Question

A type of reaction we will study is that having a very small K value (K << 1). Solving for equilibrium concentrations in an equilibrium problem usually requires many mathematical operations to be performed. However, the math involved in solving equilibrium problems for reactions having small K values (K << 1) is simplified. What assumption is made when solving equilibrium concentrations for reactions having small K values? Whenever assumptions are made, they must be checked for validity. In general, the “5% rule” is used to check the validity of assuming that x (or 2x, 3x, and so on) is very small compared to some number. When x (or 2x, 3x, and so on) is less than 5% of the number the assumption was made against, then the assumption is said to be valid. If the 5% rule fails, what do you do to solve for the equilibrium concentrations?

Short answer

We need to assume that x is very small when compared to the K value (K<<1). For this considered reaction, the assumption is valid. We need to solve the expression by using the method of successive approximations.

Step-by-step solution

Setup the ICE table for the equilibrium reaction under the given conditions.

Consider the following equilibrium reaction at a specific temperature.

$2{\mathrm{CO}}_2\left(\mathrm{g}\right)\rightleftharpoons 2\mathrm{CO}\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right)\mathrm{K}=2\times {10}^{-6}$

The equilibrium constant expression for the above reaction is as follows:

$\mathrm{K}=\frac{{\left[\mathrm{CO}\right]}^2\left[{\mathrm{O}}_2\right]}{{\left[{\mathrm{CO}}_2\right]}^2}$ ...(1)

The assumption is based on the fact that K is substantially smaller than 1. The equilibrium mixture for these reactions will not include many products; instead, the equilibrium mixture will contain predominantly reactants. If we define the concentration change in terms of algebraic variable, x as the concentration of a reactant that must react in order to reach equilibrium, then x must be a small number because K is a small number.

We don't assume x = 0 since we need to know the value of x to solve the problem. Instead, we'll focus on the ICE table's equilibrium row.

The initial concentration of CO₂ is assumed to be 0.40 M.

Setup the ICE table:

$$\begin{gathered} 2{\mathrm{CO}}_2\left(\mathrm{g}\right)\rightleftharpoons 2\mathrm{CO}\left(\mathrm{g}\right)+{\mathrm{O}}_2\left(\mathrm{g}\right) \\ \mathrm{InitiaL}\left(\mathrm{M}\right)0.4000 \\ \mathrm{Change}\left(\mathrm{M}\right)-2\mathrm{x}+2\mathrm{x}\mathrm{x} \\ ----------------------- \\ \mathrm{Equlibrium}\left(\mathrm{M}\right)0.40-2\mathrm{x}+2\mathrm{x}\mathrm{x} \end{gathered}$$

Insert these equilibrium concentrations from the ICE table and the equilibrium constant value into equation (1) as shown below.

$2.0\times {10}^{-6}=\frac{{\left(2\mathrm{x}\right)}^2\left(\mathrm{x}\right)}{{\left(0.40-2\mathrm{x}\right)}^2}$

Make the assumption for the equilibrium concentration terms from the ICE table.

An significant assumption may be made for the reactants (or products) with equilibrium concentrations of (negative for reactant). Because K<<1 is tiny (x<<1), the assumption is that subtracting x from some starting concentration will make little or no change. That is, we assume that ; that is, we assume that a substance's initial concentration equals its equilibrium concentration.

The expression becomes,

$2.0\times {10}^{-6}=\frac{{\left(2\mathrm{x}\right)}^2\left(\mathrm{x}\right)}{{\left(0.40\right)}^2}$

Solve for x.

x=0.0043

Therefore, the change in concentration, x is 0.0043 M.

Use the 5% rule to check the validity of assumption that we made.

$$\begin{gathered} \mathrm{percent}\mathrm{error}=\frac{0.0043}{40}\times 100\% \\ =1.08\% \end{gathered}$$

This assumption simplifies the algebra and generally results in a value of x that is within 5% of the real value of x. In this example, the percent error is 1.08% which is less than 5%. The assumption is valid.

When the 5% rule fails, the problem must be solved precisely or using the successive approximation approach.