Fertilizer application rates are typically given in terms of lbs/acre. When we describe fertilizers we discuss it in terms of ppm. However, it is important to keep in mind the overall mass of fertilizer in smaller water volumes will be less in shorter/smaller tanks or "long" aspect ratio tanks. Even generally rectangular tanks will have disparities of application rates in the range of 20% (60p vs 90p). Therefore, it may become more difficult to maintain "low nutrient" systems in these smaller tanks because your plants will bottom out on nutrients faster when compared to taller tanks.

Thus to maintain the same rate of growth (mass gain) in smaller tanks while avoiding stunting it is better to err on the side of higher ppm of fertilizer.

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At a total loss where this going.
A 20ppm reading of any compound in any volume of water is still 20 ppm.

Of course the tank volume would mean more of a said compound would be required to equal the same ppm.

Hope I said that right?

If you normalize fertilizer rates W.R.T. plantable surface area, there is a large discrepancy between tank sizes. This may be particularly important for nutrients we maintain in low levels i.e. in nitrate limitation, trace nutrient dosing.

For example: in a tall tank that is densely planted, dosing 2ppm nitrates each dose you will probably have sufficient nitrogen. In a tank of equal foot print that is planted to the same coverage but is half as tall, 2ppm nitrate could become insufficient. This effect could be more significant for nutrients we add in the ppb range.

Aquarium depth - length – height

10 gallon
10” – 20” – 12”, planted surface area 200 square inch, 200 stem plants, 10 ppm NO3, 617 mg KNO3, 3 mg KNO3 per plant

125 gallon
18” – 72” – 24”, planted surface area 1296 square inch, 1296 stem plants, 10 ppm NO3, 7715 mg KNO3, 6 mg KNO3 per plant

Clearly, due to the aquarium height and therefore the total water volume, plants have 2 x more available nitrogen in 125 gallon aquarium than in a 10 gallon aquarium when the nitrate water column concentrations are the same, 10 ppm.

Like @Maryland Guppy, I’m struggling to understand the implication. I don’t think we can use surface area in the way that they do in agriculture. I think that lbs/acre is used for application convenience. Soil test reports usually reference ppm and then convert to the quantity needed to adjust. In a tank, particularly with stems, we have a 3D environment, as though in agriculture you could grow everything underground and look for saturation levels in the soil strata.

If you are saying that a shorter tank may have more relative plant mass, because of the third dimension (height), then isn’t ppm still valid? Seems to me that a better approach would be to determine plant mass as a function of available space, but I don’t know how you would do that.

@Edward has a good example of this. Area is a good analog for plant mass in most cases anyways. Nitrates may seem like an unconvincing example because we can test for it. But for traces and infrequently tested nutrients it becomes more interesting.

I have enough going on as it is to bother adding this to the equation!

Then one must consider plant height, if really tall it could be a stem (x2)
Some species are hogs and some others not so much.

I think what OP is trying to say is that aquarium volume vs plant volume matters. I've never heard of this or seen it proven but it is a novel idea. My biggest defense against this mattering is that ppm is ppm. It doesn't matter how big the tank is, the ratio to fertilizers running over the leaves is still the same if the ppm is the same, no matter the tank size.

What could change that? If for some reason the length of time between fertilizers running over plant leaves matters (due to more water between them). I'm not sure what else a larger aquarium volume vs plant volume would effect.

What I completely disagree with is this:
This makes no sense. If there are 3 plants in a 20 gallon and 6 plants in a 40 gallon with fert ppm being the same, total mass (the raw tsps) of the fertilizer floating around likely doesn't matter. Put another way, putting 3 tsps into 20 gallons and 6 tsps into 40 gallons doesn't make the 20 gallon tank any worse off. The ratios remain the same.

NOW, if you have 6 plants in a 20 gallon and 6 plants in a 40 gallon, that will cause a fert requirement difference. So again, I think we need to talk about aquarium mass vs plant mass. I think OP is getting lost a bit.

And finally, if you are reading this and still not following, think of it this way: a gas engine needs gas to run. It sucks down gas at a certain rate to do that. Everytime you add a new plant to your aquarium, you just added another gas engine. More plants = more fertilizers required. Tank size has nothing to do with this unless you mashed a [censored][censored][censored][censored] ton of plants into a 10 gallon so much so that they take up more space than the water. Then we could be talking about a nutrient transfer issue due to flow, etc.

No... why would this matter? The same amount of plants are touching the same amount of fertilizer in both scenarios. What you might mean is that the plants will suck up the nutrient faster in the smaller one. That makes sense.

Imagine a 60f vs 60p planted with glosso. Which tank bottoms out of nutrients first if both are added with 5ppm nitrates? What about 1ppb nickel?

Are you asking if having more glosso in one tank requires more ferts than another one with less glosso? If so, then yes. 5ppm will drain quicker in a tank with more plants than ones with less plants.

Ok so yeah, I think you are confused. You are over complicating trying to state that having more plants requires more ferts.

Neither should bottom out unless you've fallen asleep and missed your dosing schedule completely.

I reckon if one targets water column levels very low this could be an issue.
Full EI users would never see this issue.

My water column NO3 target is 20ppm.
If I were to test my water (2xmonth) and detect 5ppm NO3 I would immediately add 15ppm.

Again, you are missing the point. Say for example you are running lean nitrogen and only testing twice a month. It is much more likely you bottom out in between doses or shortly after dosing in a smaller tank than a larger one. Same goes for micros which we cannot measure at all and add incredibly sparingly.

There is twice as much fertilizer per plant in double height aquarium despite the same water column fertilizer ppm concentration.

I would think the idea would be to not bottom out.
Found my tanks to be predictable for required dosing.
Measuring Fe as an idicator for the other micros is not an issue and reliable.

Testing twice a month just makes sure my weekly dosing is in line.

Are you attempting nitrogen depravation in your tanks?

I get this part!

Or would there really be if stems are tall and branching out?
Stem grows, branches, has several side shoots, is it still a stem count of 1?

I feel like I'm taking crazy pills when I read this thread.

Can we all at least agree that one plant sucks up less ferts than two plants over the course of a week? This also means that 1ppm of nitrate will last longer in a tank with one plant vs one with two plants. In fact, it might even last twice as long.

I believe op is trying to say this. He is referring to carpet plants in a square tank vs a rectangle. All this is saying is that there is more plants in the rectangle tank than the square. Or to wrap this up, the rectangle has more plants consuming more ferts than the square tank. 1ppm will last less as long in the rectangle tank because it has more plants.

More plants = ferts dissapear sooner. Even if they stay at the same ppm after a fert dosing.