Parts is Parts

Posted on 2015-12-10 by Randy Bancroft

By The Metric Maven

One Autumn night I was awoken at 2:00 AM by the shrill report of the carbon monoxide (CO) detector in my bedroom. I was slightly concerned, but mostly annoyed. This CO detector (and another identical unit) had been prone to setting off their alarm even when I suspected nothing was wrong. When I had a new water heater installed a few years back, the company gave me a new CO detector for my basement. I gladly replaced the CO detector that cried wolf with the new unit. So far it has never sounded an alarm. What I dislike about both units is they have no units displayed. They either shriek or don’t shriek. I determined that, if possible, I was going to purchase a new CO detector which provides measurement information. I found a new CO detector, and it indicated that it has a readout, in parts per million (PPM).

The instructions indicated that a low level of CO is 0 to 50 PPM, a mid level of CO is 50 to 100 PPM, and a high level of CO is above 100 PPM. The detector alarm sounds above 100 PPM. These are all nicely defined values, but I’ve never been sure about parts per million as a “unit.” I assume parts per million of carbon monoxide is the number of parts of CO in a million parts of other stuff. It seems like it’s telling me something, but not quite. I recall a late friend who was a political cartoonist expressed great concern to me that a toxic substance had been found in biscuit mix in parts per billion. I wasn’t sure what to think, a billion is a very big number (1 000 000 000), I wondered if this was even something of concern without any investigation, but with a simple estimate. If the Earth’s diameter (12.75 Mm) is divided by a billion, one could drop it into the cap of some ball point pens (12.75 mm). I wondered out loud if sea snake venom, or plutonium in this dilution would be deadly. It seemed like parts per trillion would be almost vanishing in their concentration.

The deadliest snake in the world is said to be the Inland Taipan which is found in Australia. When it delivers a bite (often repeatedly) it injects between 44 and 110 milligrams of venom. The median lethal dose in mice is 25 micrograms/Kg. Below is a comparison of the lethality of a sample of snakes. The units typically used are milligrams/Kg, but I used micrograms to produce integer numbers (see Naughtin’s Laws).

Inland Taipan (Australia)            25 μg/Kg
Beaded Sea Snake                                                             164 μg/Kg
Indian Cobra    565 μg/Kg
Eastern Diamond Back Rattlesnake (North America)                           11 400 μg/Kg

But what is this in parts per million?—I have no idea. Parts per million in air can be the number of grams of a substance for every million grams of mass. This is parts per million by mass. It can also be one milliliter of gas for every million milliliters of air. This is parts per million by volume. A third choice is 1 gram of gas for every million milliliters of air. This is parts per million by mass per volume. I think I feel justified at my confusion. Given the units are mass over mass milligrams/Kilogram is a factor of one million, so the LD50 values appear to be:

Inland Taipan (Australia)    0.025 PPM
Beaded Sea Snake                                                                                     0.164 PPM
Indian Cobra                                                                                                0.565 PPM
Eastern Diamond Back Rattlesnake (North America)                              11.400 PPM

So if these doses were increased by one thousand, they would be parts per billion, and have the same values as found in the first table.

Inland Taipan (Australia)                                                                                 25 PPB
Beaded Sea Snake                                                                                      164 PPB
Indian Cobra                                                                                                 565 PPB
Eastern Diamond Back Rattlesnake (North America)                            11 400 PPB

It appears that the political cartoonist was right, parts per billion can be a problem, but which parts per billion? In this case it’s parts per billion by mass. In the case of my CO detector it just states parts per million, so which PPM? I’m quite sure that “parts is parts” does not apply. Also there are two different “versions” of the words used for magnitude descriptions, which are called long and short scale.

In the case of gasses, it makes sense to have a standard temperature and pressure, and express values with metric units. This is often done with mass over a given volume which is a density. It is possible to convert 50 PPM CO to an equivalent value of 58 milligrams per cubic meter. One hundred parts per million is about 115 milligrams per cubic meter. These values are close enough that the CO levels of concern could be 0 to 50 mg/m³ for low levels, 50 to 100 mg/m³ for mid levels, and 100 mg/m³ and above for high levels of CO. I can imagine a milligram spread over a cubic meter, but an alternative could clearly be a version of grams per liter. In both cases one can visualize the quantities involved more intuitively. Milligrams are a common dosage mass for aspirin and other over the counter products.

Carbon dioxide (CO₂) is a greenhouse gas, and its concentration has been measured since the late 1950s at Mauna Loa, Hawaii. The value is generally given in parts per million, which has very little meaning for me. Here is a current graph of the Keeling Curve:

The values would make a lot more sense to my intuition if the graph were in mass per cubic meter. Assuming the analysis presented here is correct, we can re-plot the monthly data (I could not find the seasonally corrected data) in terms of milligrams per cubic meter, assuming 380 ppmv is 684 mg/m³:

I can immediately estimate that the amount of CO₂ per cubic meter has increased by about 100 milligrams per cubic meter since 1982 (http://spacemath.gsfc.nasa.gov/data/KeelingData.xls). The upper graph has a value of about 315 ppmv in 1960, which is approximately 567 milligrams per cubic meter. So just since measurements have been kept, the amount of CO₂in the air has increased by about 130 mg/m³. I have some intuitive understanding of this increase, and it seems like a lot!

Sulfur Dioxide (SO₂) is another common pollutant. A blogger named Rei has been measuring the amount of sulfur dioxide with pollutant meters located around the Baroabunga volcano in Iceland. His blog Barobunga: “Like Being In An Enclosed Space With A Diesel Engine.” first describes human safety limits, and context levels in micrograms per cubic meter from 20 micrograms per cubic meter to the highest recorded concentration, 1000 µg/m³, taken next to Russian smelters. He observed this graph on 2014-09-10:

Rei had the reaction I would have: “could that be right?” Perhaps the equipment calibration was off?—or some other problem had occurred. After checking the equipment, it was determined that these measurements are accurate. The measured levels peaked at almost 2600 µg/m³. This plume descended on a nearby town and began to produce respiratory distress. (tip of the hat to Helen for posting a link to this article) I’ve always been doubtful about the use of parts per million, billion, trillion and so on as providing any understandable quantitative information. This article by Rei re-ignited my interest in this question.

The Wikipedia page on sulfur dioxide shows that the amount of the substance released into the air has decreased from 28.3 Megagrams in 1970 to 17.1 Mg in 1999 (the last year data is provided). Knowing the 24 hour safety limit for the World Heath Organization is 20 µg/m³ one can see that 17 Megagrams released in the the air above the US is a significant amount of pollution. One aim of this blog is to examine the clearest expression of numerical magnitudes possible to convey information in the most accessible way possible. It seems that it might be best to go with expressions which contain metric units over those which are in terms of normalized values (i.e. PPM). Determining the best method of expression in the case of CO and SO₂ is not just an academic exercise, peoples lives can be lost from misinterpretation. Mass of pollutant per cubic meter of air looks like a good way to express these values, but this is a serious enough question that it should be the subject of rigorous study.

EU NOx Pollution Limits — click to enlarge

In September of 2015 Volkswagen admitted that it had rigged car emissions tests using software. This fraudulent activity affected about 11 million cars. New Scientist in a 2015-10-03 article entitled The Curious Case of NOx Pollution stated: “If this practice is widespread, it could help explain why NOx emissions in many European countries continue to overshoot targets.” The figure to the left is from the article. The limits are in micrograms per cubic meter. Perhaps the world is beginning to change and use this more accessible way of numerical expression. Unfortunately the U.S. has not embraced the metric system and abandoned its medieval units in the 21st century. If it did, we all might breath a little easier.

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:

The first book is titled: Our Crumbling Invisible Infrastructure. It is a succinct set of essays that explain why the absence of the metric system in the US is detrimental to our personal heath and our economy. These essays are separately available for free on my website, but the book has them all in one place in print. The book may be purchased from Amazon here.

The second book is titled The Dimensions of the Cosmos. It takes the metric prefixes from yotta to Yocto and uses each metric prefix to describe a metric world. The book has a considerable number of color images to compliment the prose. It has been receiving good reviews. I think would be a great reference for US science teachers. It has a considerable number of scientific factoids and anecdotes that I believe would be of considerable educational use. It is available from Amazon here.

The third book is called Death By A Thousand Cuts, A Secret History of the Metric System in The United States. This monograph explains how we have been unable to legally deal with weights and measures in the United States from George Washington, to our current day. This book is also available on Amazon here.

Wishing Upon a Star

Posted on 2015-11-30 by Randy Bancroft

Alpha-Centauri-Wikimedia-Commons — Alpha Centauri (Wikimedia Commons)

By The Metric Maven

Bulldog Edition

A wish can be a supernatural request which is granted by a supernatural talisman. The song, When You Wish Upon a Star, when modulated onto an electromagnetic (radio/light) wave, that is traveling in a vacuum, moves at 300 Megameters per second. This is only true if the light is traveling in a vacuum (we’ll get back to that), and space is a pretty good vacuum. Einstein was rather clear about the fact that information cannot be propagated faster than the speed of light. This means that any receiving star (other than the Sun) would have to wait years to know that a wish was requested of it.

The Alpha Centauri star system is the closest and it would take light a little over four years for a supernatural request to arrive, so your wish would be delayed by at least that amount of time. Alpha Centauri is also only seen in the U.S. for very short periods of time, and only at latitudes which are south of Houston Texas and is practically invisible. Assuming Alpha Centauri is the ineffective talisman that I expect it is, one would have to wait about eight-years for a non-reply. If you wish on a star that takes light over 75 years or so to arrive, well, then you will not be alive to receive the non-reply. Unless you plan to live to 150 years of age. The odds of that happening are not good.

Astronomers like to conflate time and distance into a strange and exotic sounding description called a light-year. Each of the multitude of stars we view at night has light that emanated at a different time, and so when a star is farther and farther away in distance, we witness how it looked longer and longer ago. Every star has a unique time delay associated with it. The further we look out into the Universe, the farther back in time we see.

When you look at any object or person, you do not see them instantaneously. If a person is 500 mm from you, the light you see has taken about 1.67 nanoseconds to impact your retina. The person is therefore 1.67 light-nanoseconds away from you. If you see an erupting volcano that is 1000 meters distant, the image seen by your eyes has a “distance” of 3.33 light-microseconds. Standing in Denver Colorado, Pike’s Peak (which is visible from Denver), is about 160 Km distant or 533 light-microseconds. Which has more meaning in terms of distance?—160 Kilometers or 533 light-microseconds? This is not really fair one might argue. As far as a person is concerned, this amount of time is instantaneous, and so it makes perfect sense to use distance and forget about the propagation speed of light.

When does a product of the speed of light and time begin to be a distance that makes some sense? There are a lot of choices:

Light-Second 300 Mm (Megameters)

Light-minute 18 Gm (Gigameters)

Light-hour 1.08 Tm (Terameters)

Light-day 25.92 Tm (Terameters)

Light-week 181.44 Tm (Terameters)

Light-month 725.76 Tm (Terameters)

Light-year 9.46 Pm (Petameters)

Light-Century 946 Pm (Petameters)

When the New Horizons probe was near Pluto, it took about four hours for the radio signal to propagate from the Earth to the spacecraft. It was not typically said that the probe was 4 light-hours from the Earth. Why not use light-hours if the conflation of light-speed and distance is so useful? A light second is a 3000 hour long (100 Km/hr) drive, or 3000 car-hours. It is also 7.5 times around the Earth. A light-minute is not enough distance to traverse from one planet to the next in our solar system. The light hour is a distance from the Sun to a point between Jupiter and Saturn. The light-day, light week and light month are all well short of our nearest star system, Alpha Centauri. A light century (which no one generally uses) is 100 light years. Betelgeuse is over six times this far, and it can be called a nearby star. The length across the Milky Way galaxy is about 100 000 to 180 000 light-years. Our closest galaxy is Andromeda and it is 2 500 000 light-years distant. The observable universe is about 91 000 000 000 light-years. It is hard to see that this single “unit,” the light-year, is really descriptive over the large dynamic range of the Universe. Enormous numbers cannot be visualized, but they can be categorized, which gives them more intrinsic relative meaning. The metric system is quite useful for accomplishing exactly that.

Furthermore, the light-year has a built-in assumption about what year is used. According to Wikipedia: “As defined by the International Astronomical Union (IAU), a light-year is the distance that light travels in vacuum in one Julian year.” My favorite engineering reference for unit definition has this entry:

The options given for a light year length are:

Anomalistic Light Year: 9.460 980 Petameters

Julian Light Year: 9.460 730 Petameters

Siderial Light Year: 9.460 895 Petameters

Tropical Light Year: 9.460 528 Petameters

There are two questions that in my view are rather separate: 1) How far away is an object based on a linear measurement? 2) How long does it take an electromagnetic wave to get from there to here (or vice-versa)? Astronomers might argue that the light-year is really the best description in their view, but when one looks at a star there is no way to really grasp the amount of time or distance. They all look very similar. The first question one probably wants to know is: “how far is that star?” rather than “how long does an electromagnetic wave take to arrive?”

There is another apparent problem. Suppose I were to ask: what is the radius of the Sun? One might immediately say it is 696 000 Kilometers, but I could also argue that it’s about 100 000 light-years, or 1000 light-centuries in extent! Light does not always travel at 300 000 meters/second, it can travel slower than this value when a dielectric medium is present, such as plastic, glass or gas. It takes a photon about 100 000 years to make its way from the Sun’s center to its surface. The photon also loses energy (changes frequency) as it works its way through stellar plasma, but light is a general term for an electromagnetic wave, and its frequency is not specified by astronomers. They just say “light,” so if a photon is just one millimeter inside of the event horizon of a black hole, would its distance to any other body in the universe, in light years, be infinite?—or even possess an imaginary distance? Is this a legitimate use of a light-year as a “measurement unit?” Well, no, it is not. Astronomers define a light-year in a vacuum, but Wikipedia also calls it an informal unit and claims it is a length, and should not be confused with time—even though time is in the name of the “unit.” The light-year reminds me of Saturday Night Live’s Shimmer Floor Wax, it’s both a floor wax and a dessert topping. Some astronomers have been less than enthusiastic about the light-year as a “unit.” According to Wikipedia:

The light-year unit appeared, however, in 1851 in a German popular astronomical article by Otto Ule.^[18] The paradox of a distance unit name ending on year was explained by Ule by comparing it to a hiking road hour (Wegstunde). A contemporary German popular astronomical book also noticed that light-year is an odd name.^[19] In 1868 an English journal labelled the light-year as a unit used by the Germans.^[20] Eddington called the light-year an inconvenient and irrelevant unit, which had sometimes crept from popular use into technical investigations.^[21]

Astronomers define a light year as the distance light travels in a year in a vacuum; but there is another unit which is defined as the distance light travels in a given amount of time in a vacuum. It is the meter, and it’s the base linear measurement value of the metric system. The meter does not have any unit of time in its name, and so it would alleviate the time confusion immediately. Astronomers who might not be familiar with this unit can convert it to 3.33564 light-nanoseconds for clarity. The metric system also has a unique unit of time, the second. One can use metric prefixes with it to describe intervals of time. It’s about time, it’s about space, but only one at a time, unless it’s a relative place.

Postscript: And Then There Were Two? I have been informed that Myanmar has quietly continued to pursue metrication:

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system: