About the examples

The following examples



Note: If you wish to replicate the R code below, then you will need to copy and paste the following commands in R first (to make sure you have all the packages and their dependencies):

install.packages("install.load")
# install the install.load package

install.load::install_load("ramify")
# install and/or load the packages and their dependencies


install.packages("import")
# install the import package

import::from(pracma, mldivide)
# import mldivide from the pracma package



1) Balance the following equation


sodium hydrogen carbonate + hydrochloric acid yields sodium chloride + carbon dioxide + water

NaHCO3 + HCl —> NaCl + CO2 + H2O


Written in a form to create the matrix using ramify as found in the Chemistry Fundamentals Program (UNC-Chapel Hill):

Na: 1a + 0b - 1c - 0d = 0e
H: 1a + 1b - 0c - 0d = 2e
C: 1a + 0b - 0c - 1d = 0e
O: 3a + 0b - 0c - 2d = 1e
Cl: 0a + 1b - 1c - 0d = 0e


Set up the matrix in R and then solve


library(ramify)

A <- mat("1, 0, -1, 0, 1; 1, 1, 0, 0, 1; 1, 0, 0, -1, 1; 3, 0, 0, -2, 1; 0, 1, -1, 0, 1")
# The ones are added to the last column of A to make it square.

A
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    0   -1    0    1
## [2,]    1    1    0    0    1
## [3,]    1    0    0   -1    1
## [4,]    3    0    0   -2    1
## [5,]    0    1   -1    0    1
b <- mat("0; 2; 0; 1; 0")

b
##      [,1]
## [1,]    0
## [2,]    2
## [3,]    0
## [4,]    1
## [5,]    0
coeffs1 <- inv(A) %*% b * det(A)
# determines the values of the other coefficients

coeffs1
##      [,1]
## [1,]   -3
## [2,]   -3
## [3,]   -3
## [4,]   -3
## [5,]    0
deta1 <- det(A)
# determines the value of the last coefficient

deta1
## [1] -3

a = 3
b = 3
c = 3
d = 3
e = 3

Thus, the final solution is 3NaHCO3 + 3HCl —> 3NaCl + 3CO2 + 3H2O



2) Balance the following equation (UNC-Chapel Hill)


iron (II) chloride + sodium phosphate yields iron (II) phosphate + sodium chloride

FeCl2 + Na3(PO4) —> Fe3(PO4)2 + NaCl


Written in a form to create the matrix using ramify as found in the Chemistry Fundamentals Program (UNC-Chapel Hill):

Fe: 1a + 0b - 3c = 0d
Cl: 2a + 0b - 0c = 1d
Na: 0a + 3b - 0c = 1d
(PO4): 0a + 1b - 2c = 0d


Set up the matrix in R and then solve


library(ramify)

A <- mat("1, 0, -3, 1; 2, 0, 0, 1; 0, 3, 0, 1; 0, 1, -2, 1")
# The ones are added to the last column of A to make it square.

A
##      [,1] [,2] [,3] [,4]
## [1,]    1    0   -3    1
## [2,]    2    0    0    1
## [3,]    0    3    0    1
## [4,]    0    1   -2    1
b <- mat("0; 1; 1; 0")

b
##      [,1]
## [1,]    0
## [2,]    1
## [3,]    1
## [4,]    0
coeffs2 <- inv(A) %*% b * det(A)
# determines the values of the other coefficients

coeffs2
##      [,1]
## [1,]   -3
## [2,]   -2
## [3,]   -1
## [4,]    0
deta2 <- det(A)
# determines the value of the last coefficient

deta2
## [1] -6

a = 3
b = 2
c = 1

Thus, the final solution is 3FeCl2 + 2Na3(PO4) —> 1Fe3(PO4)2 + 6NaCl



3) Balance the following equation (Andersen)


methane + oxygen yields carbon dioxide + water

CH4 + O2 —> CO2 + H2O


Written in a form to create the matrix using ramify as found in Andersen:

C: 1a + 0b = 1c + 0d
H: 4a + 0b = 0c + 2d
O: 0a + 2b = 2c + 1d


Set d = 1 to complete the system of equations

Set up the matrix in R and then solve


library(ramify)

A <- mat("1, 0, -1, 0; 4, 0, 0, -2; 0, 2, -2, -1; 0, 0, 0, 1")

A
##      [,1] [,2] [,3] [,4]
## [1,]    1    0   -1    0
## [2,]    4    0    0   -2
## [3,]    0    2   -2   -1
## [4,]    0    0    0    1
b <- mat("0; 0; 0; 1")

b
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    0
## [4,]    1
coeffs3 <- inv(A) %*% b
# determines the values of the coefficients

coeffs3
##      [,1]
## [1,]  0.5
## [2,]  1.0
## [3,]  0.5
## [4,]  1.0
coeffs3 <- coeffs3/min(coeffs3)
# obtains the coefficients as whole integers

coeffs3
##      [,1]
## [1,]    1
## [2,]    2
## [3,]    1
## [4,]    2

a = 1
b = 2
c = 1
d = 2

Thus, the final solution is 1CH4 + 2O2 —> 1CO2 + 2H2O



4) Balance the following equation (Andersen)


dichromate ion + iron (II) ion + hydrogen ion yields chromium (III) ion + iron (III) ion + water (Purdue University)

Cr2O7-2 + Fe2 + H1 —> Cr3 + Fe3 + H2O


Written in a form to create the matrix using ramify as found in Andersen:

Cr: 2a + 0b + 0c - 1d - 0e - 0f = 0
O: 7a + 0b + 0c - 0d - 0e - 1f = 0
Fe: 0a + 1b + 0c + 0d - 1e - 0f = 0
H: 0a + 0b + 1c + 0d - 0e - 2f = 0
Charge: -2a + 2b + 1c - 3d - 3e - 0f = 0


Set f = 1 to complete the system of equations

Set up the matrix in R and then solve


library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("2, 0, 0, -1, 0, 0; 7, 0, 0, 0, 0, -1; 0, 1, 0, 0, -1, 0; 0, 0, 1, 0, 0, -2; -2, 2, 1, -3, -3, 0; 0, 0, 0, 0, 0, 1")

A
##      [,1] [,2] [,3] [,4] [,5] [,6]
## [1,]    2    0    0   -1    0    0
## [2,]    7    0    0    0    0   -1
## [3,]    0    1    0    0   -1    0
## [4,]    0    0    1    0    0   -2
## [5,]   -2    2    1   -3   -3    0
## [6,]    0    0    0    0    0    1
b <- mat("0; 0; 0; 0; 0; 1")

b
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    0
## [4,]    0
## [5,]    0
## [6,]    1
coeffs4 <- mldivide(A, b)  # solve using pracma mldivide (left division or \)
# determines the values of the coefficients

coeffs4
##           [,1]
## [1,] 0.1428571
## [2,] 0.8571429
## [3,] 2.0000000
## [4,] 0.2857143
## [5,] 0.8571429
## [6,] 1.0000000
coeffs4 <- coeffs4/min(coeffs4)
# obtains the coefficients as whole integers

coeffs4
##      [,1]
## [1,]    1
## [2,]    6
## [3,]   14
## [4,]    2
## [5,]    6
## [6,]    7

a = 1
b = 6
c = 14
d = 2
e = 6
f = 7

Thus, the final solution is 1Cr2O7-2 + 6Fe2 + 14H1 —> 2Cr3 + 6Fe3 + 7H2O



5) Balance the following equation (Andersen)


diphosphorus tetraiodide + white phosphorus + water yields phosphonium iodide + phosphoric acid (Purdue University & Wikipedia)

P2I4 + P4 + H2O —> PH4I + H3PO4


Written in a form to create the matrix using ramify as found in Andersen:

P: 2a + 4b + 0c - 1d - 1e = 0
I: 4a + 0b + 0c - 1d - 0e = 0
H: 0a + 0b + 2c + 4d - 3e = 0
O: 0a + 0b + 1c + 0d - 4e = 0


Set e = 1 to complete the system of equations

Set up the matrix in R and then solve


library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("2, 4, 0, -1, -1; 4, 0, 0, -1, 0; 0, 0, 2, -4, -3; 0, 0, 1, 0, -4; 0, 0, 0, 0, 1")

A
##      [,1] [,2] [,3] [,4] [,5]
## [1,]    2    4    0   -1   -1
## [2,]    4    0    0   -1    0
## [3,]    0    0    2   -4   -3
## [4,]    0    0    1    0   -4
## [5,]    0    0    0    0    1
b <- mat("0; 0; 0; 0; 1")

b
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    0
## [4,]    0
## [5,]    1
coeffs5 <- mldivide(A, b)  # solve using pracma mldivide (left division or \)
# determines the values of the coefficients

coeffs5
##         [,1]
## [1,] 0.31250
## [2,] 0.40625
## [3,] 4.00000
## [4,] 1.25000
## [5,] 1.00000
coeffs5 <- coeffs5/min(coeffs5)
# attempt to obtain the coefficients as whole integers

coeffs5
##      [,1]
## [1,]  1.0
## [2,]  1.3
## [3,] 12.8
## [4,]  4.0
## [5,]  3.2
coeffs5 <- coeffs5 * 10
# obtains the coefficients as whole integers

coeffs5
##      [,1]
## [1,]   10
## [2,]   13
## [3,]  128
## [4,]   40
## [5,]   32

a = 10
b = 13
c = 128
d = 40
e = 32

Thus, the final solution is 10P2I4 + 13P4 + 128H2O —> 40PH4I + 32H3PO4



6) Balance the following equation (Sen)


potassium nitrate + carbon yields potassium carbonate + carbon monoxide + nitrogen

KNO3 + C —> K2CO3 + CO + N2


Written in a form to create the matrix using ramify as found in Sen:

K: 1a + 0b - 2c - 0d - 0e
N: 1a + 0b - 0c - 0d - 2e
O: 3a + 0b - 3c - 1d - 0e
C: 0a + 1b - 1c - 1d - 0e


Set up the matrix in R and then solve to obtain similar results to the MATLAB cheminpsolver from Sen


library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("1, 0, -2, 0, 0; 1, 0, 0, 0, -2; 3, 0, -3, -1, 0; 0, 1, -1, -1, 0")

b <- mat("0; 0; 0; 0")


m <- dim(A)[1]

m
## [1] 4
n <- dim(A)[2]

n
## [1] 5
P <- rbind(cbind(A, zeros(m, n)), cbind(eye(n), -eye(n)))
# non-homogeneous system element

P
##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
##  [1,]    1    0   -2    0    0    0    0    0    0     0
##  [2,]    1    0    0    0   -2    0    0    0    0     0
##  [3,]    3    0   -3   -1    0    0    0    0    0     0
##  [4,]    0    1   -1   -1    0    0    0    0    0     0
##  [5,]    1    0    0    0    0   -1    0    0    0     0
##  [6,]    0    1    0    0    0    0   -1    0    0     0
##  [7,]    0    0    1    0    0    0    0   -1    0     0
##  [8,]    0    0    0    1    0    0    0    0   -1     0
##  [9,]    0    0    0    0    1    0    0    0    0    -1
l <- rbind(zeros(m, 1), ones(n, 1))
# non-homogeneous system element

l
##       [,1]
##  [1,]    0
##  [2,]    0
##  [3,]    0
##  [4,]    0
##  [5,]    1
##  [6,]    1
##  [7,]    1
##  [8,]    1
##  [9,]    1
# Py = l

coeffs6 <- mldivide(P, l, pinv = FALSE)

coeffs6
##                [,1]
##  [1,]  2.000000e+00
##  [2,]  4.000000e+00
##  [3,]  1.000000e+00
##  [4,]  3.000000e+00
##  [5,]  1.000000e+00
##  [6,]  1.000000e+00
##  [7,]  3.000000e+00
##  [8,] -1.845734e-15
##  [9,]  2.000000e+00
## [10,]  0.000000e+00
coeffs6 <- coeffs6[1:5]

coeffs6
## [1] 2 4 1 3 1
coeffs6 <- coeffs6/min(coeffs6)
# obtains the coefficients as whole integers

coeffs6
## [1] 2 4 1 3 1

a = 2
b = 4
c = 1
d = 3
e = 1

Thus, the final solution is 2KNO3 + 4C —> 1K2CO3 + 3CO + 1N2



Works Cited

P. K. Andersen and G. Bjedov, Department of Freshman Engineering, Purdue University, “Chemical Stoichiometry Using MATLAB” by https://bloqm.files.wordpress.com/2015/03/estequiometria-com-matlab.pdf

Purdue University College of Science Chemical Education Division Groups Division of Chemical Education, “Nomenclature”, http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch2/names.html

S. K. Sen, Hans Agarwal, Sagar Sen, “Chemical equation balancing: An integer programming approach”, Mathematical and Computer Modelling 44 (2006) 678-691, http://www.irisa.fr/triskell/members/sagarsen/papers/folder.2009-12-15.3064396439/ploneexfile.2009-12-15.5737828636/attachment_download/file

UNC-Chapel Hill Department of Chemistry – Chemistry Fundamentals Program, “Balancing Equations Using Matrices”, https://www.shodor.org/unchem/math/matrix/ and https://www.shodor.org/unchem/math/matrix/balance3.html.

Wikimedia Foundation, Inc. Wikipedia, 27 May 2016, “Phosphine”, https://en.wikipedia.org/wiki/Phosphine.



---
title: "Using R to Balance Chemical Reactions (Stoichiometry)"
author: "Irucka Embry, E.I.T."
date: "`r Sys.Date()`"
output:
html_document:
mathjax: default
---

<br />
<br />

# About the examples

The following examples

<br />
<br />

Note: If you wish to replicate the R code below, then you will need to copy and paste the following commands in R first (to make sure you have all the packages and their dependencies):

```{r eval = FALSE}
install.packages("install.load")
# install the install.load package

install.load::install_load("ramify")
# install and/or load the packages and their dependencies


install.packages("import")
# install the import package

import::from(pracma, mldivide)
# import mldivide from the pracma package
```

<br />
<br />

# 1) Balance the following equation

<br />

sodium hydrogen carbonate + hydrochloric acid yields sodium chloride + carbon dioxide + water

NaHCO~3~ + HCl ---> NaCl + CO~2~ + H~2~O

<br />

Written in a form to create the matrix using `ramify` as found in the Chemistry Fundamentals Program (UNC-Chapel Hill):

Na: 1a + 0b - 1c - 0d = 0e
<br />
H: 1a + 1b - 0c - 0d = 2e
<br />
C: 1a + 0b - 0c - 1d = 0e
<br />
O: 3a + 0b - 0c - 2d = 1e
<br />
Cl: 0a + 1b - 1c - 0d = 0e
<br />

<br />

Set up the matrix in R and then solve

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)

A <- mat("1, 0, -1, 0, 1; 1, 1, 0, 0, 1; 1, 0, 0, -1, 1; 3, 0, 0, -2, 1; 0, 1, -1, 0, 1")
# The ones are added to the last column of A to make it square.

A


b <- mat("0; 2; 0; 1; 0")

b


coeffs1 <- inv(A) %*% b * det(A)
# determines the values of the other coefficients

coeffs1


deta1 <- det(A)
# determines the value of the last coefficient

deta1
```

a = `r abs(coeffs1[1])`
<br />
b = `r abs(coeffs1[2])`
<br />
c = `r abs(coeffs1[3])`
<br />
d = `r abs(coeffs1[4])`
<br />
e = `r abs(deta1[1])`

### Thus, the final solution is `r abs(coeffs1[1])`NaHCO~3~ + `r abs(coeffs1[2])`HCl ---> `r abs(coeffs1[3])`NaCl + `r abs(coeffs1[4])`CO~2~ + `r abs(deta1[1])`H~2~O

<br />
<br />

# 2) Balance the following equation (UNC-Chapel Hill)

<br />

iron (II) chloride + sodium phosphate yields iron (II) phosphate + sodium chloride

FeCl~2~ + Na~3~(PO~4~) ---> Fe~3~(PO~4~)~2~ + NaCl

<br />

Written in a form to create the matrix using `ramify` as found in the Chemistry Fundamentals Program (UNC-Chapel Hill):

Fe: 1a + 0b - 3c = 0d
<br />
Cl: 2a + 0b - 0c = 1d
<br />
Na: 0a + 3b - 0c = 1d
<br />
(PO~4~): 0a + 1b - 2c = 0d
<br />

<br />

Set up the matrix in R and then solve

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)

A <- mat("1, 0, -3, 1; 2, 0, 0, 1; 0, 3, 0, 1; 0, 1, -2, 1")
# The ones are added to the last column of A to make it square.

A


b <- mat("0; 1; 1; 0")

b


coeffs2 <- inv(A) %*% b * det(A)
# determines the values of the other coefficients

coeffs2


deta2 <- det(A)
# determines the value of the last coefficient

deta2
```

a = `r abs(coeffs2[1])`
<br />
b = `r abs(coeffs2[2])`
<br />
c = `r abs(coeffs2[3])`

### Thus, the final solution is `r abs(coeffs2[1])`FeCl~2~ + `r abs(coeffs2[2])`Na~3~(PO~4~) ---> `r abs(coeffs2[3])`Fe~3~(PO~4~)~2~ + `r abs(deta2[1])`NaCl

<br />
<br />

# 3) Balance the following equation (Andersen)

<br />

methane + oxygen yields carbon dioxide + water

CH~4~ + O~2~ ---> CO~2~ + H~2~O

<br />

Written in a form to create the matrix using `ramify` as found in Andersen:

C: 1a + 0b = 1c + 0d
<br />
H: 4a + 0b = 0c + 2d
<br />
O: 0a + 2b = 2c + 1d
<br />

<br />

Set d = 1 to complete the system of equations

Set up the matrix in R and then solve

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)

A <- mat("1, 0, -1, 0; 4, 0, 0, -2; 0, 2, -2, -1; 0, 0, 0, 1")

A


b <- mat("0; 0; 0; 1")

b


coeffs3 <- inv(A) %*% b
# determines the values of the coefficients

coeffs3

coeffs3 <- coeffs3 / min(coeffs3)
# obtains the coefficients as whole integers

coeffs3
```

a = `r abs(coeffs3[1])`
<br />
b = `r abs(coeffs3[2])`
<br />
c = `r abs(coeffs3[3])`
<br />
d = `r abs(coeffs3[4])`

### Thus, the final solution is `r abs(coeffs3[1])`CH~4~ + `r abs(coeffs3[2])`O~2~ ---> `r abs(coeffs3[3])`CO~2~ + `r abs(coeffs3[4])`H~2~O

<br />
<br />

# 4) Balance the following equation (Andersen)

<br />

dichromate ion + iron (II) ion + hydrogen ion yields chromium (III) ion + iron (III) ion + water (Purdue University)

Cr~2~O~7~^-2^ + Fe^2^ + H^1^ ---> Cr^3^ + Fe^3^ + H~2~O

<br />

Written in a form to create the matrix using `ramify` as found in Andersen:

Cr: 2a + 0b + 0c - 1d - 0e - 0f = 0
<br />
O: 7a + 0b + 0c - 0d - 0e - 1f = 0
<br />
Fe: 0a + 1b + 0c + 0d - 1e - 0f = 0
<br />
H: 0a + 0b + 1c + 0d - 0e - 2f = 0
<br />
Charge: -2a + 2b + 1c - 3d - 3e - 0f = 0
<br />

<br />

Set f = 1 to complete the system of equations

Set up the matrix in R and then solve

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("2, 0, 0, -1, 0, 0; 7, 0, 0, 0, 0, -1; 0, 1, 0, 0, -1, 0; 0, 0, 1, 0, 0, -2; -2, 2, 1, -3, -3, 0; 0, 0, 0, 0, 0, 1")

A


b <- mat("0; 0; 0; 0; 0; 1")

b


coeffs4 <- mldivide(A, b) # solve using pracma mldivide (left division or \)
# determines the values of the coefficients

coeffs4

coeffs4 <- coeffs4 / min(coeffs4)
# obtains the coefficients as whole integers

coeffs4
```

a = `r abs(coeffs4[1])`
<br />
b = `r abs(coeffs4[2])`
<br />
c = `r abs(coeffs4[3])`
<br />
d = `r abs(coeffs4[4])`
<br />
e = `r abs(coeffs4[5])`
<br />
f = `r abs(coeffs4[6])`

### Thus, the final solution is `r abs(coeffs4[1])`Cr~2~O~7~^-2^ + `r abs(coeffs4[2])`Fe^2^ + `r abs(coeffs4[3])`H^1^ ---> `r abs(coeffs4[4])`Cr^3^ + `r abs(coeffs4[5])`Fe^3^ + `r abs(coeffs4[6])`H~2~O

<br />
<br />

# 5) Balance the following equation (Andersen)

<br />

diphosphorus tetraiodide + white phosphorus + water yields phosphonium iodide + phosphoric acid (Purdue University & Wikipedia)

P~2~I~4~ + P~4~ + H~2~O ---> PH~4~I + H~3~P~O~4

<br />

Written in a form to create the matrix using `ramify` as found in Andersen:

P: 2a + 4b + 0c - 1d - 1e = 0
<br />
I: 4a + 0b + 0c - 1d - 0e = 0
<br />
H: 0a + 0b + 2c + 4d - 3e = 0
<br />
O: 0a + 0b + 1c + 0d - 4e = 0
<br />

<br />

Set e = 1 to complete the system of equations

Set up the matrix in R and then solve

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("2, 4, 0, -1, -1; 4, 0, 0, -1, 0; 0, 0, 2, -4, -3; 0, 0, 1, 0, -4; 0, 0, 0, 0, 1")

A


b <- mat("0; 0; 0; 0; 1")

b


coeffs5 <- mldivide(A, b) # solve using pracma mldivide (left division or \)
# determines the values of the coefficients

coeffs5

coeffs5 <- coeffs5 / min(coeffs5)
# attempt to obtain the coefficients as whole integers

coeffs5

coeffs5 <- coeffs5 * 10
# obtains the coefficients as whole integers

coeffs5
```

a = `r abs(coeffs5[1])`
<br />
b = `r abs(coeffs5[2])`
<br />
c = `r abs(coeffs5[3])`
<br />
d = `r abs(coeffs5[4])`
<br />
e = `r abs(coeffs5[5])`

### Thus, the final solution is `r abs(coeffs5[1])`P~2~I~4~ + `r abs(coeffs5[2])`P~4~ + `r abs(coeffs5[3])`H~2~O ---> `r abs(coeffs5[4])`PH~4~I + `r abs(coeffs5[5])`H~3~P~O~4

<br />
<br />

# 6) Balance the following equation (Sen)

<br />

potassium nitrate + carbon yields potassium carbonate + carbon monoxide + nitrogen

KNO~3~ + C ---> K~2~CO~3~ + CO + N~2~

<br />

Written in a form to create the matrix using `ramify` as found in Sen:

K: 1a + 0b - 2c - 0d - 0e
<br />
N: 1a + 0b - 0c - 0d - 2e
<br />
O: 3a + 0b - 3c - 1d - 0e
<br />
C: 0a + 1b - 1c - 1d - 0e
<br />

<br />

Set up the matrix in R and then solve to obtain similar results to the MATLAB cheminpsolver from Sen

<br />

```{r, warning = FALSE, message = FALSE, tidy = TRUE}
library(ramify)


import::from(pracma, mldivide)
# import mldivide from the pracma package


A <- mat("1, 0, -2, 0, 0; 1, 0, 0, 0, -2; 3, 0, -3, -1, 0; 0, 1, -1, -1, 0")

b <- mat("0; 0; 0; 0")


m <- dim(A)[1]

m


n <- dim(A)[2]

n


P <- rbind(cbind(A, zeros(m, n)), cbind(eye(n), -eye(n)))
# non-homogeneous system element

P


l <- rbind(zeros(m, 1), ones(n, 1))
# non-homogeneous system element

l


# Py = l

coeffs6 <- mldivide(P, l, pinv = FALSE)

coeffs6

coeffs6 <- coeffs6[1:5]

coeffs6

coeffs6 <- coeffs6 / min(coeffs6)
# obtains the coefficients as whole integers

coeffs6
```

a = `r abs(coeffs6[1])`
<br />
b = `r abs(coeffs6[2])`
<br />
c = `r abs(coeffs6[3])`
<br />
d = `r abs(coeffs6[4])`
<br />
e = `r abs(coeffs6[5])`

### Thus, the final solution is `r abs(coeffs6[1])`KNO~3~ + `r abs(coeffs6[2])`C ---> `r abs(coeffs6[3])`K~2~CO~3~ + `r abs(coeffs6[4])`CO + `r abs(coeffs6[5])`N~2~

<br />
<br />

## Works Cited

P. K. Andersen and G. Bjedov, Department of Freshman Engineering, Purdue University, "Chemical Stoichiometry Using MATLAB" by  https://bloqm.files.wordpress.com/2015/03/estequiometria-com-matlab.pdf

Purdue University College of Science Chemical Education Division Groups Division of Chemical Education, “Nomenclature”, http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch2/names.html

S. K. Sen, Hans Agarwal, Sagar Sen, "Chemical equation balancing: An integer programming approach", *Mathematical and Computer Modelling* 44 (2006) 678-691, http://www.irisa.fr/triskell/members/sagarsen/papers/folder.2009-12-15.3064396439/ploneexfile.2009-12-15.5737828636/attachment_download/file

UNC-Chapel Hill Department of Chemistry -- Chemistry Fundamentals Program, "Balancing Equations Using Matrices", https://www.shodor.org/unchem/math/matrix/ and https://www.shodor.org/unchem/math/matrix/balance3.html.

Wikimedia Foundation, Inc. Wikipedia, 27 May 2016, “Phosphine”, https://en.wikipedia.org/wiki/Phosphine.

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