Activity Overview Students will balance chemical equations using systems of equations, inverse matrices, and determinants.

Transcription

1 Getting Strte with Alger Blne Me ID: XXXX By Pt Flynn Time require 60 minutes Ativity Overview Stuents will lne hemil equtions using systems of equtions, inverse mtries, n eterminnts. Conepts Inverse mtrix metho for solving systems of equtions Teher Preprtion This tivity is intene for use with stuents who re enrolle or were enrolle in high shool hemistry. If this is not the se, then some introutory work nees to e one with the stuents in terms of hemil equtions n the lning of them. Seek the help of your shool s hemistry instrutor for help in this. In hemistry, mny stuents enounter iffiulties lning hemil equtions even when they re following steps outline y their instrutor. The purpose of this tivity is to provie the stuents metho to use when ll else fils. Stuents shoul lrey e fmilir with mtrix representtion for system of equtions s well s the onept of the eterminnt. Mthemtil prerequisites from Alger inlue fmilirity with ifferent methos of solving systems of equtions, inluing rief exposure to the use of mtries in oing so. The sreenshots on pges 5 emonstrte expete stuent results. Refer to the sreenshots on pges 6 n 7 for preview of the stuent TI-Nspire oument (.tns file). To ownlo the stuent.tns file n stuent worksheet, go to eution.ti.om/exhnge n enter XXXX in the quik serh ox. Clssroom Mngement This tivity is intene to e omplete iniviully, with the teher filitting instrution. Use the following pges s neee to ssist the lss in ompleting the tivity. If time is limite, you my hoose to hve stuents omplete Prolems 1 n in lss, ssigning the exerises from Prolem 3 s homework. There is no stuent worksheet for this tivity. The stuent TI-Nspire oument provies ple for stuents to reor their nswers. Alterntively, you my wish to hve the lss reor their nswers on seprte sheets of pper, or just use the questions pose to engge lss isussion. The TI-Nspire solution oument AlgAt_BlneMe_Soln_EN.tns shows the expete results of working through the tivity. TI-Nspire Applitions Clultor, Notes 008 Texs Instruments Inorporte Pge 1

2 Getting Strte with Alger Prolem 1 Blning Al O AlO 3 Prolem 1 wlks stuents through how to lne the hemil eqution Al O Al O. There re no tions to e performe y stuents for the first 3 three steps; they shoul just e sure they re n unerstn eh step. Step 1: Assign vrile oeffiient to eh of the three terms in the eqution: Al O AlO3 Step : Trnslte the hemil eqution into mthemtil system of equtions, s shown in the sreenshot to the right. There shoul e n eqution for eh element in the hemil eqution. If neee, remin stuents tht the oeffiients in these two equtions re erive from the numer of toms of the element in eh hemil ompoun. Step 3: Write the system of equtions s mtrix eqution. For stuents who re omfortle with using mtries to solve equtions, this mtrix eqution shoul look fmilir, exept for the pperne of the slr,. Tell stuents tht vlue will e hosen for in the next step. Step : Pge 1.6 explins tht the solute vlue of the eterminnt of the oeffiient mtrix shoul e use s the vlue of. Using the eterminnt ensures tht the vlues foun for n in the next step re integers. Ask stuents why it mkes sense to use the solute vlue of the eterminnt ( must e positive integer; it represents numer of moleules.) Stuents my rell tht the eterminnt of mtrix A is given y et( A), ut stuents n use the mthemtil pilities of the Notes pplition to evlute the eterminnt. Instrut them to hol own the SHIFT key (g) while they rrow over the expression to selet it, n hoose Evlute Seletion from the Ations menu. Pressing will return the vlue of ; here,. 008 Texs Instruments Inorporte Pge

3 Getting Strte with Alger Step 5: To solve this eqution for, oth sies of the eqution must e multiplie y the inverse of the oeffiient mtrix. Remin stuents tht mtrix multiplition is not ommuttive, n must e one in the orer shown. Stuents shoul sustitute for n then they n use Evlute Seletion on the seletion shown to the right to etermine. 3 Step 6: Now, stuents nee to sustitute the vlues they otine for,, n, n hek tht the vlues inee lne the hemil eqution. Al 3 O AlO3 Aoring to these vlues, there re luminum toms on the left-hn sie n = luminum toms on the right-hn sie, so the luminum is lne. Similrly, there re 3 = 6 oxygen toms on the left n 3 = 6 oxygen toms on the right. Therefore, the hemil eqution is lne. Prolem Blning CH6 O CO HO Prolem sks stuents to lne the hemil eqution CH6 O CO HO. Note tht this eqution hs four terms n three elements, so it will require system of three equtions, rther thn two. For this exmple, stuents will omplete more of the require steps themselves. Step 1: Assign vrile oeffiient to eh of the four terms in the eqution. Step : Trnslte the hemil eqution into mthemtil system of equtions, s shown. On pge.3, stuents will fin there is n extr step require here. The terms shoul e sutrte from oth sies in orer to isolte the term on the right-sie of the eqution. Stuents n opy (/ + C) n pste (/ + V) their equtions from pge. to pge.3 inste of retyping them.) After mnipultion, the equtions shoul e s shown elow. C: H: O: Texs Instruments Inorporte Pge 3

4 Step 3: Write the system s mtrix eqution. Here, the oeffiient mtrix is 3 3 mtrix, s shown to the right. Step : Fin the vlue of. This oeffiient mtrix hs eterminnt of 1, so this is n exmple of why you shoul use the solute vlue of the eterminnt. Agin, stuents n use opy/pste n the mthemtil pilities of the Notes pplition to fin this vlue. Step 5: The vlues for,, n my then e foun s shown elow. Getting Strte with Alger Step 6: Now, the vlues of,,, n my e sustitute k into the originl hemil eqution. However, stuents re not rey to hek their solutions. Be sure stuents unerstn the informtion on pge.6 regring reuing the oeffiients in hemil y ny ommon ftors. Stuents shoul then hek these reue vlues, s efore, on pge.7. Prolem 3 More prtie Prolem 3 provies stuents with eight exerises to prtie lning hemil equtions using inverse mtries n eterminnts. Stuents my nswer iretly in the stuent TI-Nspire oument (see TI-Nspire solution oument) or on seprte sheets of pper. Solutions to eh prolem n given elow. 1. Mg P Mg3P. H N NH 3 Mg P Mg3P Mg: P: 0 0 H N NH3 H: N: ; 1 ; 6 1Mg P Mg3P, whih reues to: 6Mg P Mg3P 6H N NH3, whih reues to: 3H N NH3 008 Texs Instruments Inorporte Pge

5 Getting Strte with Alger 3. P O P O5 P: 0 0 O: ; 0 P 0O 8 P O5, whih reues to: P 5O P O5. S O SO6 S: O: ; 8 6 8S 6O SO6, whih reues to: S 3 O SO6 5. KO CO KCO3 O K: O: 3 3 C: FeO3 CO Fe CO Fe: O: C: ; 1 3; 3 KO CO K CO3 3 O FeO3 3CO Fe 3 CO 7. C3H8 O CO HO C: H: O: ; 0 1 C3H 8 0O 1CO 16 HO, whih reues to: C3H8 5O 3CO HO 8. CH3Cl O CO HO e HCl C: e H: e Cl: e O: 0 1 0e e ; e CH 3Cl 10O 8CO HO HCl, whih reues to: CH 3Cl 5O CO H O HCl 008 Texs Instruments Inorporte Pge 5

6 Getting Strte with Alger Blne Me ID: XXXX (Stuent)TI-Nspire File: AlgAt_BlneMe_EN.tns 008 Texs Instruments Inorporte Pge 6

7 Getting Strte with Alger 008 Texs Instruments Inorporte Pge 7