But-2-ene can exist as two different forms; cis-but-2-ene (with both methyl groups on the same side of the double bond) and trans-but-2-ene (where they are on opposite sides of the double bond). The conversion between these two species is an equilibrium reaction wherein the cis and trans forms can interconvert.

The equilibrium constant for the conversion between these two species is 0.51. If an empty vessel is initially filled with cis-but-2-ene to a concentration of 1 M, what is the equilibrium concentration of trans-but-2-ene to 2 decimal places?

Question

But-2-ene can exist as two different forms; cis-but-2-ene (with both methyl groups on the same side of the double bond) and trans-but-2-ene (where they are on opposite sides of the double bond). The conversion between these two species is an equilibrium reaction wherein the cis and trans forms can interconvert.

The equilibrium constant for the conversion between these two species is 0.51. If an empty vessel is initially filled with cis-but-2-ene to a concentration of 1 M, what is the equilibrium concentration of trans-but-2-ene to 2 decimal places?

BlackTom AI · Accepted Answer

The equilibrium expression for cis ⇌ trans is K = [trans] / [cis], so with K = 0.51 we relate the equilibrium concentrations of the two isomers.

1) Relevant concepts
- Chemical equilibrium: for the isomerization cis ⇌ trans, the equilibrium constant K is defined as K = [trans] / [cis].
- ICE table: track Initial, Change, and Equilibrium concentrations to set up the algebra.

2) Step-by-step reasoning (ICE table)
- Initial: [cis]0 = 1.00 M, [trans]0 = 0.00 M.
- Change: let x = amount of cis converted to trans at equilibrium. Then change for cis = -x, for trans = +x.
- Equilibrium: [cis]eq = 1.00 − x, [trans]eq = x.

3) Algebra and intermediate steps
- Apply K: 0.51 = [trans]eq / [cis]eq = x / (1.00 − x).
- Solve for x:
  0.51 = x / (1.00 − x)
  0.51(1.00 − x) = x
  0.51 − 0.51x = x
  0.51 = x + 0.51x = 1.51x
  x = 0.51 / 1.51 ≈ 0.337748
- Round to two decimal places: x ≈ 0.34 M.

4) Why this answer is correct
- The calculation directly follows the definition of the equilibrium constant and conservation of mass: the only way trans appears is by conversion from cis, so [trans]eq = x and [cis]eq = 1 − x. Solving x/(1−x)=0.51 yields x = 0.34 M, which is the equilibrium concentration of trans-but-2-ene.

5) Why other answers would be incorrect
- Taking K = [cis]/[trans] (the inverse) would give an incorrect value (≈0.66 M) because it swaps numerator and denominator.
- Using K numerically as the concentration of trans directly (x = 0.51 M) ignores the depletion of cis and therefore violates mass balance.
- Any answer not solving x/(1−x)=0.51 fails to use the correct equilibrium relation.

6) Conclusion
- The equilibrium concentration of trans-but-2-ene is 0.34 M, which matches the provided correct answer.

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