Tag Archives: drugs in waterways

Drugs in waterways: derivative mix

We return to naproxen (sold under the brand name Aleve). Naproxen is my "drug of choice" for these worksheets because it apparently occurs in a lot of our waterways and its decay is pretty well understood. Last time we discussed naproxen in particular, we looked at a function k_{phot}(z) that gives the rate of photolysis for naproxen at a depth z , the rate at which the substance breaks down in the presence of sunlight. There are a few different ways that substances like naproxen, ciprofloxacin (an antibiotic), cocaine, or bisphenol-A get taken out of waterways: breakdown in sunlight, breakdown by organic processes, or sedimentation. Naproxen breaks down easily in sunlight but it doesn't like to be filtered by sand or settle out into sediment even when the water is treated with ferric sulfate to make coagulation happen.

The linked abstract is for a paper about a pilot-scale drinking water purification plant, looking at how water from the River Vantaa could be used for drinking water if the groundwater source for Helsinki, Finland, were to be rendered unusable. Remember that groundwater usage is increasing enormously across the world, and so our nice clean aquifers are overtaxed in many locations. We should not waste so much water (agriculture and lawns, folks!) but will also need to learn a lot about how surface water can be purified so that we can drink it again.

The worksheet below has a mix of derivative and rate of change questions. It asks about some derivatives that require the chain rule (quotient rule and exponential function rule combined) and it also asks students, at the end, to switch variables and look at how the rate of photolysis changes as turbidity changes. After every heavy rain a lot of sediment enters a river and then settles out over time. Development and construction can also change turbidity substantially: digging up a lot of trees and plants to expose dirt allows a lot of that dirt to run off. Agriculture also has its role, as during the planting season fields can be vulnerable to erosion and run-off.

Chain Rule: Photolysis of Naproxen

If you're in a position to work with a science teacher or run experiments yourself, I found a fun page on experiments with turbidity appropriate to junior to senior high school students (and what college student wouldn't mind playing with mud, really?). This could make a cool big brother/big sister activity: high school seniors do the math and the freshmen or junior high students do some experiments on turbidity. In addition, there's a World Water Monitoring project and day (September 18) that you could join.

Drugs in our waterways: the worksheet

As promised, here comes the worksheet for drugs in waterways. As I mentioned yesterday, this was the worksheet hardest to construct so far.

I did a lot of reading on pharmaceuticals in waterways and I liked the paper "Attenuation of Wastewater-Derived Contaminants in an Effluent-Dominated River," from 2006. It discusses the importance of biodegradation and photolysis in breaking down naproxen (brand name Aleve) in the Trinity River, which is full of water from a wastewater treatment plant much of the time. It's got some great graphs and a good set of supplementary calculations and information, which allowed me to pick out the pre-calculus problem inherent in the analysis.

The worksheet is set up as three pages that might be usable somewhat independently: the first page sets up the problem and asks about domain and range, the second asks about limits, and the third asks about piecewise functions. When used in a classroom, you're going to want to know that students will need calculators and could perhaps profitably use graphing software. (A little hand-graphing is good for them, though, as it's so useful later on.)


  • limits,
  • continuity, and
  • piecewise functions.

Worksheet on Photolysis of Naproxen

First page: challenge them to think symbolically about domain and range, and physically. Negative depth in the river doesn't make sense. They don't need to know the values of alpha or k_{surf}, though they'd prefer to!

Second page: I've deliberately given a limit (as z goes to zero) that will not be easy using limit rules (unless there's a trick I haven't seen). It is very easy using L'Hopital's rule or a power series expansion of the exponential so tell students they will soon have tools to do this rigorously. For now, they can use numerical calculation and grapple with the idea of "limit." Students also already have the value at zero, given implicitly in problem 3 by the scientists. We have a removable discontinuity at z=0, always a somewhat challenging concept for students. The limit as z goes to infinity can be done with limit rules.

Third page: Piecewise functions! Some students like 'em, others hate 'em, and many mostly understand but don't know how to write 'em. Coach students through correct 'mathematical grammar' for piecewise functions. There's also a chance to practice writing nice coherent sentences explaining why a function is continuous at the end of the page. At every university or college I've taught at, students have wanted more examples of how to write mathematics -- this is a good chance!

Along the path to this worksheet, I learned why water is blue!

Ok. Off to enjoy some sunlight now. I'm going to wear my mineral sunscreen so that I don't have to worry about avobenzone sloughing off of me into my local waterways... Once limits and continuity are covered, we're only a few steps away from the definition of derivative.

Drugs in our waterways: teaser

This is definitely one of the more complex topics we'll be discussing on the blog. Physics can be so clean -- bodies of rock moving through space, atmospheric gas escaping -- but biology, especially when we start talking about ecosystems, can be so messy! I mean literally as well as figuratively: people who work on measuring the levels of pharmaceuticals in our waterways have to wade through muck, dig through algae, and get in boats!

The presence of pharmaceuticals and endocrine-disrupting compounds in our waterways is an issue that's only becoming more pressing. I'll concentrate on Minnesota because that's where I live. Remember that flap about water bottles containing bisphenol-A? Maybe you switched your water bottle -- but you can't get away from the fact that small quantities can now be found in over 40 percent of Minnesota lakes [1]. Don't take antidepressants at the moment? Well, maybe you'll get a little when you're swimming: venlafaxine (sold as Effexor) is found in 9.4 percent of stream water samples analyzed by the Minnesota Pollution Control Agency in 2010 [2] and amitriptyline is found in almost 30 percent of lakes randomly sampled in 2012 [1]. Disturbingly, cocaine is found in over 30 percent of lakes, too -- what are Minnesotans doing?! [1] And you'd think we'd have fewer mosquitos here given that DEET, the insect repellent, is found in over 70 percent of our lakes! [1]

While the amounts we're talking about are very tiny, they still disrupt fish and mussel life and reproduction. Strangely, fish exposed to some antidepressants are more aggressive predators. Frogs exposed to birth control chemicals can reverse sex. We don't really know what happens to humans exposed to frequent low-level chemical concentrations of this type, although there are disturbing preliminary results relating endocrine disruptors to obesity, diabetes, and endometriosis. We do know that the pharmaceutical industry is hugely important economically, and there is conversation around making the industry greener.

Ok: enough geeking out about the prevalence and importance of drugs in our waterways. Where's the math?

I'm working out a worksheet that will discuss photolysis, the breakdown of chemicals due to light exposure. The rate of photolysis depends on the clearness of the water at a given depth in the photic zone (the zone that light can reach), and this rate constant can be modeled by a

  • composition of a rational function and an exponential function.
  • This function gives a rate, even though we haven't done any differentiation.
  • I'm hoping to work in a page on limits as well.

However, this post is super-long already -- so I'll  stop! The worksheet should be up today or tomorrow (Tuesday).

[1] Pharmaceuticals and Endocrine Active Chemicals in Minnesota Lakes, released May 2013, online at MPCA site on water. Dramatic graph on page 6.

[2] Pharmaceuticals and Personal Care Products in Minnesota’s Rivers and Streams: 2010, released April 2013, online at MPCA site on water. Pages 2-3 discuss venlafaxine.