed, P is pressure, q_max is the maximum capacity of the surface and K describes how strongly the molecule likes the surface.

So the intuitive correspondence is:

Game armour	corresponds to	Adsorption
Hit size	↔	Pressure
Damage blocked	↔	Amount adsorbed
Maximum damage one hit can lose	↔	Maximum surface capacity
Armour runs out of headroom	↔	Surface saturates

That is already enough to see the analogy. The PhD training does, unfortunately, require me to spoil it slightly: hit size is a per-event amount of damage, not an intensive variable like pressure. The correspondence is purely mathematical and has no physical meaning.

The nicer surprise is that this is not just two curves that happen to look vaguely similar after enough coffee. Rearranging the armour expression for actual damage blocked make the structure explicit:

For the nerds: the exact correspondence

In this particular armour model, D ↔ P, while A/5 plays two roles: it is both the maximum amount of damage one hit can lose and the hit size at which the armour reduction is 50%. In Langmuir language those correspond to q_max and 1/K.

They are not equal by accident. Their product, q_maxK = (A/5)(5/A), fixes the initial slope at exactly 1: tiny hits are blocked almost entirely, and armour can never block more than the hit itself. That is the Henry's-law (linear, low-pressure) region, forced here by a physical constraint rather than fitted, and the 90% cap then trims that slope to 0.9 in the game.

That also shows where the analogy stops. In a Langmuir model, capacity and affinity are independent parameters. In this armour equation they are tied together by the single quantity A/5.

The damage-reduction fraction A/(A + 5D) also has a neat analogue: mathematically it corresponds to the fraction of Langmuir sites that are still empty, 1 − θ. The little squares in the interactive are just a visual joke built around that structure; no game is modelling microscopic sites in a breastplate.

What if chemistry designed the armour instead?

So far I have been working backwards: start with a game mechanic, then notice that it looks like adsorption.

The sensible thing to do at this point is stop.

Unfortunately, a PhD trains you to ask one more question.

What happens if we go in the opposite direction? Pick an adsorption isotherm first, rename pressure as armour investment, turn the response into damage reduction, and see what falls out. None of the four systems below belongs to Path of Exile or, as far as I know, to any game studio. They are deliberately imaginary.

Reverse the experiment

Let an adsorption isotherm balance your game

Four imaginary armour systems. Same 90% maximum reduction. Very different incentives to keep stacking armour.

Langmuir Sensible soft cap Sips Build breakpoint Freundlich Needs a hard cap BET Just one more layer

Imaginary armour law R = 0.9 A / (K + A)

Armour investment, A

7,000

Damage reduction 52%

Effective health 2.1×

Next +1,000 armour +2.9 pts

How this would feel to play Always useful, increasingly expensive

Early armour matters immediately, but each extra point buys a little less protection than the one before it.

Langmuir is the well-behaved one. Armour helps immediately, then every additional point helps a little less.

Sips is where the things get interesting. Its exponent is a balance knob: pushing it above one and small armour investments are underwhelming, then the curve wakes up around a breakpoint. Now the system is quietly telling the player: if you are going to invest in armour, please do it properly.

Freundlich is a good reminder that an equation can be perfectly useful in its original context and still become troublesome when you transplant it into a game. The power law has no natural saturation ceiling, so the damage-reduction system needs a 90% cap bolted on from outside. At some point the game designer has to become the boundary condition.

And then there is BET.

Brunauer, Emmett and Teller extended the adsorption picture to multiple molecular layers rather than stopping after one. BET, 1938 ↗ If you have ever sat through a surface-science lecture, you can probably see the problem already.

For the game version, I map armour onto a relative-pressure-like variable, calculate a BET-style loading score, and then squeeze that score back into a bounded 0–90% damage-reduction scale. That second step is pure game-design glue. Without it, BET is doing exactly what it was built to do and has no interest in whether my fictional knight is overpowered.

The result is armour with stacking enthusiasm. In adsorption, molecules can form layers on top of an already adsorbed layer. In this imaginary RPG, the corresponding strategy is apparently to respond to wearing armour by wearing more armour on top of the armour.

At some point you are no longer a knight. You are a packed bed with a sword.

And that is the useful point hiding under the joke. Equations are not neutral once you put them inside a game. The shape of the equation decides whether the player is rewarded for taking a little armour, committing to a breakpoint, riding a power law into a cap, or turning themselves into a unit operation.

Chemical engineering in the wild

The bit I enjoyed most in this activity was not simply that a video game contains an equation that looks like Langmuir adsorption. It was how quickly the shape felt familiar.

Somewhere between studying chemical engineering, I stopped thinking of these equations as belonging only to reactors, catalysts or adsorption experiments. They have became patterns.

That is probably the transferable part of the degree. Chemical engineering is introduced with distillation columns, heat exchangers, reactors and a lot of piping. But after a while it becomes a way of looking at systems.

Apparently that now includes watching a fictional knight get hit with a very large sword on Instagram and thinking, hmm, Langmuir.

There are worse outcomes from graduate school.