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U4735 Environmental Science for Decision Makers

Basic Mathematical Tools for Environmental Science.

Below are some import analytical tools that we will be using during the semester. Most of the semester will be spent applying these tools to environmental science questions, but for the first several weeks we will spend some time becoming familiar with these tools.

• Multiplying: just multiply the numerators and denominators together:

• Dividing: invert the second fraction and multiply as above:

• Adding and subtracting: find the common denominator:

• Exponents are useful for expressing numbers in scientific notation (sometimes called exponential notation), for units, and for exponential growth problems (all covered below). The number 10² ("squared") just stands for 10x10. In another case, 2³ ("cubed") is 2x2x2, or 8. In this case, the number 2 is called the base. The most common numbers used as bases are 2, e (» 2.7183), and 10 (corresponding to base 2, base e, and base 10).
• A negative sign in an exponent means the number should be one over itself. 10^-3 is 1/10³, or 1/1000, or 0.001. Note, this is often used in units: m s^-1 is the same as m/s (meters per second). Any number to the zero power is one (x⁰ = 1).
• Multiplying exponents-to multiply two exponents with the same base, just add the exponents. For example, 10³x10^-1 is 10² (1000x0.1=100). Numbers with different bases cannot be directly multiplied. To divide, just subtract the exponents.
• Exponents raised to higher powers-if 10⁴ is cubed (i.e., raised to the third power: (10⁴)³), then the exponents are multiplied together: (10⁴)³ = 10¹².
• Exponents are not limited to whole numbers. For example, 10^3.34is 2.187x10³. (A calculator is necessary to make the conversion. There should be a key labeled 10^x on your calculator. Type in 3.34, push that key, and you should get 2187.7616...)
• A useful summary:

• Scientific notation is just a quick way of writing large (or small) numbers. The following table should give an idea how this works:

• Scientific notation has two important advantages:

1. It is a compact way of writing large numbers. 4,700,000,000 is cumbersome to write, and 4.7x10⁹ is much shorter. Along the same lines, it is easier to work with numbers in scientific notation. For example, to multiply 4.7x10⁹ by 3.0x10³, just multiply 4.7 by 3.0 (14) and add the exponents together (9+3=12). The final answer is 1.4x10¹³. See below why it isn't 1.41x10¹³. Also, note that 14 is converted to 1.4x10¹, and the 1 is added to 12 to get 13.
1. Scientific notation exactly specifies the number of significant figures (see below).

• Most numbers used in this class are only approximations of actual values. For example, the world's population in 1983 is given as 4.7x10⁹ in Spherical Cow. The authors of the study could have calculated the number to be 4,719,390,207. Unfortunately, population is difficult to estimate, and they felt that only the first two numbers were known with certainty. These two numbers are called significant figures, and only these numbers are reported.
• In calculations, the number with the most uncertainty determines the significant figures in the answer. For example, to calculate the average population density, the population (4.7x10⁹ people) is divided by the ice-free land surface (1.33x10¹⁴ m²). The answer from a calculator is 3.5338x10^-5 (and even this I rounded off). The area has three significant figures, but the population only has two significant figures. Therefore, the answer should have only two significant figures: 3.5x10^-5 people per square meter (luckily, this is a very small number). For multiplication and division, the number with the least number of significant figures determines the number of significant figures in the answer.
• When adding or subtracting two numbers, the situation involves lining up the numbers. For example, adding 4.05 to 10.001 gives 14.05. 14.051 would be incorrect, because the thousands place in 4.05 was unknown:
  4.05?
  + 10.001?
  14.05?
  Note that the rule for multiplication doesn't apply in addition and subtraction (rounding to 14.1, three significant figures, is unnecessary, because the hundreds place was known for each number).
• Scientific notation removes ambiguity on the number of significant figures. How many significant figures does 1,000 have? It could be one, two, three, or four. (If the number was reported as 1,000.0, then the answer would be five.) If we knew that the number has two significant figures, the best way to write it would be 1.0x10³.

• See Appendix I in Spherical Cow for a list of units and their abbreviations. Also, there is a table of unit prefixes there and conversions. We will try to stick to the mks (meter, kilogram, second) variation of the metric system, but there will also be some problems in cgs (centimeter, gram, second), which is favored by chemists. Also, it will be necessary in some cases to start with data that's in the English system (e.g., first you'll need to convert feet into meters).
• Units are often combined with exponents. For example, squared meters are represented by m². How many squared meters (m²) are there in a squared kilometer (km²)? First, 1 km = 10³ m. Therefore 1 km² is (10³ m)² or 10⁶ m².
• Often in problems, various units will cancel out. The exponents on units follow the same rules as introduced above. For example, if population density (people m^-2) is multiplied by area (m²), then the result is just people (m^-2 x m² = m⁰ = 1).
• Unit conversion-the easiest way to keep track of units is to use ratios. For example, how many square meters are in one acre (an English measure of area)? Since 1 acre = 43560 ft², the ratio 43560 ft² / 1 acre is equal to one. By using ratios that have the desired unit over the current unit, a series of conversions can be made:

• Parts Per Million (or Billion, Trillion, etc.)-This unit is used throughout environmental science, but there are some inconsistencies. First, it is important to know parts per million of what. For example, CO₂ in the atmosphere is measured in PPM by volume. For example, if the measured CO₂ concentration is 360 PPMV (the extra "V" stands for volume), then 1 liter of air contains 360x10^-6 liters of CO₂. To add to the confusion, this same number could be reported as 360 m mol CO₂ per mol of air. Why is this the same? Because number of moles are directly proportional to the volume (by the ideal gas law), the units are equivalent. (Remember that 1 m mol = 10^-6 mol.) For concentrations in soils and water, PPM usually means per weight (really mass). For example, lead in soil might be reported as 10 ppb (equivalent to 10 mg lead per kg of dry soil).

• This example of population growth lays out the basic principles of exponential growth. First, consider a population that takes a certain amount of time to double in size (say T=2 years). Secondly, assume that the population always takes the same amount of time to double, no matter no large it grows (this is the definition of exponential growth). At time t=0, the population is 1,000. How large is the population after four years (T=4 years)? After the second year, the population doubles to 2,000. At the end of the fourth year, the population would be 4,000. Mathematically, would could express this exponential growth as:

• Instead of specifying the growth rate as the number of doubles per year, the number e is introduced. The symbol e (sometimes typed as exp) stands for a number that is approximately 2.7183... (like p , it keeps going). The advantage of using e as a base is that the corresponding rate, r, is given by the percent growth rate. The equation is analogous to the one given above:

• Finally, note that the exponent is dimensionless (i.e., does not have units). The growth rate is % per year and the time is in years. For example, 5% per year multiplied by 10 years gives 0.5 (0.05 yr^-1x10 yr = 0.5). To make the actual calculation, your calculator should have a key labeled e^x. For example, take N(0)=1000, r=5% yr^-1, and t=10. The population, N(10), is:

• Often it is necessary to solve a problem where either the growth rate or the number of years is unknown. The concept used in this case is the natural log (denoted by either log_e or ln). This function is the inverse of taking e raised to some power. Therefore, ln (e^x) = x. You should be able to take the natural logs of numbers on your calculator.
• Consider the following problem. The population in 1983 was 4.7x10⁹. The growth rate in Spherical Cow was calculated to be 1.9% yr^-1. From this information, when will the population equal 6.0x10⁹? Starting with the original equation:

• Another useful summary:

Base 10 logs are very similar to natural logs. Just as ln x is the inverse of e ^x, log₁₀ x (usually the 10 is omitted: log x) is the inverse of 10 ^x. For example, to solve the problem 10 ^x = 3.54, take the log of both sides to get x = log 3.54. Using your calculator, you'll find that x = 0.549. In general:

where n is the base of both the log and the exponent.

Spherical Cow provides a good explanation of box models on page 23. Here is a quick summary of their terminology and assumptions:

• stock is the amount of "stuff" in the box (usually either a volume, a mass, or a concentration) [M]
• flow rate is the amount of "stuff" coming either into or out of the box (this should be the same units as stock, except divided by time (e.g., per year, per day, etc)) [F_in, F_out]
• steady state means that the flow rates and the stock are not changing with time (i.e., F_in = F_out)
• Residence time is defined as

Residence time is often used to figure out how long a pollutant will stay in the lake after it is no longer being introduced. For example, say that a lake has 10 mg of mercury per cubic meter. The residence time of water in the lake is 10 years. If no more mercury is being introduced into the lake, how long will the mercury take to go away? How much mercury will be left after 10 years?

The mercury will never go away completely. Even though the volume of water in the lake has been replace after 10 years, the mercury has been diluted, but some still remains because the new water that comes into the lake is contaminated with the old mercury. In fact, this is an exponential decay problem (just like population growth, except with a negative sign):

Where C denotes concentration, t is time, and T is the residence time.

Why does this work? Like population growth, there is an analogy of e with 2. You might guess that half the mercury has been removed after 10 years, and this is almost correct. Again, think about marbles. If there are 10 marbles in a bowl, and one is added and one (randomly selected) is removed once per minute, how long does the average marble stay in the bowl? The answer is the residence time, 10 minutes. But what about individual marbles? The chance is one in ten that a marble will be removed after one minute, and there is approximately a 35% percent chance that it will stay in longer than 10 minutes. It is not far from wrong to say that half the marbles will stay in longer than the expected time and that half will be removed quicker. Instead of using 2 as the base, e gives the exact solution. (The reason is the same as with population growth-compound interest, or in this case compound decay.)

• In the case of multiple inputs and/or outputs, it is the sum of all that must equal zero. Often, inputs are considered to be positive values, and outputs are negative. The only caution is to be consistent, because the signs are reversed in some situations (it just depends on what you define to be the box).