2.1: A "Simple" Chemical Transformation

Figure 2-1. Photosynthesis, a process that we all learn about in elementary school, is a complex sequence of steps by which electrons are “harvested” from water by chlorophyll in green plants to build carbohydrates from carbon dioxide molecules. While the process is amazingly complex, the sum of all the individual steps can be summarized in a simple balanced equation.

(Photo credit: Asif Mridha, Wikimedia/CC SA 4.0)

 

 

 

 

 

To begin our foray into the quantitative side of chemistry, we’ll use a reaction that should be at least somewhat familiar to everyone reading this: photosynthesis. This process, by which plants convert carbon dioxide, CO₂, and water, H₂O, into oxygen gas, O₂, and a type of sugar called glucose, C₆H₁₂O₆, is simple only in the sense that it can be neatly summarized by the following chemical equation (eq. 2.1.1):

\[ \ce{CO2 + 6 H2O   ->   C6H12O6 + 6 O2} \]                  

In the production of glucose, plants use an exquisite system of proteins and other types of organic molecules to “harvest” electrons that are initially associated with water and use them to make the complex carbon skeleton of the sugar (glucose) as well as its numerous bonds between carbon and hydrogen atoms. Much of modern chemistry, and a large portion of this book, is devoted to understanding how electrons are shared between atoms and using those insights to understand and predict chemical reactivity. For now, however, we will be content to simply consider the formulas of the compounds involved in photosynthesis and not the details of how they are formed.

The equation above contains a surprising amount of information. Reading it literally, it translates to the following: six molecules of carbon dioxide and six molecules of water are converted to one molecule of glucose and six molecules of oxygen gas. Let’s dissect the equation to illustrate the various aspects of the information embedded in it. We can begin with the four individual chemical species, or compounds, represented. Unlike elements, the smallest particle of a compound (that retains the properties of the compound itself) is not an individual atom but a molecule, a collection of atoms having well-defined ratios and structures, like linoleic acid in the last chapter. Water, glucose, oxygen gas [1], and carbon dioxide are all compounds. Every molecule of carbon dioxide has two oxygen atoms and one carbon atom, represented by the symbols O and C, respectively, hence the molecular formula, CO₂ [2]. By the same logic, each molecule of glucose, C₆H₁₂O₆, is composed of six carbon atoms, twelve hydrogen atoms, and six atoms of oxygen. With respect to how the four compounds are represented in the above equation, we read it from left to right: compounds on the left of the arrow are referred to as reactants, the compounds that exist before any reaction takes place, and those on the right are products, the compounds into which the reactants’ component atoms are rearranged through the chemical reaction. 

Equation 2.1.1 is balanced, meaning that the number of atoms of each element are exactly the same on both sides. Why do we need to balance chemical equations? Because it reflects a central idea in chemistry. Specifically, it reflects the fact that matter is conserved in all chemical reactions. That is, while the component atoms are rearranged in the formation of new compounds, the number of each type at the end of a reaction is always equal to that before the reaction commenced. This is referred to as the Law of Conservation of Mass, and its discovery by French chemist Antoine Lavoisier (Figure 2-2) is widely considered to be a turning point in the development of chemistry — a paradigm-shift, to employ an overused but in this case apt expression — that turned the field into a modern science [3]. Chemical equations are balanced by using numerical coefficients that precede each species: these numbers indicate the quantitative relationships that exist between the molecules of reactants and products. To illustrate how this equation is balanced, we’ll focus on the number of oxygen atoms that are represented in the reactants and products: if the equation is properly balanced, the total number of oxygen atoms must be the same on each side. In the reactants there are two oxygen atoms in each molecule of carbon dioxide and one in each molecule of water and, since there are six molecules of each compound, a total of eighteen total oxygen atoms are present. There must therefore be eighteen oxygen atoms in the products; six are in the one molecule of glucose and there are two in each of the six molecules of oxygen gas, O₂. One could also show that the atoms of carbon (six total) and hydrogen (twelve total) are also the same in the reactants and products. The equation therefore reflects the fact that mass is conserved during photosynthesis. 

 

 

 

Figure 2-2. Detail from a portrait of Antoine (1743-1795) and Marie Anne Lavoisier (1758-1836) by Jacques Louis David (1788). Together they investigated the nature of chemical reactions and, through their painstaking and careful measurements of the masses of reactants and products of many chemical reactions, were the first to articulate the Conservation of Mass. For more information on the portrait itself, see https://www.metmuseum.org/art/collection/search/436106. (Open Access)

 

 

 

 

 

Because the ability to correctly balance chemical equations is of fundamental importance in chemistry (if not all of science), we present a systematic approach to doing so in the box below.

 

 

Balancing Chemical Equations: A Few Tips

The ability to balance chemical equations is perhaps the most fundamental skill in doing any work in chemistry. No matter what you do in a chemically related endeavor, describing in quantitative terms how reactions take place is critical. The goal is always the same: you must make the number of atoms of each element the same on both sides of the equation.

Here a few key ideas related to the details of chemical bookkeeping:

• Chemical equations can only be balanced by changing the coefficients that appear before each species in the equation; if no coefficient is given it should be interpreted as having a value of exactly one.

• Never change the subscripts of any compound; these specify the identity of the reactants and products. If they are changed, you are fundamentally changing the meaning of the equation. Recall, if no subscripts are included for a given element, that means there is one atom of that element in the molecule.

• The number of atoms of a given element is determined by multiplying its subscript in a given formula by the coefficient the precedes that compound; if an element appears in more than one compound on either the reactant or product side, you must add the number of atoms of that type over all of the compounds in which it appears. For example, if “5 H₂O” appears in an equation, there are 10 hydrogen atoms and 5 oxygen atoms associated with the water molecules. 

• If any compound includes parentheses in its formula, you must multiply all of the subscripts inside the parentheses by the subscript outside the parentheses. To illustrate, Fe(NO₃)₃ has 1 iron atom, 3 nitrogen atoms, and 9 oxygen atoms, whereas “2 Fe(NO₃)₃” has 2 iron atoms, 6 nitrogen atoms, and 18 oxygen atoms.

Let’s use the photosynthesis reaction to illustrate how you use the above points to actually balance the equation. The unbalanced equation, which shows the reactants and products with no coefficients, is:

\[ \ce{CO2 + H2O   ->   C6H12O6 + O2} \nonumber \]

How to balance the above? The following steps provide a good approach for balancing any chemical equation:

1. Begin by looking for elements that appear only one time on each side of the equation. Two such elements are present here:

a) carbon appears only one on the reactant side (in CO₂) and once on the product side (in C₆H₁₂O₆)

b) hydrogen appears only once on the reactant side (in H₂O) and once on the product side (in C₆H₁₂O₆)

2. Balance those elements identified in Step 1 by changing the coefficients in front of the compounds as needed. As written, there is one carbon atom on the left side of the equation, but six on the right. The coefficient for CO₂ must therefore be 6 to balance carbon atoms. Six carbon dioxide molecules have the same number of carbon atoms as one molecule of glucose.
   
   Moving on to hydrogen, there are two hydrogen atoms on the left side of the unbalanced equation but twelve on the right. The coefficient for water must therefore also be 6 to balance the hydrogen atoms.

We pause here to write the partially balanced (meaning still unbalanced, but better than when we started) equation:

\[ \ce{6 CO2 + 6 H2O   ->   C6H12O6 + O2} \nonumber \]

As written, carbon and hydrogen are both balanced, but oxygen is not. There are 18 oxygen atoms on the left and only eight on the right. We finish balancing the equation in the next step.

3. If possible, balance the remaining element(s) by changing the coefficients that do not affect the number of atoms of elements you’ve already balanced. In this case, the side that is deficient in oxygen (the right), has such a species: O₂. You can make the total number of oxygen atoms on the right equal 18 by making the coefficient in front of O₂ equal to 6. This gives us the fully balanced equation that was included in the text at the start of this section: 

 \[ \ce{6 CO2 + 6 H2O   ->   C6H12O6 + 6 O2} \nonumber \]

To summarize: the best way to balance equations is to start with elements that appear in only compounds in the reactants and products, then move on to the elements that appear multiple times.

 

Worked Example 

Problem 2-1.

2,2,4-trimethylpentane is a major component of gasoline. The complete combustion of this compound is described by the following unbalanced chemical equation. 

\[ \ce {C8H18 + O2 -> CO2 + H2O} \nonumber \]

Balance the above equation.

 

Solution

Both carbon and hydrogen are present in only one compound each in the reactants and products, so we start there. Oxygen appears in both products so we will deal with it last. 

• Carbon: there are 8 carbon atoms on the left and only one on the right; the coefficient for CO₂ is therefore 8.

• Hydrogen: there are 18 hydrogen atoms on the left and 2 on the right; the coefficient for H₂O is therefore 9.

The partially balanced equation is now: 

\[ \ce {C8H18 + O2 -> 8 CO2 + 9 H2O} \nonumber \]

• Oxygen: there are 25 oxygen atoms on the right and only 2 on the left. Because the compound of oxygen is O₂, we must multiply the subscript, 2, by a term such that the product is 25. The coefficient is therefore ²⁵/₂ and the fully balanced equation is:

 ​​​​​​\[ \ce {C8H18 + \frac{25}{2} O2 -> 8 CO2 + 9 H2O} \nonumber \]

Postscript: many students are under the impression that fractional coefficients, such as the ²⁵/₂ above, are “not allowed”. Not true! As we explain further below, chemical reactions usually involve huge collections of atoms and molecules. And these numbers represent ratios, not necessarily individual molecules. For example, if we had a droplet consisting of 1,000,000 molecules of the 2,2,4-trimethylpentane (a drop so tiny it wouldn’t be visible to the naked eye or even with most microscopes), its complete combustion would require 12,500,000 molecules of oxygen. There is nothing wrong or improper about using fractions to balance equations (regardless of what your middle school science teacher may have told you!)

 

Exercise 

Problem 2.2: Balance the following equations: 
 

1. \( \ce{C4H10 + O2 -> CO2 + H2O} \)

2. \( \ce{C2H5OH + O2 -> CO2 + H2O} \)

3. \( \ce{TiCl4 + H2O -> TiO2 + HCl} \)

4. \( \ce {Fe + HNO3 -> Fe(NO3)2 + H2} \)  

5. \( \ce{Na + Cl2 -> NaCl} \)

 

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Notes.

[1]. One point of clarification may be helpful in the case of oxygen, O₂: individual oxygen atoms, that is, oxygen atoms in their elemental form, are highly reactive; they are much more stable when joined by a chemical bond to another oxygen atom. Thus, the commonly encountered form of oxygen is the diatomic molecule, O₂ and, as such, it is considered a compound, not a simple element.

[2]. By convention, a symbol not followed by a number means that there is only one atom of the element represented; thus there is only one carbon atom in CO₂. At the macroscopic level, the fixed atomic ratios give rise to the fact that samples of pure compounds, regardless of their source or history, always consist of the same mass percentages of their component elements, a phenomenon referred to as The Law of Definite Proportions, first articulated by French chemist Joseph Proust (1754-1825). It should be noted that there are exceptions to this law, however, but their number is limited as is their relevance to most of general, organic and biochemistry.

[3]. Antoine and Marie Anne Lavoisier are fascinating historical figures. An excellent book that describes their work in the context of the times they lived in is Lavoisier in Year One: The Birth of a New Science in the Age of Revolution by Madison Smartt Bell (2005). Oxygen, a play by Roald Hoffman and Carl Djerassi (2001) focuses more on the role of Marie Anne Lavoisier; it provides a fascinating glimpse into the role of the sexes in the practice of science in past and present times.