14.10
The Small x Assumption
If a reaction has a small equilibrium constant, the equilibrium position favors the reactants. In such reactions, a negligible change in concentration…
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.
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Q1: When can you use the small x assumption to solve equilibrium problems?
The small x assumption applies when the equilibrium constant is small and initial reactant concentrations are sufficiently high, resulting in negligible conversion to products. This approximation simplifies calculations by treating the change in concentration as negligible. However, you must verify that x is less than 5% of the initial concentration to confirm the assumption's validity.
Q2: How do you verify that the small x assumption is valid?
After solving for x using the simplified equilibrium expression, calculate the percentage change by dividing x by the initial concentration and multiplying by 100. If this percentage is less than 5%, the assumption is valid. If x exceeds 5% of the initial concentration, you must solve the problem again using the quadratic formula without making the approximation.
Q3: What happens if the small x assumption produces an invalid result?
If x is greater than 5% of the initial concentration, the assumption is invalid and you cannot use the simplified equilibrium expression. Instead, you must rearrange the equilibrium expression into quadratic form and solve using the quadratic formula to obtain accurate equilibrium concentrations.
Q4: Why does a small equilibrium constant favor using the small x assumption?
A small equilibrium constant indicates that the reaction produces very little product at equilibrium, meaning only a small fraction of reactants convert. When combined with high initial reactant concentrations, this ensures the change in concentration remains negligible, making the small x assumption mathematically sound and simplifying calculations significantly.
Q5: How does the small x assumption simplify the equilibrium expression?
The assumption allows you to eliminate x from the denominator of the equilibrium expression. For example, if K = x²/(0.66 − x), you approximate (0.66 − x) as 0.66, yielding K = x²/0.66. This avoids the need to expand and solve a quadratic equation, making the calculation much faster and more straightforward.
Q6: What does the ICE table show in small x assumption problems?
The ICE table organizes Initial concentrations, Change in concentration (x), and Equilibrium concentrations for each species. It provides a systematic way to track how concentrations shift during the reaction and helps set up the equilibrium expression. The table clearly shows where the small x approximation is applied by treating the change as negligible.
Q7: In the carbonyl dichloride example, why was x only 0.0018% of the initial concentration?
Carbonyl dichloride has an extremely small equilibrium constant (2.2 × 10⁻¹⁰), meaning the reaction barely proceeds forward. With a high initial concentration of 0.66 M and such a small K value, only a tiny amount of reactant decomposes. The resulting x value of 1.2 × 10⁻⁵ M represents this minimal conversion, validating the small x assumption.