Lecture Notes: Simplifying Algebra with the X's Small Approximation

Introduction

• Discusses methods to simplify algebra using the "x's small" approximation.
• Applicable in certain equilibrium problems where changes in reaction are small.
• Provides guidelines for recognizing when the approximation is applicable.

X's Small Approximation

• Definition: Assumes changes in equilibrium reactions are small enough to be negligible.
• Usage: Used when reactions are close to equilibrium.
• Conditions for Use:
  • Small equilibrium constant with high reactant concentration or pressure.
  • Large equilibrium constant with high product concentration or pressure.

Example Scenario

• Equation: 2H2S in equilibrium with H2 and S2.
• Initial Conditions: 0.0250 M H2S, equilibrium constant 1.67 x 10^-7 at 800°C.
• Calculation:
  • Initial assumption that x is small.
  • Formulated equilibrium constant expression.
  • Results in x^3, solved with approximation.
  • X calculated as 2.97 x 10^-4, validated as a good approximation (1.19% relative change).

When X's Small Fails

• Condition: If x is greater than 5% of the added/subtracted value, the assumption fails.
• Solution: Use successive approximations.

Successive Approximations

• Procedure:
  • Recalculate x by plugging back into the equation iteratively.
  • Continue until values converge satisfactorily under 5%.
• Example:
  • Adjusted initial concentration to 2.5 x 10^-4.
  • Initial x found as 1.38 x 10^-5 but fails the 5% test.
  • Through iterations, converged to x = 1.28 x 10^-5, validated as correct.

Conclusion

• Use of X-Value: Calculated x used to find final equilibrium concentrations.
• Checking Work: If multiple iterations are needed (X6, X7), recheck calculations.
• Exam Conditions: Typically, should not need more than 6 iterations.