Computer Applications in Engineering and Construction Programming Assignment #4 Chemical equilibrium using root-finding techniques

Transcription

1 CVEN Computer Applications in Engineering and Construction Programming Assignment #4 Chemical equilibrium using root-finding techniques Date distributed : 9/30/2015 Date due : 10/9/2015 at 8:00 a.m. Upload your memorandum and program code (in one Word file) to turnitin.com by the date and time listed above. Demonstrate your work in the memordandum text, and follow the directions in the course syllabus. 1 Introduction The composition of air in the Earth s atmosphere includes the dominant gases, nitrogen and oxygen, and many trace gases, including argon, carbon dioxide, and many others. Carbon dioxide is weakly soluble in water and behaves like an acid when dissolved. This makes the natural rain in a clean atmosphere slightly acidic as the carbon dioxide dissolves into the rain droplets. The chemical reactions in the rain droplet can be described by the following equilibrium reactions: CO 2 (g) CO 2 (aq) CO 2 (aq) + H 2 O H 2 CO 3 H 2 CO 3 H + + HCO3 HCO3 H+ + CO3 2 H 2 O H + + OH Carbon dioxide dissolves into water and then reacts with water nearly to completion to form H 2 CO 3, carbonic acid. The carbonic acid dissociates in solution to form HCO3, bicarbonate, and CO3 2, carbonate, each time also releasing a hydrogen ion. Finally, water also dissociates into its ionic components, H +, the hydrogen ion, and OH, the hydroxyl group. Each of these equilibrium equations has an associated equilibrium constant, the form of which depends on the reaction and common usage. Because the dissolved CO 2 reacts nearly fully in water to form H 2 CO 3, the solubility constant is usually written for the equivalent reaction CO 2 (g) + H 2 O H 2 CO 3 (aq) This dissolution reaction is governed by Henry s law, and at equilibrium, the concentrations of CIVIL 3136 TAMU College Station, Texas (979) FAX (979)

2 Table 1: Composition of carbon dioxide in water Components H + H 2 CO 3 H 2 0 Species H 2 CO HCO CO H OH Recipe CO H 2 O CO 2 (g) and H 2 CO 3 (aq) are related by the Henry s law coefficient, giving p CO2 = [H 2CO 3 ] K H where p CO2 is the partial pressure of carbon dioxide in the atmosphere and the [ ]-brackets indicate concentration. Here, we will use units of atm for pressure and M for concentration, where M is the molar concentration, given by moles of solute in moles of solvent; in this case, moles of carbonic acid per mole of water. Note that M is dimensionless in this unit system. The equilibrium constants for the two dissociation reactions for carbonic acid and bicarbonate are given by acidity constants. In the present case, they are K a1 = [H+ ][HCO 3 ] [H 2 CO 3 ] K a2 = [H+ ][CO 2 3 ] [HCO 3 ] (3) where K a1 and K a2 are dimensionless. The dissociation of water is normally related to the ion product of water, given by K w = [H + ][OH ] (4) where K w is also dimensionless. These equations are all coupled, so that the concentration [H + ] in one reaction must also be the same value in all other reactions at equilibrium. To solve this system of reactions, we must define the fundamental components of the system, the recipe for the mixture, and identify the individual species in each of the equations. The fundamental components in the water side of the system are H +, H 2 CO 3, and H 2 O, from which each of the remaining chemical species in the reaction equations can be assembled. The recipe for this mixture is simply carbon dioxide plus water. Table 1 shows how each of the species present in the mixture and the mixture recipe can be formed from the fundamental components. In the table, the numbers give the numbers of each of the fundamental components needed to form the chemicals listed in the left-hand column. (1) (2) 2

3 Each column of Table 1 represents a mass balance for each of the fundamental components of the system. Hence, to determine the total hydrogen ion content, we must add the first column together: T OT H + = [HCO 2 ] 2[CO2 3 ] + [H+ ] [OH ] = 0 (5) If we rearrange equations 1 through 4 to solve for the components of the above equation in terms of [H + ] and p CO2, then we obtain the governing equation for the hydrogen ion (after multiplying by minus one) K a1 K H p CO2 [H + ] + 2 K a1k a2 K H p CO2 [H + ] 2 [H + ] + K w [H + ] = 0 (6) In practice, we usually refer to the hydrogen ion concentration by the ph, which is defined as ph = log 10 [H + ] (7) In this programming assignment, we will explore methods to solve Equation 6 to compute the ph of rain water using root-finding techniques. 2 Assignment 2.1 ph of Natural Rain Water To find the ph of natural rain water for a given pco 2 concentration, it is necessary to solve Equation 6 for [H + ]. The equilibrium constants to use for this exercise are found by experiment and are given by K H,CO2 = M/atm K a1,co2 = K a2,co2 = K w = Do each of the following: 1. Write a Matlab function that returns the value of the LHS of Equation 6 given the inputs of ph and pco 2. For the root-finding problem, pco 2 will be constant, and ph will be the independent variable we adjust during each step of the root finding algorithm. 2. Check your Matlab function using hand calculations. Compute the correct result using a hand calculator for the case pco 2 = atm (280 ppm) and ph = 7. Note that the result will not be zero. Compute the corresponding result using your Matlab function. Document the results of your hand calculation check and the output from your function in the laboratory memorandum. 3. Write a separate Matlab function to implement the bisection method to solve for the root of Equation 6. Write a third Matlab.m-file to contain the script that will handle solving the whole root-finding problem. Define the required error criterion so that your program can 3

4 find the ph accurate to two decimal places; you should pass this criterion as input to your bisection method function. Test your code by solving for the ph given pco 2 = 280 ppm. In your laboratory memorandum, discuss how you set the upper and lower bounds for the bisection method, the error criterion you used, the solution for this case, and the number of iterations required to converge on the solution. Compare results at this stage to a separate set of hand calculations for the real root of the equation. 4. Plot the ph of rain droplets as a function of pco 2 for a range spanning 280 ppm to 420 ppm. Atmospheric CO 2 concentration has been measured on Mauna Loa in Hawaii for over 50 years. Look at the data available at: and discuss your plot in light of this trend in the text of your laboratory memorandum. 2.2 Acid Rain due to SO 2 Anthropogenic emission of SO 2, sulfur dioxide, to the atmosphere occurs primarily through fossil fuel combustion (coal, natural gas, etc.) and can also lead to lower ph in rain. Some reactions of SO 2 with hydrogen peroxide and ozone lead to sulfuric acid, H 2 SO 4, but reactions with pure water follow a similar set of equilibrium reactions to those for CO 2. These latter reactions are described by the following SO 2 (g) + H 2 O H 2 SO 3 (aq) H 2 SO 3 H + + HSO3 HSO3 H+ + SO3 2 H 2 O H + + OH The equilibrium constants for these reactions are different from those for CO 2 due to the different nature of the sulfur molecule, and are given by K H,SO2 = 1.23 M/atm K a1,so2 = K a2,so2 = 10 7 K w = For this set of parameter values, do each of the following: Show that the equation for [H + ] for SO 2 dissolution has the same form as Equation 6. Modify your function from item 1 above so that the equilibrium constants are provided as input and generalize the comments in your function to indicate that the partial pressure input can be for any gas that behaves in a similar way. Use your function and the built-in Matlab solver fzero to find the ph for rainwater due to SO 2 dissolution over the range of partial pressures pso 2 = 1 ppb to 10 ppb. You should 4

5 write a new driver program that defines the solubility constants and interacts with the fzero command. Be sure that ph is returned with at least two decimal places of accuracy, as before. Use fzero and a new definition of the root-finding problem to determine the pso 2 concentration that gives the same ph as a raindrop subject to 280 ppm CO 2. Think about what the relevant dependent variable is and what the appropriate root-finding equation would be. Discuss the results of each step in this section as part of your laboratory memorandum. 3 Acknowledgements This programming assignment was adapted from a similar assignment designed by Professor Cahill. A Using Matlab s fzero Command If the function for which you would like to find the root has only one input variable, then the normal usage of fzero can be used: xr = fzero(myfun, x0) where myfun is the name of your function file and x0 is an initial guess for the root. If, however, the function myfun takes additional inputs, then there are two main ways to do this. The way demonstrated in the Matlab help is to use anonymous functions. This has the disadvantage that if you change the value of the input parameters after the definition of the anonymous function, their values are not updated in the function unless you repeat the definition of the anonymous function. To avoid this problem, Matlab also allows you to pass the additional input parameters to the function through a list of variables at the end of the call to fzero. To do this, you must also pass the structure of options. A simple solution is to create an options structure using the default options. As an example, assume your function takes two inputs: myfun(x, A). You must be sure that the first input to the function is the independent variable you will adjust to find the root. Then, you can pass A to myfun from fzero as follows: opts = optimset; xr = fzero( myfun, x0, opts, A) Note that myfun must be enclosed in single quotes for this to work. The function optimset is a built-in Matlab function that returns the default options structure. Leave off the ; to see what is contained in this structure. An alternative syntax is to precede the function name by opts = optimset; xr = fzero(@myfun, x0, opts, A) This works for any number of input variables: you just list all of them starting from the position of A in the above two commands. This method works for most Matlab built-in functions that take user-defined functions as input, including fminsearch and the suite of ODE solvers. 5