Aquarium physics for children: water, light and activities

Bubbles rising from a round aquarium air stone — Bubbles rising from an aquarium air stone.Nimmerwelk · CC BY-SA 4.0 · Resized; uncropped.

Where the physics is

Seven ideas, and where each one is written out.

Read the sentence, then follow the link. The linked pages are written for adults, which is deliberate: they carry the equations and the measured numbers, and you are going to need them for the exercises below.

Gas exchange

A wide puddle breathes faster than a deep bucket holding the same water

Oxygen only gets into water across the top, where the air touches it. So a tank that is wide and shallow takes oxygen in quickly, and a tall narrow one takes it in slowly, even when both hold exactly the same amount of water. How much water there is turns out not to matter at all. Only the average depth does.

The physics of an aquarium does the derivation and prints the time constant. What nobody tells you about plants says the same thing in one line, and easy setups is what happens when you design around it.

Diffusion

A molecule crossing still water is unbelievably slow

Left alone, a dissolved gas spreads by bumping randomly from molecule to molecule. In air that is fast. In water it is roughly ten thousand times slower, and every underwater leaf sits inside a thin still film that the carbon dioxide has to crawl across before the leaf can use it. This is why moving the water helps a plant more than adding more carbon dioxide to it.

Chemistry prints both diffusion coefficients. Materials and water explains what the still film does to a leaf.

Light down through water

Water eats red first, and it does it by multiplying, not subtracting

Every centimeter of water takes the same fraction of the light away, not the same amount, so the light left falls off in the curve that halves and halves again. Red light is taken fastest and blue light slowest, which is why deep water looks blue and why a red plant near the bottom never photographs the way it looks in your hand.

Optics carries the attenuation law, the absorption of pure water color by color, and what happens to it in Lake Sevan. Find a plant has how much light a leaf needs before it breaks even.

Refraction

There is a mirror on the underside of the surface, and a round window in the middle of it

Light bends when it crosses from water into air. Past a certain angle it stops crossing altogether and bounces straight back down, so a fish looking up sees the whole outside world squeezed into a bright circle overhead, and sees the tank reflected everywhere outside that circle. The same bending makes the tank look shallower from above than it really is.

Optics has the critical angle, the circle, and the factor by which a tank looks shallower when you look down into it.

Heat

Water is very hard to warm up, which is why the swing is small and the mean is stubborn

It takes a lot of energy to raise the temperature of water by one degree, so a deep tank barely moves during a day while a shallow one rides up and down with the sun. Evaporation is the other half: every gram of water that leaves the surface carries heat away with it, which is why a tub in a dry Armenian summer stays far cooler than people expect, and why the same water indoors does not.

The physics of an aquarium has the swing equation and the energy balance. Temperature planning and the windowsill build are the same physics with a fan and a window added.

Flow

The middle of the tank is churning and the surface of a leaf is glass-calm

Whether water mixes itself or slides in smooth sheets depends on how fast it moves and how big the thing it moves past is. Out of the filter pipe it is genuinely churning. By the time it reaches a leaf three centimeters across it is sliding smoothly, and around a single moss filament stickiness has won completely. Stirring the middle of a tank does not stir the skin of a leaf.

The physics of an aquarium prints the flow numbers for a pipe, a tank, a leaf and a moss frond, and explains why turnover rate predicts almost nothing.

Pressure and load

A full tank presses on the floor less hard than you do standing up

Pressure is weight divided by the area it stands on. A tank gets heavier as it gets bigger, but its feet get wider at the same time, so the pressure under the cabinet stays almost the same from a small tank to a very large one. The question a floor actually cares about is not crushing but bending, and bending depends on where in the room you put it.

Floor loading and glass thickness has the masses, the pressures, the building code figures and the correction the site had to make to its own earlier arithmetic.

Altitude, which is the Armenian one

Air is thinner here, so our water holds less oxygen than the books say

How much gas water can hold depends on how hard the air above it is pushing. Yerevan stands about 1,113 meters above the sea, Gyumri higher still at about 1,513 m, and Meghri lowest at about 627 m¹. Every dissolved oxygen table printed in an aquarium book is a sea level table, so in Yerevan it is roughly a tenth too generous, and the gap is widest on the hottest days.

The physics of an aquarium carries the correction and the table for all three towns.

A planted tank photographed across the waterline, broad leaves growing out into the air above and the same plants continuing below over dark rock and pale sand, with a neon tetra among them.Tommy Sze · Unsplash License · Resized; uncropped.

How to use them

Context, given, find. In that order, every time.

Each exercise starts with a situation, then lists the numbers you are handed, then asks the question. The numbers in a Given block are the conditions of that problem, the way they are in any school book. Anything else you need, a relation or a measured constant, is on a page this site has already published, and the exercise tells you which one.

Work in meters

Convert first, then compute

Liters are convenient and meters are consistent. A thousand liters is a cubic meter, so 180 liters is 0.18 cubic meters. Doing the conversion before anything else removes most of the mistakes people make in these problems.

What the tag means

The calculator you need

Each exercise is tagged with the buttons it actually needs. Most want nothing but multiplication and division. Five want the e^x and ln keys, and one wants tan. Nothing here needs anything beyond a school calculator.

Check yourself

Several answers are already printed somewhere

A few of these reproduce a number in a table on the linked page. When yours matches to the last digit shown, you did it right. When it does not, the disagreement is worth more than the answer, so write down where you think it came from.

Exercises

Twelve problems about real tanks.

The first few take one step. The later ones take three or four, and one of them asks you to decide something rather than to compute it.

Exercise 1

multiply and divide

Two tubs in the same yard

Anahit and her grandmother set up two water tubs on the balcony in Malatia, on the same morning, with the same water from the same tap. One is a long shallow trough. The other is a tall plastic barrel, cut down. They put exactly the same amount of water in each. By August the fish in the barrel are gasping at the surface in the early morning and the fish in the trough are not, and nobody has fed them differently or put more fish in one than the other.

Water in each container: 180 L
Trough, inside floor: 0.90 m × 0.50 m
Barrel, inside floor: 0.60 m × 0.30 m
Gas transfer velocity, both: k = 0.5 m/day

Find:

the mean depth of the water in each container, in meters;
the time constant of re-oxygenation for each, in hours;
how many times faster the trough refills its oxygen than the barrel does.

The two relations you need, mean depth from volume and surface, and the time constant from mean depth and the gas transfer velocity, are on the physics of an aquarium.

Exercise 2

multiply and subtract

The same tank in two towns

Two cousins keep the same 54 liter tank, bought from the same shop on the same day. One lives in Gyumri and one lives in Meghri. Both run their tanks at 25 degrees, both keep the same fish, and both insist their water is identical. One of them is right about the fish and wrong about the water.

Tank volume, each: 54 L
Water temperature, each: 25 °C
Towns: Gyumri and Meghri
Both tanks: fully saturated with oxygen

Find:

the dissolved oxygen concentration at saturation in each town, in mg/L;
the total mass of dissolved oxygen in each full tank, in milligrams;
how much more oxygen, as a percentage, the Meghri tank holds.

The saturation table corrected for each town's station pressure is on the physics of an aquarium. You need no formula for this one, only the right row and the right two columns.

Exercise 3

multiply and divide

Correcting a sea level book for Yerevan

A borrowed aquarium book from the 1980s prints a dissolved oxygen table and says nothing about altitude, because it was written at sea level for readers at sea level. You want the 20 degree figure for a tank in Yerevan. A friend says just scale it by the ratio of the pressures. That is nearly right, and the reason it is only nearly right is that water vapour in the air above the surface takes up part of the pressure and does not thin out with altitude the way the dry gases do.

Water temperature: 20 °C
Saturation vapour pressure of water at 20 °C: 23.4 hPa²
Standard sea level pressure: 1013.25 hPa
Yerevan station pressure: look it up

Find:

the saturation concentration of dissolved oxygen in Yerevan at 20 °C, in mg/L, using the altitude correction properly;
the answer your friend's simple pressure ratio would have given;
the difference between the two, and say which way the shortcut is wrong.

Then check your first answer against the printed table on the same page. They should agree.

The correction formula, the Yerevan station pressure and the sea level value at 20 °C are all on the physics of an aquarium.

Exercise 4

needs e^x and ln

The morning after a hot night

A shallow tub stands in the shade of a walnut tree in a courtyard in Yerevan. Through a still July night the fish and the bacteria have been using oxygen faster than the surface could replace it, and at sunrise the water is badly down. There is no pump and no air stone. The question the owner actually has is whether to run out and buy one, or whether the morning will fix it by itself.

Water volume: 120 L
Water surface area: 0.75 m²
Water temperature: 25 °C, steady
Where: Yerevan
Gas transfer velocity: k = 0.3 m/day, still and sheltered
Dissolved oxygen at sunrise: 3.6 mg/L

Find:

the mean depth and the time constant, in hours;
how long after sunrise the water reaches 6.0 mg/L, in hours;
the dissolved oxygen concentration twelve hours after sunrise, in mg/L.

The recovery curve, the time constant and the saturation value for Yerevan at 25 °C are on the physics of an aquarium. Rearranging the curve to give a time rather than a concentration is the part you do yourself.

Exercise 5

needs tan

What the fish can see of the room

A child lies on the floor beside a tank and notices that the underside of the water surface is a mirror almost everywhere, except for one bright round patch directly above the fish, through which the ceiling lamp and her own face are visible. She wants to know how big that patch is, and whether the fish can see the whole room through it or only part of it.

Fish depth below the surface: 0.30 m
Tank inside width, front to back: 0.60 m
Water: clean and still

Find:

the radius of the bright circle on the surface, in centimeters;
its diameter;
whether the circle fits inside the tank, and say what the fish sees in the direction where it does not.

The angle that decides where the mirror starts is on optics. The circle's radius is that angle and the depth in a right-angled triangle, which is yours to draw.

Exercise 6

multiply and divide

The slope that looks flatter than it is

Tigran builds a substrate slope, low at the front and high at the back, and works entirely by looking down into the open tank from above, which is how almost everyone does it. When the tank is filled and he looks at it from the front, through the glass, the slope he was so pleased with looks disappointingly flat. He has not made a mistake with the sand. He has made one with the physics.

Real water depth at the front: 36 cm
Real water depth at the back: 48 cm
Viewing: straight down from above, through the water surface

Find:

how deep the water appears at the front and at the back, seen from above;
how large the height difference of the slope appears, in centimeters;
how tall he would have to build the real difference so that it looks like 12 cm from above.

The factor by which water makes a depth look smaller is printed on optics, with the worked example of a 40 cm tank.

Exercise 7

needs e^x and ln

Why the red plant at the bottom is not the red plant at the top

A tall tank has the same red stem plant growing at the surface and at the substrate. Under the same lamp, the one at the bottom photographs brown. The water is clear, there is no tint and no algae bloom, so the only thing between the lamp and the lower leaves is 45 centimeters of clean water. That turns out to be enough.

Depth from surface to substrate: 0.45 m
Water: pure, no color and no particles
Colors to compare: 700 nm deep red and 420 nm blue

Find:

the half depth of each of the two colors, in meters;
the percentage of each color that still reaches the substrate;
how many times more deep red is lost over that 45 cm than blue is.

The attenuation law, the half depth relation and the absorption coefficient of pure water at each wavelength are all on optics.

Exercise 8

needs e^x and ln

The stoneworts that used to grow at nineteen meters

Before the lake was drawn down, meadows of stoneworts grew on the floor of Lake Sevan at nineteen meters, which means enough light was reaching them there to live on. They do not grow there now. The water above them has changed: dissolved organic matter and suspended algae absorb light that used to get through, and the published thermal model of the lake uses a single extinction coefficient to describe it.

Lake: Sevan, present day
Extinction coefficient: the one the published lake model uses — look it up
Depth the stoneworts once grew at: 19 m

Find:

the depth at which one percent of the surface light is left, in meters;
the percentage of surface light reaching 19 m today;
the depth at which half the surface light is left.

The attenuation law and the extinction coefficient for Sevan are both on optics. What the plants did with that light is on the Lake Sevan page.

Exercise 9

needs e^x and ln

Will a leaf on the bottom pay for itself?

A leaf uses energy all day and all night just to stay alive, and it makes energy only while there is light on it. Below a certain brightness it spends more than it earns and slowly starves, however healthy it looks on the day you plant it. This tank has slightly tea-coloured water from a piece of driftwood, which absorbs more light than clean water does, and the owner wants to know whether a carpet plant will hold at the bottom.

Light just under the surface: 120 µmol m⁻² s⁻¹ of PAR
Measured total attenuation coefficient of this water: 1.4 per meter
Depth from surface to substrate: 0.45 m

Find:

the PAR reaching the substrate;
whether that is above or below the measured range in which a submerged leaf just breaks even;
the depth at which the light falls to the bottom of that measured range, and say whether any part of this tank is below it.

Then answer in one sentence: would you put a demanding carpet plant on this substrate, and why?

The attenuation law is on optics. The measured break-even light range for a submerged leaf is on find a plant, and again on materials and water.

Exercise 10

multiply and divide

How far the tub swings in a July day

A wide open tub stands in a yard outside Yerevan through July. The owner has a thermometer in it and is startled by how far the reading travels between dawn and late afternoon. His neighbor has a deeper tub in the same yard, in the same sun, and hers barely moves. Neither of them has done anything clever. The difference is entirely the depth of the water, because the same sunlight has to warm a taller column.

Sunlight taken in by the water: 29.9 MJ per m² over the day
First tub, mean depth: 0.20 m
Second tub, mean depth: 0.60 m
Both: unshaded, same yard, same day

Find:

the daily temperature swing of each tub, in kelvin;
the mean depth that would hold the swing to 4 K or less;
and answer in a sentence: does making the tub deeper make the water cooler, or only steadier?

The swing equation, the fraction of the day's energy it uses and the volumetric heat capacity of water are all on the physics of an aquarium. Your first answer should land on a value that page has already printed for a 0.20 m tub.

Exercise 11

multiply and divide

The neighbor downstairs

A family in a Soviet-era block in Yerevan wants a 240 liter tank in the living room. A relative who once worked in construction tells them the floor is rated for far less than the tank will weigh and that the whole idea is dangerous. The relative has quoted a real number from a real code. The question is whether it is the right number to compare against.

Tank: 240 L
Grand total mass, water, glass, substrate, hardscape and cabinet: 339 kg
Cabinet footprint: 0.60 m²
Room floor area: 12 m²
Gravity: 9.81 m/s²

Find:

the pressure under the cabinet, in kilopascals;
how many times that is the residential live load figure in the code Armenia inherited;
the pressure if the same mass were spread evenly over the whole room, and what percentage of the code figure that is.

Then say, in two sentences, which of those two comparisons is the fair one and why.

The pressure relation, the code figure and the reason the two comparisons differ are on floor loading and glass thickness. Your first and third answers should both match numbers printed there.

Exercise 12

multiply and divide

Three milliseconds, or twenty four seconds

Every underwater leaf wears a thin film of water that will not move, however hard the filter is blowing at the far side of the tank. Carbon dioxide has to cross that film by diffusion alone before the leaf can do anything with it. The same molecule crossing the same distance of still air would arrive almost instantly. That comparison is the whole reason underwater plants are carbon-starved and land plants are not.

Thickness of the still film on the leaf: 0.3 mm
Rough time to diffuse a distance L: t = L² / (2D)³
Diffusing substance: carbon dioxide

Find:

how many times larger the diffusion coefficient of carbon dioxide is in air than in water;
the time to cross 0.3 mm of still water, in seconds;
the time to cross 0.3 mm of still air, in seconds.

Then answer: if the filter doubles the water speed at the leaf and halves the thickness of that film, by what factor does the crossing time fall?

Both diffusion coefficients are printed on chemistry. What the film does to a leaf is on materials and water.

Armenian hydrometeorological service, WMO Climate Normals 1991 to 2020, as submitted to WMO and archived by NOAA NCEI, Region VI Armenia. Station heights and station pressures: Yerevan Arabkir at 1113 m, Gyumri at 1513 m, Meghri at 627 m. The derived station pressures used in the exercises are our own, and are set out on the physics of an aquarium.
Buck, A.L. (1981). New equations for computing vapor pressure and enhancement factor. Journal of Applied Meteorology 20(12): 1527 to 1532. The saturation vapour pressure of water at 20 °C given in Exercise 3. DOI 10.1175/1520-0450(1981)020<1527:NEFCVP>2.0.CO;2
Berg, H.C. (1993). Random Walks in Biology, expanded edition. Princeton University Press. The order-of-magnitude diffusion time given in Exercise 12, t = L²/2D, is the standard one-dimensional random walk result set out there.