Skip to main content
Engineering LibreTexts

6.5: Problems

  • Page ID
    44497
  • Problems marked with the symbol \(\ddagger\) will be difficult to solve without the use of computer software.

    Section 6.1

    1. By “counting atoms” provide at least one version of a balanced chemical equation based on

    \[? \ce{C2} \ce{H6} + { ?} \ce{O2} \to { ?} \ce{ CO} + { ?} \ce{C2} \ce{H4O} + { ?} \ce{H2O} + \ce{CO2} \nonumber\]

    that is different from the two examples given in the text.

    Section 6.2

    2. Construct an atomic matrix for the following set of components: Sodium hydroxide (Na OH), methyl bromide (\(\ce{CH3Br}\)), methanol (\(\ce{CH3OH}\)), and sodium bromide (NaBr).

    3. Construct an atomic matrix for a system containing the following molecular species: \(\ce{NH3}\), \(\ce{O2}\), \(\ce{NO}\), \(\ce{N2}\), \(\ce{H2O}\), and \(\ce{NO2}\). Find the rank of this matrix.

    4. Begin with the statement that mass is neither created nor destroyed by chemical reaction

    \[\sum_{A = 1}^{A = N}r_{A} =0 \nonumber\]

    and use it to derive Equation \((6.2.8)\). Be careful to state any restrictions that might be necessary in order to complete the derivation.

    5. The rank of a matrix is conveniently determined using the row reduced form of the matrix. Consider the atomic matrix given by Eq. 2 of Example \(6.2.1\) and use elementary row operations to develop the row reduced echelon form of that matrix.

    6. Using Eq. \((6.2.8)\), show how to obtain Eqs. 4 in Example \(6.2.1\).

    7. Use elementary row operations to express Eq. 5 of Example \(6.2.1\) in terms of the row reduced echelon form of the atomic matrix. Indicate how Eqs. 6 are obtained using the row reduced echelon form.

    8. First find the rank of the atomic matrix developed in Problem 3. Next, choose \(\ce{N2}\), \(\ce{H2O}\), and \(\ce{NO2}\) as the pivot species and develop a solution for \(R_{\ce{NH3}}\), \(R_{\ce{O2}}\) and \(R_{\ce{NO}}\).

    9. Express the atomic matrix in Eq. 12 of Example \(6.2.2\) in row reduced echelon form. Use that form to express \(\mathscr{R}_{\ce{C2H6}}\) and \(\mathscr{R}_{\ce{C2H4}}\) in terms of \(\mathscr{R}_{\ce{H2}}\).

    10. Represent Eqs. 13 of Example \(6.2.2\) in the form of the pivot theorem illustrated by Eq. 7 of Example \(6.2.1\).

    11. In this problem you are asked to consider the complete combustion of methanol, thus the molecular species under consideration are

    \[\ce{CH3OH} , \quad \ce{O2} , \quad \ce{H2O} , \quad \ce{CO2} \nonumber\]

    Develop the atomic matrix in row reduced echelon form, and use Axiom II with \(\ce{CO2}\) as the pivot species in order to determine the rates of production, \(R_{ \ce{CH3OH}}\), \(R_{ \ce{O2} }\), \(R_{ \ce{H2O}}\) in terms of \(R_{ \ce{CO2} }\). Express your results in the form analogous to Eq. 7 of Example \(6.2.1\).

    12. Consider the complete combustion of methane to produce water and carbon dioxide. Construct the atomic matrix in row reduced echelon form, and show that Axiom II can be used to express the rates of production of methane, oxygen and water in terms of the single pivot species, carbon dioxide. Express your results in the form of the pivot theorem illustrated by Eq. 7 of Example \(6.2.1\).

    13. For the molecular species listed in Problem 2, determine the ratio of rates of production given by

    \[\frac{R_{\ce{NaOH}} }{R_{\ce{NaBr}} } , \quad \frac{R_{\ce{CH3Br}}}{R_{\ce{NaBr}} } , \quad \frac{R_{\ce{CH3OH}} }{R_{\ce{NaBr}} } \nonumber\]

    14. The production of alumina, (\(\ce{NaAlO2}\)), from bauxite, \([\ce{Al(OH)3}]\), requires sodium hydroxide (\(\ce{NaOH}\)) as a reactant and yields water (\(\ce{H2O}\)) as product. For this system, determine the ratio of the rates of production given by

    \[\frac{R_{ \ce{Al(OH)3}} }{R_{\ce{NaAlO2}} } , \quad \frac{R_{ \ce{NaOH}} }{R_{ \ce{NaAlO2} } } , \quad \frac{R_{\ce{H2O}} }{R_{ \ce{NaAlO2}} } \nonumber\]

    15. At \(T=20\) C, the rate of disappearance (i.e., the net rate of production) of methyl bromide in the reaction between methyl bromide and sodium hydroxide (Problems 2 and 12) is

    \[- R_{ \ce{CH3Br}} =0.2 \ { mol/m}^{ 3} \ { s} \nonumber\]

    The molecular species involved in this reaction are sodium hydroxide (NaOH), methyl bromide (\(\ce{CH3Br}\)), methanol (\(\ce{CH3OH}\)) and sodium bromide (NaBr). In this problem you are asked to

    (a). Determine the rate of production of sodium hydroxide, methanol, and sodium bromide in kmol/m\(^{3}\)s.

    (b). Determine the rates of production for all components in kg/m\(^{3}\)s.

    (c). Show that mass is neither created nor destroyed by this chemical reaction.

    (d). Verify that atomic species are neither created nor destroyed by this chemical reaction.

    16. Methanol can be synthesized by reacting carbon monoxide and hydrogen over a catalyst. The rate of production of methanol at 400 K is of \(r_{\ce{CH3OH}} = 0.035 \ kg/m^3 s\). Determine the rate of production in kmol/m\(^{3}\)s for all components of the synthesis reaction.

    17. Given an atomic matrix of the form

    \[\mathbf{A} = \begin{bmatrix} 1 & 2 & 1 & 4 \\ 0 & 2 & 4 & 2 \\ 0 & 1 & 5 & 4 \\ 0 & 1 & 3 & 2 \end{bmatrix}\nonumber\]

    indicate how one can obtain a row reduced matrix of the form

    \[\mathbf{A^{\prime\prime}} = \begin{bmatrix} 1 & 2 & 1 & 4 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}\nonumber\]

    18. For the partial oxidation carbon described in Example \(6.2.3\), explore the use of pivot species other than CO and \(\ce{CO2}\). This will lead to five more possibilities in addition to those given by Eqs. 7 in Example \(6.2.3\).

    19. Use elementary row operations to verify that Eq. 4 leads to Eq. 5 in Example \(6.2.4\).

    20. Express Eqs. 6 of Example \(6.2.4\) in the form of the pivot theorem as illustrated by Eq. 9 in Example \(6.2.3\).

    21. Meta-cresol sulfonic acid is a reddish brown liquid that is used as a pharmaceutical intermediate in the manufacture of disinfectants and in the manufacture of friction dust for disk brakes. This chemical is produced along with water by the sulfonation of meta-cresol with sulfuric acid. In this problem you are first asked to prove that the rank of the atomic matrix is three. Then show that the net rates of production of meta-cresol (\(\ce{C7H8O}\)), sulfuric acid (\(\ce{H2SO4}\)) and meta-cresol sulfonic acid (\(\ce{C7H8O4S}\)) can be expressed in terms of the net rate of production of water (\(\ce{H2O}\)).

    22. In Sec. 6.2.6 we illustrated how to construct a column/row partition of a matrix equation, and here you are asked to repeat that type of analysis for a row/row partition of the following matrix equation:

    \[\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \\ a_{51} & a_{52} & a_{53} \end{bmatrix} \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \\ c_5 \end{bmatrix}\label{1a}\]

    If the \(5 \times 3\) matrix is partitioned according to

    \[\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ a_{41} & a_{42} & a_{43} \\ \hdashline a_{51} & a_{52} & a_{53} \end{bmatrix}\label{2a}\]

    indicate how Equation \ref{1a} must be partitioned. Clearly indicate how the partitioned version of Equation \ref{1a} leads to the detailed results associated with Equation \ref{1a} that are given by

    \[ a_{11} b_{1}+a_{12} b_{2}+a_{13} b_{3}=c_{1} \]

    \[a_{21} b_{1}+a_{22} b_{2}+a_{23} b_{3}=c_{2} \]

    \[a_{31} b_{1}+a_{32} b_{2}+a_{33} b_{3}=c_{3} \]

    \[a_{41} b_{1}+a_{42} b_{2}+a_{43} b_{3}=c_{4} \]

    \[a_{51} b_{1}+a_{52} b_{2}+a_{53} b_{3}=c_{5} \]

    Section 6.3

    23. In Example \(6.3.1\), show how to obtain Eq. 3 beginning with the atomic matrix identified in Eq. 2.

    24. Show how to develop Eq. \((6.3.9)\) in terms of Eq. \((6.3.8)\) using the elementary row operations described in Sec. 6.2.5.

    25. Show how to obtain Eq. \((6.3.13)\) from Eq. \((6.3.12)\).

    26. When methane is partially combusted with oxygen, one finds the following molecular species: \(\ce{ CH4}\), \(\ce{O2}\), \(\ce{ CO}\), \(\ce{CO2}\), \(\ce{H2O}\) and \(\ce{H2}\). Determine the number of independent stoichiometric equations and comment on the restrictions concerning the choice of pivot and non-pivot species.

    Section 6.4

    27. Rucker et al.9 have studied the catalytic conversion of acetylene (\(\ce{C2H2}\)) to form benzene (\(\ce{C6H6}\)) along with hydrogen (\(\ce{H2}\)) and ethylene (\(\ce{C2H4}\)). For this system, chose benzene and ethylene as the pivot species, determine the rank of the atomic matrix, and apply the pivot theorem to determine the net rates of production of the non-pivot species, (\(R_{\ce{C2H2}}\)) and (\(R_{\ce{H2}}\)).

    28. The preparation of styrene (\(\ce{C8H8}\)) and benzene (\(\ce{C6H6}\)) from acetylene (\(\ce{C2H2}\)) has been considered by Tamaka et al.10 and for this system a visual representation of the atomic matrix is given by

    \[ \begin{align*} \text{Molecular Species } \to & \begin{matrix} \ce{C2H2} & \ce{C6H6} & \ce{C8H8} \end{matrix} \\ \begin{matrix} carbon \\ oxygen \end{matrix} & \begin{bmatrix} 2 & 6 & 8 \\ 2 & 6 & 8 \end{bmatrix} \end{align*}\]

    Determine the rank of the atomic matrix and use the pivot theorem to represent the rate of production of the non-pivot species in terms of the rate of production of the pivot species.

    29. The reaction of acetylene (\(\ce{C2H2}\)) with methanol (\(\ce{CH3OH}\)) to produce methyl ether (\(\ce{CH3OC2H3}\)) is sometimes known as Reppe chemistry11. Determine the rank of the atomic matrix for this system. Choose methyl ether as the pivot species and use the pivot theorem to represent the rates of production of the non-pivot species in terms of the rate of production of the pivot species.

    30. Given a system containing the molecular species: \(\ce{ CH4}\), \(\ce{O2}\), \(\ce{ Cl2}\), \(\ce{CH2Cl}\), \(\ce{HCl}\), \(\ce{H2O}\) and \(\ce{CO2}\). determine the rank of the atomic matrix. Use the pivot theorem to express the net rates of production of the non-pivot species, \(\ce{ CH4}\), \(\ce{O2}\), \(\ce{ Cl2}\), and \(\ce{CH3Cl}\) in terms of the net rates of production of the pivot species \(\ce{HCl}\), \(\ce{H2O}\) and \(\ce{CO2}\).

    31. In this problem we consider the catalytic oxidation (\(\ce{O2}\)) of ethane (\(\ce{C2H6}\)) to produce ethylene (\(\ce{C2H4}\)) and acetic acid (\(\ce{CH3COOH}\)) along with carbon dioxide (\(\ce{CO2}\)), carbon monoxide (\(\ce{CO}\)) and water (\(\ce{H2O}\)). This process has been studied experimentally by Sankaranarayanan et al12 in order to determine the factors affecting the selectivity (see Chapter 7) of ethylene and acetic acid. Make use of the pivot theorem to demonstrate that one must measure (as one possibility) the net rates of production for \(\ce{CH3COOH}\), \(\ce{C2H4}\), \(\ce{H2O}\) and \(\ce{CO}\) and in order to predict the net rates of production for \(\ce{C2H6}\), \(\ce{CO2}\) and \(\ce{O2}\).

    32. From Figure 6.32 we see that \(\alpha\)-butylene has a very different structure than isobutylene.

    6.4.1.PNG
    Figure \(\PageIndex{1}\): Isomers of butylene

    Thus we expect that reactions involving these two molecules might be quite different. However, in terms of stoichiometry, these two molecules are indistinguishable, and this means that we need to be careful in constructing the atomic matrix. For the combustion of a mixture of \(\alpha\)-butylene and isobutylene, we can express the atomic matrix as

    \[\begin{align*} \text{Molecular Species } \to & \begin{matrix} \ce{C4}\text{H}_8^{\alpha} & \ce{C4}\text{H}_8^{iso} & \ce{H2O} & \ce{CO2} & \ce{CO} \end{matrix} \\ \begin{matrix} carbon \\ hydrogen \\ oxygen \end{matrix} & \begin{bmatrix} 4 & 4 & 0 & 1 & 1 \\ 8 & 8 & 2 & 0 & 0 \\ 0 & 0 & 1 & 2 & 0 \end{bmatrix} \end{align*}\]

    Take \(\ce{CO2}\) and \(\ce{C4}\text{H}_8^{iso}\) as the pivot species in order to obtain expressions for \(R_{\ce{C4H8}}^{\alpha}\), \(R_{\ce{CO}}\) and \(R_{\ce{H2O}}\) in terms of \(R_{\ce{CO2}}\) and \(R_{\ce{C4H8}}^{iso}\).

    33. A complex system of optical isomers involves ortho-cresol (\(\ce{C7H8O}^{OC}\)), meta-cresol (\(\ce{C7H8O}^{MC}\)), water (\(\ce{H2O}\)), sulfuric acid (\(\ce{H2SO4}\)), ortho-cresol-sulfonic acid (\(\ce{C7H8O4S}^{OCS}\)), and meta-cresol sulfonic acid (\(\ce{C7H8O4S}^{MCS}\)). Use Axiom II to develop three constraints for the net rates of production associated with these six molecular species.


    1. Aris, R. 1965, Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.↩

    2. Schema: an outline, diagram, plan, or preliminary draft, Webster’s New World Dictionary, Simon & Schuster, New York, 1988.↩

    3. Amundson, N.R. 1966, Mathematical Methods in Chemical Engineering: Matrices and Their Application, page 51, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.↩

    4. Truesdell, C. and Toupin, R.A. 1960, The Classical Field Theories, Handbuch der Physik, Springer-Verlag, Berlin.↩

    5. Kolman, B. 1997, Introductory Linear Algebra, Sec. 4.6, Sixth Edition, Prentice-Hall, Upper Saddle River, New Jersey,↩

    6. Noble, B. 1969, Applied Linear Algebra, Prentice-Hall, Inc., Englewood Cliffs, New Jersey,↩

    7. Kolman, B. 1997, Introductory Linear Algebra, Sixth Edition, Prentice-Hall, Upper Saddle River, New Jersey.↩

    8. Kvisle, S., Aguero, A. and Sneeded, R.P.A. 1988, Transformation of ethanol into 1,3-butadiene over magnesium oxide/silica catalysts, Applied Catalysis, 43, 117-121.↩

    9. Rucker, T.G., Logan, M.A., Gentle, T.M., Muetterties, E.L. and Somorjai, G.A. Conversion of acetylene to benzene over palladium single-crystal surfaces. 1. The low-pressure stoichiometric and high-pressure catalytic reactions, J. Phys. Chem. 90, 2703–2708, 1986.↩

    10. Tanaka, M., Yamamote, M. and Oku, M., Preparation of styrene and benzene from acetylene and vinyl acetylene, USPO, 2723299, 1955.↩

    11. Walter Reppe, 1892–1969.↩

    12. Sankaranarayanan, T.M., Ingle, R.H., Gaikwad, T.B., Lokhande, S.K., Raja. T., Devi, R.N., Ramaswany, V., and Manikandan, P., Selective oxidation of ethane over Mo-V-Al-O oxide catalysts: Insight to the factors affecting the selectivity of ethylene and acetic acid and structure-activity correlation studies, Catal. Lett., 121: 39–51, 2008.↩

    • Was this article helpful?