Equations and Mass Relationships in Biology/Ecological (Biological) Stoichiometry - ChemPRIME

# Equations and Mass Relationships in Biology/Ecological (Biological) Stoichiometry

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Ecological (Biological) Stoichiometry

A new approach to studying the relationships in ecological or biological systems has been developed by R.W. Sterner, J.J. Elser, and others [1] [2]. It is called ecological or biological stoichiometry because it focuses on the chemical requirements of each trophic level, in addition to energy requirements. Proponents say that "Ecological stoichiometry recognizes that organisms themselves are outcomes of chemical reactions and thus their growth and reproduction can be constrained by supplies of key chemical elements [especially carbon (C), nitrogen (N) and phosphorus (P)]".[3]

An algal bloom is explained by ecological stoichiometry of nitrate and phosphate nutrients

For example, by writing a chemical equation for photosynthesis in oceanic algae, we can predict which nutrients (nitrogen as nitrate, NO3-, phosphorus as hydrogen phosphate, HPO42-, etc.) are required for algae growth, and what products result from algal respiration.

106 CO2(g) + 16 NO3-(aq) + HPO42-(aq) + 122 H2O(l) + 18 H+(aq) ↔

C106H263O110N16P1(s) + 138 O2(g)           (1)

The equation above does not include some minerals, like potassium ion (K+), which may both consumed and produced in the processes, and the "formula" for algae (C106H263O110N16P1) does not represent a single molecule, but just the overall composition of the algae (one might call it an "average" molecular formula)[4]. It illustrates the elemental homeostasis exhibited by plants and animals--their elemental composition is fixed, regardless of the composition of their environment, so if the environment does not provide the correct elements in the correct ratios, the plants or animals do not thrive.

The double arrow indicates that the reaction may produce the products on the right from the reactants on the left during photosynthesis, or may proceed in the reverse direction during respiration. The abbreviations in parentheses indicate the state of the species; (g) indicates a gas, (l) a liquid, (s) a solid, and species which are dissolved in aqueous (water) solutions are denoted (aq).

This balanced chemical equation not only tells how many molecules of each kind are involved in a reaction, it also indicates the amount of each substance that is involved. Equation (1) says that 106 CO2 molecules can react with 122 H2O molecules, 16 NO3- ions, 1 HPO42- and 18 H+ ions to give 1 C106H263O110N16P1"molecule" and 138 O2 molecules. It also says that 106 mol of CO2 molecules can react with mol of 122 H2O molecules, 16 mol of NO3- ions, 1 mol of HPO42- and 18mol of H+ ions to give 1 mol of C106H263O110N16P1"molecule" and 138 mol of O2 molecules.

The balanced equation does more than this, though. It also tells us that 2 × 16 = 32 mol NO3- will react with 2 × 1 = 2 mol HPO42-, and that ½ × 16 = 8 mol NO3- requires only ½ × 1 = ½ mol HPO42-. In other words, the equation indicates that exactly 16 mol NO3- must react for every 1 mol HPO42- consumed. For the purpose of calculating how much HPO42- is required to react with a certain amount of NO3- therefore, the significant information contained in Eq. (1) is the ratio

$\frac{\text{1 mol HPO}_{\text{4}}^{\text{2-}}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}$

We shall call such a ratio derived from a balanced chemical equation a stoichiometric ratio and give it the symbol S. Thus, for Eq. (1),

$\text{S}\left( \frac{\text{HPO}_{\text{4}}^{\text{-}}}{\text{NO}_{\text{3}}^{\text{-}}} \right)=\frac{\text{1 mol HPO}_{\text{4}}^{\text{2-}}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}~ ~ ~ ~ \text{(2)}$

Thus for algae to grow, 1 HPO42- ion must be present for every 16 NO3- ions. Often, this amount of HPO42- is not present, so algae cannot grow. But if a good source of HPO42- is added (for example, as modern detergent runoff), the stoichiometric ratio is established and an algae bloom results.

The word stoichiometric comes from the Greek words stoicheion, “element,“ and metron, “measure.“ Hence the stoichiometric ratio measures one element (or compound) against another.

EXAMPLE 1 Derive all possible stoichiometric ratios from Eq. (1)

Solution Any ratio of amounts of substance given by coefficients in the equation may be used:

$\text{S}\left( \frac{\text{NO}_{3}^{\text{-}}}{\text{O}_{\text{2}}} \right) =\frac{\text{16 mol NO}_{3}^{\text{-}}}{\text{138 mol O}_{\text{2}}}$      $\text{S}\left( \frac{\text{O}_{\text{2}}}{\text{NO}_{\text{3}}^{\text{-}}} \right)$=$\frac{\text{138 mol O}_{\text{2}}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}$

$\text{S}\left( \frac{\text{CO}_{2}}{\text{NO}_{\text{3}}^{\text{-}}} \right)$=$\frac{\text{106 mol CO}_{2}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}$      $\text{S}\left( \frac{\text{O}_{\text{2}}}{\text{H}_{\text{2}}\text{O}} \right)=\frac{\text{138 mol O}_{\text{2}}}{\text{122 mol H}_{\text{2}}\text{O}}$

$\text{S}\left( \frac{\text{CO}_{2}}{\text{H}_{\text{2}}\text{O}} \right)=\frac{\text{106 mol CO}_{2}}{\text{122 mol H}_{\text{2}}\text{O}}$      $\text{S}\left( \frac{\text{NO}_{\text{3}}^{\text{-}}}{\text{H}_{\text{2}}\text{O}} \right)=\frac{\text{16 mol NO}_{\text{3}}^{\text{-}}}{\text{122 mol H}_{\text{2}}\text{O}}$

There are many more stoichiometric ratios, each of which relates any two of the reactants or products. Eq. (2) gives one of them.

When any chemical reaction occurs, the amounts of substances consumed or produced are related by the appropriate stoichiometric ratios. Using Eq. (1) as an example, this means that the ratio of the amount of CO2 consumed to the amount of N3 - consumed must be the stoichiometric ratio S(CO2/NO3-):

$\frac{n_{\text{CO}_{\text{2}}\text{ consumed}}}{n_{\text{NO}_{\text{3}}^{\text{-}}\text{ consumed}}}=\text{S}\left( \frac{\text{CO}_{2}}{\text{NO}_{\text{3}}^{\text{-}}} \right)=\frac{\text{106 mol CO}_{2}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}$

Similarly, the ratio of the amount of O2 produced to the amount of CO2 consumed must be

S(O2/CO2):

$\frac {n_ {\text{O}_{\text{2}}\text{ produced}}} {n_{\text{CO}_{\text{2}}\text{ consumed}}}~ ~ ~ =~ ~ ~ ~ ~ \text{S}\left( \frac{\text{O}_{\text{2}}}{\text{CO}_{\text{2}}} \right)~ ~ ~ = ~ ~ ~\frac{\text{138 mol O}_{\text{2}}}{\text{106 mol CO}_{2}}$

In general we can say that

$\text{Stoichiometric ratio }\left( \frac{\text{X}}{\text{Y}} \right)=\frac{\text{amount of X consumed or produced}}{\text{amount of Y consumed or produced}}\text{ (3}\text{a)}$

or, in symbols,

$\text{S}\left( \frac{\text{X}}{\text{Y}} \right)=\frac{n_{\text{X consumed or produced}}}{n_{\text{Y consumed or produced}}}\text{ (3}\text{b)}$

Note that in the word Eq. (3a) and the symbolic Eq. (3b), X and Y may represent any reactant or any product in the balanced chemical equation from which the stoichiometric ratio was derived. No matter how much of each reactant we have, the amounts of reactants consumed and the amounts of products produced will be in appropriate stoichiometric ratios.

EXAMPLE 2 Find the amount of oxygen produced when 3.68 mol CO2 is consumed according to Eq. (1).

Solution The amount of oxygen produced must be in the stoichiometric ratio S(O2/CO2) to the amount of carbon dioxide consumed:

$\text{S}\left( \frac{\text{O}_{\text{2}}}{\text{CO}_{\text{2}}} \right)~ ~=~ ~\frac{n_{\text{O}_{\text{2}}\text{ produced}}}{n_{\text{CO}_{\text{2}}\text{ consumed}}}$

Multiplying both sides bynCO2 consumed, we have

$n_{\text{O}_{\text{2}}\text{ produced}}=n_{\text{CO}_{\text{2}}\text{ consumed}}\times \text{S}\left( \frac{\text{O}_{\text{2}}}{\text{CO}_{\text{2}}} \right)=\text{3}\text{.68 mol CO}_{\text{2}}\times \frac{\text{138 mol O}_{\text{2}}}{\text{106 mol CO}_{\text{2}}}=\text{4}\text{.79 mol O}_{\text{2}}$

This is a typical illustration of the use of a stoichiometric ratio as a conversion factor. Example 2 is analogous to Examples 1 and 2 from Conversion Factors and Functions, where density was employed as a conversion factor between mass and volume. Example 2 is also analogous to Examples 2.4 and 2.6, in which the Avogadro constant and molar mass were used as conversion factors. As in these previous cases, there is no need to memorize or do algebraic manipulations with Eq. (3) when using the stoichiometric ratio. Simply remember that the coefficients in a balanced chemical equation give stoichiometric ratios, and that the proper choice results in cancellation of units. In road-map form

$\text{amount of X consumed or produced}\overset{\begin{smallmatrix} \text{stoichiometric} \\ \text{ ratio X/Y} \end{smallmatrix}}{\longleftrightarrow}\text{amount of Y consumed or produced}$

or symbolically.

$n_{\text{X consumed or produced}}\text{ }\overset{S\text{(X/Y)}}{\longleftrightarrow}\text{ }n_{\text{Y consumed or produced}}$

When using stoichiometric ratios, be sure you always indicate moles of what. You can only cancel moles of the same substance. In other words, 1 mol CO2 cancels 1 mol CO2 but does not cancel 1 mol O2.

The next example shows that stoichiometric ratios are also useful in problems involving the mass of a reactant or product.

EXAMPLE 3 Equation (1) may be used to calculate the ratio of nitrate to phosphate in a fertilizer that is optimal for algal growth. Phosphate fertilizers may be in liquid form, containing phosphoric acid[5], while nitrate fertilizers may contain solid KNO3. If 100 g of KNO3 is applied to a small pond, what is the stoichiometric mass of H3PO4 required for optimal algae growth?

Solution The problem asks that we calculate the mass of H3PO4 from a mass of KNO3. As we learned in Example 2 of The Molar Mass, the molar mass can be used to convert from the mass of KNO3 to the amount of KNO3. We can thus calculate the amount of KNO3. The amount of NO3- will be the same (there is 1 mol of NO3- in 1 mol of KNO3). We can then use a stoichiometric ratio to calculate the amount of HPO42- that would stoichiometrically react with the given mass of NO3-. This is the same as the amount of H3PO4, and we can use the molar mass to convert the amount to the mass.

This requires the stoichiometric ratio

$\text{S}\left( \frac{\text{HPO}_{\text{4}}^{\text{-}}}{\text{NO}_{\text{3}}^{\text{-}}} \right)=\frac{\text{1 mol HPO}_{\text{4}}^{\text{2-}}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}~ ~ ~ ~ \text{(2)}$

The amount of KNO3 in 100 g is

n (mol) = m (g) / M (g/mol) = 100 g / 101.1 g/mol = 0.989 mol

The amount of H3PO4 required is then

$n_{\text{HPO}_{\text{4}}^{\text{2-}}}$ = $n_{\text{NO}_{\text{3}}^{\text{-}}\text{ consumed}}\text{ }\!\!\times\!\!\text{ conversion factor}$=$\text{0}\text{.989 mol NO}_{\text{3}}^{\text{-}}\times$$\frac{\text{1 mol HPO}_{\text{4}}^{\text{2-}}}{\text{16 mol NO}_{\text{3}}^{\text{-}}}=\text{0}\text{.0618 mol HPO}_{\text{4}}^{\text{2-}}$

The molar mass of H3PO4 (98.0 g/mol) is used to calculate the mass:

m (g) = n (mol) x M (g/mol) = 0.0618 mol x 98.0 g/mol = 6.06 g.

Very little phosphate is required compared to nitrate, so the phosphate in discharged washwater may well cause the eutrophication shown in the figure above when (as is often the case) plenty of nitrate is available.

With practice this kind of problem can be solved in one step by concentrating on the units. The appropriate stoichiometric ratio will convert moles of NO3- to moles of HPO42- which equals moles of H3PO4, and the molar mass will convert moles of HPO42- to grams. A schematic road map for the calculation can be written as

$m_{\text{KNO}_{\text{3}}} \text{ }\xrightarrow{M_{\text{KNO}_{\text{3}}}}\text{ }n_{\text{KNO}_{\text{3}}} \text{ }\xrightarrow{S\text{(HPO}_{\text{4}}^{\text{2-}}\text{/NO}_{\text{3}}^{\text{-}}\text{)}}\text{ }n_{\text{HPO}_{\text{4}}^{\text{2-}}}\text{ }\xrightarrow{M_{\text{H}_{\text{3}}\text{PO}_{\text{4}}}}\text{ }m_{\text{H}_{\text{3}}\text{PO}_{\text{4}}}$

## References

1. Ecological Stoichiometry: The biolgy of elements from Molecules to Biosphere; Robert W. Sterner and James J. Elser; Princeton University Press, Princeton, NJ, 2002.
2. http://en.wikipedia.org/wiki/Ecological_stoichiometry
3. http://www.plosbiology.org/article/info:doi%2F10.1371%2Fjournal.pbio.0050181
4. Ecological Stoichiometry: The biolgy of elements from Molecules to Biosphere; Robert W. Sterner and James J. Elser; Princeton University Press, Princeton, NJ, 2002, p. 30.
5. http://www.extension.umn.edu/distribution/cropsystems/dc6288.html