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 Country 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 make a measurement, you must first decide on an appropriate unit. **Units** are agreed-upon reference measurements that are standardized across measurement devices.

For example, watches tick at a rate of once per second. Watch designers want their devices to tick as closely as possible to that rate; otherwise, their product cannot fulfill its purpose—to measure time.

When you use a clock to measure the time your toaster takes to perfectly toast your bread, you can report your measurement as, for example, 100 seconds. Anyone else with the same toaster can use **any working timepiece** to reproduce your result.

**not a unit of volume**?

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

A therapeutic dose of amphetamine is \(\SI{10}{\milli\gram}\), although recreational doses can be 10 times higher.

If one-tenth of the population regularly uses amphetamine, which of the following is the **most appropriate unit** for the amount of amphetamine consumed each day in Baltimore, a city of 700,000 people?

The researchers reported their measurement not as pure mass of amphetamine, but as a **rate**: \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\) in the most contaminated stream. This tells us that \(\SI{1}{\liter}\) of water contains \(\SI{630}{\nano\gram}\) of amphetamine.

Rates relate two quantities that are measured in different units—in this case, nanograms of amphetamine and liters of water. (This kind of rate is called a **concentration**, the mass of a compound per volume of mixture.)

Rates are common even outside of scientific reports and discussion. For example, the **price per weight** of mushrooms at your local store is another example of a rate. These quantities have different units; price is measured in units of your local currency and mass of mushrooms are measured in kilograms or pounds, depending on where you live.

Amphetamine is manufactured in \(\SI{10}{\milli\gram}\) tablets. If the doctor's direction is to take **2 tablets per day**, what is the patient's total amphetamine intake during **one week**?

You can assume that the patient takes the prescribed amount each day of the week.

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?

**Ratios** are like rates, but they express a proportional relationship between two measurements that have the **same unit**. For example, the proportion of a population that has an amphetamine prescription from a doctor is a ratio (incidentally, in the U.S. in 2015 it is about 1 in 15 people). Another example of a ratio is your total monthly expenses as a fraction of your monthly income.

A natural way of talking about ratios is using fractions and percents. A relative statement like "I spent three-quarters or 75% of my income last month," is often more convenient and better understood than reciting figures from your bank statement.

The research team measured a range of amphetamine concentrations from \(3\) to \(\SI[per-mode=symbol]{630}{\nano\gram\per\liter}\).

What is the smallest measured concentration as a percent of the largest measured concentration?

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 these explorations, you will continue to develop powerful ideas in physics that you can use to understand the world around you.

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