Random Events

Double or Nothing

Thu, 06 Aug 2026 00:00:00 +0000

Consider an elementary reaction in which a reactant $\mathrm{A}$ reacts with itself to produce a product $\mathrm{P}$:

$$ \mathrm{A} + \mathrm{A} \xrightarrow{} \mathrm{P} $$

with rate coefficient $k$. The rate of consumption of $\mathrm{A}$ is then:

$$ - \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{d}t} = 2 k [\mathrm{A}]^2 $$

Now suppose we add another reactant $\mathrm{B}$, with similar reactivity to $\mathrm{A}$. In this new scenario, $\mathrm{A}$ can also react with B:

$$ \mathrm{A} + \mathrm{B} \xrightarrow{} \mathrm{R} $$

Since this additional reaction step consumes a single $\mathrm{A}$ molecule, and since $\mathrm{A}$ and $\mathrm{B}$ have similar reactivity, one might be tempted to write:

Waves Under Tension

Wed, 05 Aug 2026 00:00:00 +0000

Knowing very little about music, acoustics, or wave mechanics, I couldn’t help being impressed by this Christmas Lectures video, especially the Chladni figures.

The urge to calculate those patterns was almost immediate. The problem was that, although I was already somewhat familiar with hyperbolic conservation equations, waves are a rather different beast. So I had to start from the very beginning.

The simplest wave model is the one-dimensional wave equation:

$$ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} $$

This equation describes, for example, the vibration of a guitar string, the propagation of seismic waves during an earthquake, or the pressure fluctuations that allow us to enjoy our favorite song.

A Diacid Paradox

Sun, 26 Jul 2026 00:00:00 +0000

Suppose we have a weak mono-acid $\mathrm{HA}$ and a diacid $\mathrm{H_2D}$ that is, in all chemical respects, similar to $\mathrm{HA}$, except that it has two acid groups so far apart that they do not interact. How do the acid dissociation constants of $\mathrm{H_2D}$ relate to that of $\mathrm{HA}$?

A mono carboxylic acid and a chemically similar di-acid.

Certainly, since the two acid groups are identical, we might expect:

The Case of the 3-Hole Can

Sat, 25 Jul 2026 00:00:00 +0000

The other day, I saw this video from Prof. Julius Sumner Miller and was immediately enchanted by the question:

How does the water come out of a tall can in which three holes reside?

I knew how the efflux velocity depends on the height of the liquid, but I didn’t have a clue about the distance it would travel.

So, here is the solution.

The path of the water from a tall can.