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FUNDAMENTAL FOR BUILDING MATHEMATICAL MODELS-CONCENTRATION
Chapter Goals
Understand the concepts of extensive and intensive properties Understand various ways of expressing the concentration of a chemical
INTRODUCTION
In the last few chapter we have learnt the utility of developing fate and transport models and categorized different approaches to build them. It was seen that theory based models built using the fundamental law of mass balance offered the greatest flexibility in modeling a wide range of environmental systems. In this chapter, I will discuss some basic fundamental quantities necessary for building such models. You probably have encountered these building blocks in your introductory classes in environmental engineering and science and as such only the main concepts are presented here as a referesher.
Extensive Properties Mass, Volume and Heat
As environmental engineers and scientists interested in tracking pollutants in the environment, we would like to know how much of the pollutant is within the system. The amount of pollutant within a system is represented by its mass. The mass is a fundamental physical quantity and is formally referred to as an extensive property. The extensive property is an additive property that increases with the size of the system. Volume and heat are other extensive properties that are of interest to environmental scientists and engineers.
28 Applied Environmental Systems Modeling The mass of a substance has dimensions of [M] and is expressed using the units of grams (g). Other common measures are the kilogram (kg) [1 kg = 1000 g]; milligram (mg) [1g = 1000 mg]. Trace quantities are defined in micrograms (mg) [1000 mg = 1 mg]. While mass represents the quantity of the substance, the volume defines the space occupied by the substance. In English system, the unit of mass is slugs (lbm). The volume of a substance has dimensions of [L3] and is expressed using the units of cubic meters (m3). Other units such as the liter (L) [1000 L = 1 m3] or milliliters (mL) [1000 mL = 1 L] are used as well. Extremely small quantities of volume are often denoted using microliters (mL) [1000 mL = 1 mL]. In English system, volumes are measured in cubic feet (ft3). The acrefeet is another measure commonly used in water resources literature and is equal to 1 feet of water covering 1 acre of land (1 acre feet = 43,560 ft3). The gallon is another common measure of volume and 1 US Gallon equals 3.785 L [1 ft3 = 7.480 Gallons]. The heat of a system is measured in joules (J) in MKS and SI system of units. Calories (cal.) are the units of heat in CGS system and the British Thermal Units (Btu) are used as the measure for heat in the English system. The unit Kilo joule (kJ) is used to denote high quantities of heat [1 kJ = 1000 J]. The following conversions come in handy to convert the quantity of heat expressed in one unit into another [1 cal. = 4.1868 J; 1 Btu = 1.055056 kJ and 1 Btu = 252 cal.]. The units of ergs is used in the CGS system [1 erg = 10-10 kJ].
Intensive (size-independent) Properties
All extensive properties (mass, volume and heat) are additive and depend upon the size of the system. In contrast, an intensive property is size independent and is a quantity that is normalized to a measure of system size. For example, the density of a fluid is defined as the mass occupied by a unit volume of the fluid (at constant temperature and pressure). Thus, if we increase the volume from one unit to two units, there will be a corresponding doubling in the mass and the density of the fluid will remain unchanged. Mathematically, r=
M 2 M NM = = V 2V NV
(4-1)
Where r is the density (g/m3); M is the mass (g) and V the volume (m3) and N is an arbitrary constant. Thus, the fluid will have same density (at constant temperature and pressure) regardless of how much fluid we have within our system.
Fundamental Quantities for Building Mathematical Models-Concentration
29
Concentrations
The density is a measure of normalized mass with respect to the volume of the substance. It is a useful measure when the solute (i.e., the dissolved constituent) is the same as the solvent (i.e., the substance in which the solute is dissolved). For example, the density of water indicates the mass of water per unit volume of water (here water is both the solute and the solvent). However, in most environmental systems, we are often interested in tracking chemicals (solutes) that are present in a different solvent. Air and water are two important environmental solvents in which a variety of physical, chemical and biological constituents are transported across environmental systems of interest. The normalized mass of the solute within a unit measure of the solvent is referred to as the concentration. Just like density, the concentration is an intensive property as it is also a ratio of two extensive properties. The concentration of the solute is expressed either in mass of solute per volume of the solvent or mass of the solute per mass of the solvent units. For example, if mp milligrams of phosphorus is dissolved in X liters of water sample taken from a reservoir. The phosphorus concentration (Cp) can be expressed as: Cp
FG mg - p IJ H L - H OK
2
=
m p ( mg - p ) X (L - H 2 O )
(4-2)
In addition to air and water, solid particles are another important class of solvents that transport chemical constituents. A variety of particle matter is of interest to environmental engineers and scientists. Soil particles adsorb a variety of chemicals such as fertilizers and pesticides as they infiltrate through the soil. The amount of surficial sediments that runoff into a stream or a creek in itself is of interest in many studies. In addition, chemicals (such as phosphorus) are attached to these sediments and as such the concentrations of these chemicals are of interest as well. Example 4-1 Consider an idealized lake system depicted in Figure 4-1. The concentration of the suspended sediments is 10 mg/L. If the phosphorus concentration on the suspended sediments is 2 mg/g what is the concentration of the phosphorus in the lake in mass per volume units? Solution Notice in the example the concentration units (mg/L and mg/g) do not make explicit references to the solute and the solvent. This is a common practice in environmental sciences and engineering and also a major source of confusion and potential place for errors. Hence, when dealing with concentrations, it is better to explicitly write down what the
30 Applied Environmental Systems Modeling
Figure 4-1
Distribution of sediments containing phosphorus in a lake.
solute and the solvent are in order to avoid making mistakes. Let C be the concentration of phosphorus in the lake system in the units of mg/L. Then:
C
1 g - TSS mg - TSS mg - p mg - p 2 = 10 L - H2O L - H 2O g - TSS 1000mg - TSS
= 0.02
mg - P L - H2O
(4-3)
Alternatively, the concentration can also be expressed as C = 0.02
mg - P 1000 mg - P mg - P = 20.00 L - H 2O L - H 2O 1mg - P
(4-4)
Concentrations are sometimes expressed using ratio units such as parts per million (ppm) and parts per billion (ppb). The parts per million (ppm) tell us how many parts of the solute is in a million parts of the solvent. Similarly, the parts per billion (ppb) tells us how many parts of the solute is in a million parts of the solvent. The quantity of the solute and the solvent can be expressed in either mass or volume units leading to the following measures 1 ppmv =
1 mL - solute 1 mL - solute 1 mL - solute = = (4-5) 1000000 mL - solvent 1000 L - solvent 1 m 3 - solvent 1 mg - solute 1 mg - solute 1 mg - solute = = (4-6) 1000000 mg - solvent 1000 g - solvent 1 kg - solvent
1 ppmm =
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31
I have used the subscripts v and m to denote whether the parts per million is measured in mass or volumetric units. Many times the parts per million and parts per billion are simply expressed as ppm and ppb without indicating whether they are of mass or volume basis. Hence, these measures also add to the confusion and it is best to avoid them whenever possible. The density of water is approximately equal to 1 kg/L under ambient conditions. As such, for aqueous solutions, the parts per million can be expressed in hybrid mass per volume units as follows: 1 ppm =
1 mg - solute 1 kg - water 1 mg - solute = 1 kg - water 1 L - water 1 L - water
(4-7)
Similarly, 1 part per billion is approximately equal to 1 mg / L in aqueous solutions. The use of ppm and ppb to mean mg/L and mg / L respectively is very common in water quality applications. However caution must be exercised as a billion usually is 109 in America and 1012 in Europe. Complications also arise with these measures when concentrations of the pollutants in more than one media are tracked. For example, in air quality studies ppm usually is measured on volume/volume basis (i.e., ppmv) while in water quality studies ppm is measured on mass/volume (mg/L) basis. In addition to mass per mass and mass per volume measures, counts per unit mass and/or counts per unit volume are used in environmental literature, especially to characterize concentrations of biological For organisms. example, the concentration of e-coli in water could be stated as 20#/ 100 mL. Indicating 20 counts of e-coli were enumerated in the 100 mL water sample that was analyzed. Chemists prefer to measure the amount of matter in moles. A mole is actually the number of constituent entities, particles, atoms or molecules. It is the number of particles divided by the Avogadro's number which is equal to 6.023 x 1023. The Avogadro's number is defined as the number of atoms in 12 g of carbon-12 isotope. The molecular mass of a chemical is the mass of 1 mole of the chemical. The molecular mass of benzene (C6H6) is approximately equal to 78 g/mol. Using the moles as the basis the concentration of a chemical can be written in moles per liter (mol/L) or moles per kg (mol/kg) basis. Example 4-2 The concentration of benzene leaving the exhaust of a chemical unit was measured to be 20 ppmv. Express this concentration in mol/L units. (Assume, the density of benzene = 0.78 g/mL and the molecular weight is approximately = 78 g/mol).
32 Applied Environmental Systems Modeling
C
FG mol - benzene IJ H L - air K
=
20 mL - benzene g - benzene 0.78 1000 L - air mL - benzene 1 mol - benzene 78 g - benzene
(4-8)
F mol - benzene IJ CG H L - air K
= 2.0 10
-04
mol - benzene L - air
= 0.20
mol - benzene L - air
The following definitions from chemistry come in handy for denoting concentrations of chemicals. Molality (m): Number of moles of solute in 1 kg of solvent.
m 1 mol - solute mol = 1 kg - solvent kg
(4-9)
Molarity (M): Number of moles of solute in 1 L of solvent.
M 1 mol - solute mol = 1 L - solvent L
(4-10)
Normality (N): Number of equivalents (or milliequivalents) of solute in 1 L of solvent.
N
1 eq - solute mol = L 1 L - solvent g 1 eq - solute = eq 1 L - solvent
(4-11)
Equivalent Mass (em): The ratio of atomic mass to the number of equivalents
em
(4-12)
Where z is the equivalents per mole and is equal to the valence or the ionic charge (z is always positive). Mole Fraction (xa): The ratio of number of moles of a given solute (xa) to the total number of moles of all components in a solution. Mole fraction is particularly useful when the solvent contains more than one solute of interest (i.e., for mixtures). ca =
na na + nb + + nn
(4-13)
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33
Where xa is the mole fraction of solute a and na, nb, nn are the number of moles of solutes a, b, and n in a mixture respectively. Vapor Phase Concentrations: The vapor phase concentration of a pollutant is equal to the partial pressure exerted by that pollutant in the gaseous mixture. If the air is completely saturated with the pollutant, then this partial pressure is equal to its vapor pressure. Hence, it is common to find vapor phase concentrations expressed in units of pressure (e.g., Pa and Atm.). The vapor phase concentration expressed in units of pressure can be converted to mass over volume units as follows:
C
mol - pollutant P (atm) = L - air RT
(4-14)
Where P is the vapor pressure of the pollutant; C the concentration in mol/L and R is the Universal Gas Constant (0.0811 mol/atm-K) and T is the temperature expressed in degrees Kelvin. Other consistent set of units for pressure, gas constant and temperature can be used as well. Multiply Equation (4-14) by molecular weight (g/mol) of the pollutant to obtain concentrations in (g/L) units.
Concentration of one compound expressed in terms of another compound
There are a few instances when the concentration of one compound is expressed in terms of another. Let us explore two such situations: Mass Concentration as CaCO3: The concentration of ions in water especially Ca2+ and Mg2+ (i.e., the harness ions) are often expressed in terms of CaCO3. The molecular mass of calcium carbonate (CaCO3) is 100 g/mol. (Note: In most of this text I will drop the charges of the constituents for notational convenience). Since the ionic charge of this compound is 2 (i.e., z = 2), the equivalent mass (em) is em(CaCO3)
g 100 (g/mol) g mg or 50 = = 50 eq 2 (eq/mol ) eq meg
(4-15)
If the concentration of the substance is expressed in the units of meq./L, one could simply multiply this concentration with 50 mg/meq., to express the concentration in mg/L "as CaCO3). The following example illustrates the procedure: Example 4-3 A water sample contains 60 mg/L calcium and 40 mg/L magnesium, (1) express these concentrations in meq./L and (2) express then
34 Applied Environmental Systems Modeling in mg/L as CaCO3. (Assume the molecular mass of calcium is 40 g/mol and that of magnesium is 24.3 g/mol). We first write the concentration of Calcium (Ca) and Magnesium (Mg) in mol/L as follows:
Ca 60 mg - Ca 1 g - Ca 1 mol - Ca mol = L - water 1000 mg - Ca 40 g - Ca L
= 1.5 10 -3
Mg
mol - Ca L - water
(4-16)
40 mg - Mg 1g - Mg 1 mol - Mg mol = L - water 1000 mg - Mg 24.3 g - Ca L
= 1.65 10 -3
mol - Mg L - water
(4-17)
The next step is to express the molar concentrations calculated above in terms of equivalent mass concentrations as follows:
ca
2 eq - Ca 1000 meg mol - Ca meg = 1.5 10 -3 1 meq. L L - water 1 mol - Ca
= 3.0
meq. L
(4-18)
Mg
2 eq - Mg 1000 meq. mol - Mg meq. = 1.65 10 -3 1eq. L - water 1 mol - Mg L
= 3.3
meq. (4-19) L Once the concentrations are expressed in meq./L, the concentration of CaCO3 (Equation 4-15) can be used to express concentration in mg/L as CaCO3 as follows: Ca meq 50 mg mg as CaCO 3 = 3.0 L L meq.
= 150
mg - Ca as CaCO 3 L - water
...