> For the complete documentation index, see [llms.txt](https://daniel-soto.gitbook.io/qm/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://daniel-soto.gitbook.io/qm/common-functions/concentrations.md). # Concentrations ## Overview ### Motivation What happens if we spill an amount of a toxic chemical into a lake? How much of the chemical do we have to spill before it is dangerous? How much toxin from cyanobacteria must be released before the shellfish are dangerous to eat? To make these estimations, we can use concentrations as a tool. When you encounter the terms parts per million or parts per billion, these work exactly like percentages (parts per hundred). Many of you have a command of percentages. These concentrations are simply generalizations of the concept of percentages. Once you realize that a percentage is equivalent to parts per hundred, you will be able to understand and use parts per million and billion confidently. ### Examples of concentrations * Carbon Dioxide in the atmosphere * Blood Alcohol content * What are examples of concentrations you think are important? ### Concentration Units * Percentages * Parts per million (ppm) * Parts per billion (ppm) * Parts per trillion (ppt) Do these have dimensions? Do they have units? ## Details ### Concentration We usually define a concentration as the ratio of a quantity of interest (pollutant) to the total volume of the material containing the pollutant. $$\frac{\textrm{mass or volume of material of interest}} {\textrm{total mass or volume}}$$ ### Concentration Visualization * How many cubic centimeters in a cubic meter? * Each cubic centimeter is one millionth of a cubic meter ### Example: Blood Alcohol Concentration (BAC) $$BAC = \frac{\textrm{volume of alcohol}} {\textrm{volume of blood}}$$ How much pure ethyl alcohol is in the average blood stream at the legal limit? ### Example: Carbon Dioxide $$\frac{\textrm{volume carbon dioxide}} {\textrm{volume of atmosphere}}$$ We are currently at about 400 ppm. ### Converting between concentrations To convert between different types of fractions, you can use your existing intuition from percentages. The main difference is that instead of parts per hundred, you will likely be using parts per million or billion. ### Percentage Conversion If a concentration is 50 percent, what is the fraction (a number between 1 and 0). $$50% = \frac{50\ parts}{100\ parts} = 0.5$$ ### Parts Per Million Conversion For the concentration of CO2 in the atmosphere, it is currently 400 parts per million. $$400\ ppm = \frac{400 parts\ CO2}{10^6 parts air} = 0.0004$$ To convert this fraction to a percentage, we multiply the top and bottom by the number (100) that gives us 100 parts on the bottom. $$400\ ppm = 0.0004 = 0.0004 \frac{parts CO2}{1 part air} \cdot \frac{10^2}{10^2} = \frac{4 \cdot 10^{-2}}{10^2 parts}$$ We can do the same but this time the number is $10^{-4}$ to give us a percentage. $$400\ ppm = \frac{400 parts CO2}{10^6 parts air} \cdot \frac{10^{-4}}{10^{-4}} = \frac{4 \cdot 10^{-2}}{10^2 parts}$$ ### Dimensions and Units Concentrations are dimensionless but have specific physical meanings that you must keep track of. ### pH scale * Acids have low pH * Bases have high pH * This scale is logarithmic in base 10 * One pH is a factor of 10 of the concentration of hydrogen ions ### Concentration by weight * Sugar in a can of soda * Written ## Changing Concentrations How does the concentration change if we add more of a pollutant over time? We can add a linear model to our analysis. ### Limiting examples ## Examples ### Bathtub model of atmosphere ![Bathub Model of Atmosphere](https://github.com/dsoto/ensp-202-notes/tree/12112541a7540ac807b5e2e0939be0fd9da2140c/figures/carbon-bathtub.jpg) ### Atmospheric CO2 ![](https://github.com/dsoto/ensp-202-notes/tree/12112541a7540ac807b5e2e0939be0fd9da2140c/figures/Carbon_Dioxide_400kyr_Rev.png) ### Carbon Dioxide * [Keeling Curve Website](http://keelingcurve.ucsd.edu/) * [Scripps Oxygen Measurements](http://scrippso2.ucsd.edu/) ### Parts by volume and parts by mass * We model the atmosphere as an ideal gas * This means that the volume fraction is the same as the fraction of the number of molecules * To convert from volume to mass, we use the molar mass ### Molar mass * One mole equals 6.02 $\cdot$ 10^23^ atoms or molecules * Carbon weighs 12 grams for every mole of carbon * Oxygen weighs 16 grams for every mole of oxygen * How much does one mole of carbon dioxide weigh? * Our atmosphere weighs 29 grams for every mole ## Activities ### ETC Carbon Dioxide Concentration During class, the ETC CO2 concentration increases. Is it increasing at the rate you would expect? ### Dilution Demonstration What is the concentration of a chemical by volume after a number of 100 to 1 dilutions? We demonstrate this with ink and water in a beaker For our 100 to 1 dilutions, the concentration is $$10^{-2\cdot\textrm{dilutions}}$$ ### Molarity of mercury? ### Molarity of soda?