Lecture on Solving Equilibrium Problems with Small x Approximations

Overview

• Discussed solving equilibrium problems in chemistry.
• Focus on cases where x is assumed to be small.
• Illustrated through a specific chemical reaction example.

Initial Reaction Setup

• Initial Conditions:
  • Reactant: 0 atm
  • NO: 1 atm
  • O2: 0.5 atm
• Equilibrium Constant: Very small value, indicating a shift to the left.
• Cannot assume x is small initially due to conditions.

Process

Identifying System Behavior

• Small equilibrium constant means little product at equilibrium.
• Initial setup shows 100% product, which suggests movement back to reactants.
• Reaction shifts left to reach equilibrium.

Setting up the Equilibrium Expression

• Equilibrium constant expression leads to x cubed term.
• Cannot solve directly assuming x is small under initial conditions.
• Not close to equilibrium based on initial conditions.

Adjusting Conditions

• Strategy: Push reaction completely to the left and let it return to equilibrium.
• Assume 100% product turned into reactant due to small equilibrium constant.
• Restart the table with revised initial conditions:
  • 1 atm of reactant
  • 0 atm of product

Calculation

• Reaction moves right from new initial conditions.
• Equilibrium expression still contains x cubed, but now can assume x is small.
• Solve for x:
  • Equation: 4x^3 = 5.9 x 10^-13
  • Calculated x: 5.284 x 10^-5

Results

• Verification shows the assumption holds as the percentage error is very low (10^-3%).
• Calculated equilibrium pressure of NO: 1.057 x 10^-4 atm.
• Clarification: Pressure calculated was for NO, not NO2.

Key Considerations

• Important to verify the assumption that x is small by comparing to equilibrium constant.
• Consider limiting reactants and mole ratios in reactions.
• Ensure calculations align with chemical principles and assumptions.

Conclusion

• This method can be applied to various equilibrium problems.
• Emphasis on understanding the relationship between equilibrium constants and initial conditions.
• Correct assumptions are crucial for accurate calculations.