Physics of the Everyday
# In the House

Breaking news!—The Baltimore Sun, CNN, Fox, and others are reporting that Baltimore's streams are tainted by amphetamine. Experts blame a leaky sewer system; reporters infer illicit users are dumping drugs down the toilet. In its article, "Your Drain On Drugs", CNN reports that *six* Baltimore County streams tested positive for amphetamine.

None of the media reports cite the concentration of amphetamine the researchers measured, though the abstract of the research paper states concentrations were in the range \(3\ \textrm{to}\ 630 \si[per-mode=symbol]{\nano\gram\per\liter}\).

Medicinal amphetamines are prescribed in capsules containing milligrams—not nanograms—so how do we put this measurement in context?

As we answer this question and discuss this eye-popping story, we will introduce some of the quantitative tools and structures used to communicate measurements precisely, including rates, ratios, percents, proportions, and units of measurement.

To understand the severity of Baltimore's water problem, we first look at the **units** of the measured amphetamine concentrations. Units are agreed-upon reference measurements that are standardized across measurement devices.

For example, watches measure time by ticking at a rate of once per second. When you use a watch to measure the time your toaster oven takes to perfectly toast a slice of bread, you can report your measurement as, for example, \(60\) seconds.

If your friend's watch is poorly designed and ticks slower than once per second, what happens when he tries to reproduce your perfect slice in his identical toaster?

The unit that the Baltimore water researchers chose for reporting their measurement is nanograms per liter: \(\si[per-mode=symbol]{\nano\gram\per\liter}\). This **rate** tells us that a \(\SI{1}{\liter}\) volume of water contains \(\SI{630}{\nano\gram}\) of amphetamine.

Volume is a measure of the space contained within a boundary, like the space within a pitcher or inside of a hot-air balloon. There are several different units of volume to choose from. Which of the following is **not a unit of volume**?

Each liter of Baltimore water sampled contained some amphetamine. The researchers reported the mass of amphetamine in nanograms (\(\si{\nano\gram}\)). They likely decided that \(\si{\nano\gram}\)—which are exactly one billion times smaller than grams—were most appropriate due to the small mass of amphetamine in each water sample they analyzed. Choosing \(\si{\nano\gram}\) allowed them to express their result in a form that is **easy to say and read**: \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\).

Had they reported the measurement in \(\si[per-mode=symbol]{\gram\per\liter},\) the magnitude of the measurement would be different. What is \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\) in \(\si[per-mode=symbol]{\gram\per\liter}?\)

The units of a measurement or an estimate play a role in how a number is communicated. With this in mind, let's try to estimate the amount of amphetamine that is consumed in Baltimore on an average day.

A therapeutic dose of amphetamine is \(\SI{10}{\milli\gram}.\) If \(15\%\) of the population takes a daily \(\SI{10}{\milli\gram}\) average dose of amphetamine, how much amphetamine is taken each day in Baltimore, a city of \(700000\) people?

Now that you've estimated how much amphetamine in \(\si{\milli\gram}\) is consumed in Baltimore on a particular day, you can decide on whether \(\si{\milli\gram}\) is an appropriate unit for reporting your estimate, or whether using a different unit would make your estimate easier to communicate.

Which of these units is most scientifically appropriate for expressing the amount of amphetamine consumed in Baltimore on a particular day?

**Details & Assumptions**

- Assume you want to express the amount with the least number of \(0\)'s.
- The following conversion table may be useful: \[\begin{array}{c|c} \text{milligram} & \SI{10^{-3}}{\gram} \\ \hline \text{kilogram} & \SI{10^{3}}{\gram} \\ \hline \text{nanogram} & \SI{10^{-9}}{\gram} \\ \hline \text{gigagram} & \SI{10^{9}}{\gram} \\ \end{array}\]

There is a drawback to having so many units to choose from when expressing a measurement: we only have good intuition about how big or small a measurement is for a few of them.

For example, most people can list some objects that are about \(\SI{1}{\kilo\gram}\) (a melon, a bottle of water, a jar of pickles) but not many people can list something that's about \(\SI{1}{ng}.\) Figuring out how to describe a measurement like \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\) is important for contextualizing research. Is this amount of amphetamine enough to be harmful?

A pair of measurements \(x\) and \(y\) is called **proportional** if their quotient \(\frac{x}{y}\) is constant. The mass of amphetamine, \(m\), in a sample and the sample’s volume, \(V\), are proportional. Their ratio \(\frac{m}{V}\) is equal to the amphetamine concentration \(c\).

Due to proportionality, **doubling the volume of water** collected **doubles the mass of amphetamine** in the sample. In fact, rescaling the volume by any factor rescales the mass of amphetamine by the same factor, provided the amphetamine is mixed uniformly in the water.

An independent researcher reads about the reported amphetamine level near her home in Baltimore County, and is surprised at how low this figure is. She decides to collect several water samples from the stream to test herself. Each of her samples has a volume of **200 milliliters**.

If the reported concentration, \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter},\) is correct, what **mass of amphetamine** would the researcher expect to measure in each of her samples?

One of the reasons news outlets remove scientific measurements from articles is that placing them in context may undermine public interest in the story. It is far more lucrative to ask vague questions like "Are the nation's streams tainted with hard drugs?" without providing a quantitative answer. Let's try to **visualize** a figure like \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\), the maximum concentration measured in the Baltimore surface water study.

Suppose a stream contains an amphetamine concentration of \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\). Given that there are 1 million \(\si{\nano\gram}\) in \(\SI{1}{\milli\gram}\), **how many liters** of water would you need to collect in order to extract \(\SI{1}{\milli\gram},\) a fraction of a therapeutic dose?

What concentration of amphetamine do we expect **in wastewater** near a major metro area such as Baltimore?

We can make some rough assumptions about the population:

- The population of Baltimore County is 700,000.
- 15% of the population uses an average \(\SI[per-mode=symbol]{10}{\milli\gram}\) per day of amphetamine in some form.
- The body can only metabolize 50% of the dose of amphetamine you take, the other 50% is excreted into wastewater.
- Baltimore treats \(\SI[per-mode=symbol]{950e6}{\liter}\) per day.
- It will be helpful to know that \(\SI{1}{\milli\gram} = \SI{1e6}{\nano\gram}.\)

In this quiz, we examined a U.S. national news story about drugs present in Baltimore's surface water. Despite the hyperbole in the headlines, no news article reprinted the **basic measurement** the researchers had made.

We found the researcher's actual measurement and placed it in context by manipulating basic quantitative relations: rates, ratios and percents. We used a foundational tool called **proportional scaling** in order to conclude that their highest measured concentration was equivalent to that of one tenth a tablet of amphetamine dissolved in a hot tub.

We also estimated the amount of amphetamine we would find in a day's wastewater produced by a city the size of Baltimore using a tool called a **Fermi estimate**. This approach produces a prediction based on rough guesses about the scale of known inputs, independent of any detailed modeling. We discovered that the concentration of amphetamine in the surface water was *higher* in at least one instance than our best guess of the amphetamine concentration in the wastewater.

Throughout this course, you will continue to develop powerful ideas in physics that you can use to understand the world around you.

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