14.10 : The Small x Assumption

The small x assumption is an approximation that can be used under certain conditions to simplify solving the equilibrium expression and to avoid using the quadratic formula.

If the equilibrium constant for a reaction is small and the initial concentrations of reactants are sufficiently high, it is possible that only a small amount of reactants will be converted into products.

When these conditions are met, the equilibrium concentrations can be approximated using the assumption that the change in the initial concentrations is negligible. However, it must be confirmed that the change in concentration, x, is less than 5% of the initial concentrations to show the the assumption was valid.

The decomposition of 0.66 molar carbonyl dichloride gas produces carbon monoxide and chlorine gas, and the equilibrium constant for this reaction is 2.2 × 10⁻¹⁰.

To calculate the equilibrium concentrations, the values are tabulated in the ICE table along with the change and equilibrium concentrations.

Substituting the equilibrium concentrations in the expression, K equals x squared divided by 0.66 minus x. As K is very small, the change in the initial concentration, x, of carbonyl dichloride is expected to be negligible.

Therefore, 0.66 minus x can be approximated to 0.66. When solved, x equals 1.2 × 10⁻⁵ molar.

Here, x is only 0.0018 percent of the initial 0.66 molar concentration of carbonyl dichloride, which is far less than the 5% maximum allowed. Thus, the small x assumption is valid here.

Using the value for x in the ICE table, the equilibrium concentration of carbonyl dichloride is still 0.66 molar with significant figures, while the concentrations of carbon monoxide and chlorine are both 1.2 × 10⁻⁵ molar.

If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration may occur if the initial concentrations of reactants are high and the K_c value is small. In such circumstances, the equilibrium concentration is approximately equal to its initial concentration.ÃÂÃÂ This estimation can be used to simplify the equilibrium calculations by assuming that some equilibrium concentrations are equal to the initial concentrations. However, to make this assumption, the change in the concentration of a weak acid or base, i.e., x, must be less than 5% of its initial concentration. If x is more than 5%, then the quadratic formula needs to be used to solve the equilibrium equation.ÃÂÃÂ ÃÂÃÂ

Calculation of Equilibrium Concentrations Using an Algebra-Simplifying Assumption

What are the concentrations at equilibrium of a 0.15 M solution of HCN?

Chemical equilibrium reaction: Haber process N<sub>2</sub> + 3H<sub>2</sub> ⇌ 2NH<sub>3</sub>, dynamic balance.

Using x to represent the concentration of each product at equilibrium gives this ICE table.

	ÃÂÃÂ HCN (aq)ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ	H⁺(aq)	CN^{ÃÂ¢ÃÂÃÂ}(aq)
Initial Concentration (M)	0.15	0	0
Change (M)	ÃÂ¢ÃÂÃÂx	+x	+x
Equilibrium Concentration (M)	0.15 ÃÂ¢ÃÂÃÂ x	x	x

Substitute the equilibrium concentration terms into the K_c expression

Static equilibrium diagram; forces ΣFx=0, ΣFy=0, torque MA=0 for balance analysis and calculations.

rearrange to the quadratic form and solve for x

Gene editing diagram, CRISPR-Cas9 mechanism, illustrating DNA modification process.

Thus, [H⁺] = [CN^{ÃÂ¢ÃÂÃÂ}] = x = 8.6 ÃÂÃÂ 10^{ÃÂ¢ÃÂÃÂ6} M and [HCN] = 0.15 ÃÂ¢ÃÂÃÂ x = 0.15 M.

Note in this case that the change in concentration is significantly less than the initial concentration (a consequence of the small K), and so the initial concentration experiences a negligible change:

Rotational equilibrium, Στ=0; diagram illustrating torque balance, static system analysis.

This approximation allows for a more expedient mathematical approach to the calculation that avoids the need to solve for the roots of a quadratic equation:

Chemical structure of zeaxanthin, organic molecule diagram, used in antioxidant research studies.

The value of x calculated is, indeed, much less than the initial concentration

Static equilibrium, ΣFx=0 equation, diagram of forces and torques, educational physics concept.

and so the approximation was justified. If this simplified approach were to yield a value for x that did not justify the approximation, the calculation would need to be repeated without making the approximation.

This text has been adapted from Openstax, Chemistry 2e, Section 13.4 Equilibrium Calculations.

Tags

Small X Assumption Equilibrium Expression Quadratic Formula Equilibrium Constant Initial Concentrations Reactants Products Approximation Change In Concentration ICE Table Carbonyl Dichloride Carbon Monoxide Chlorine Gas

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