About to di(mer)

Classical Mechanics Level 4

Suppose a molecule Y \text{Y} , carboxylic acid for instance, can form dimers Y : Y \text{Y}:\text{Y} denoted as Y 2 \text{Y}_2 . The forward reaction between two molecules of Y \text{Y} to form a dimer has rate constant k f = 5 × 1 0 − 5 mM − 2 s − 1 k_f = 5\times 10^{-5} \text{ mM}^{-2} \text{ s}^{-1} , and the backward reaction of one dimer breaking into two monomers has rate constant k b = 2 mM − 1 s − 1 k_b = 2 \text{ mM}^{-1} \text{ s}^{-1} .

You prepare a solution of 100 mM 100\text{ mM} Y \text{Y} , what is the equilibrium concentration (in mM \text{mM} ) of the dimer Y 2 \text{Y}_2 ?


The answer is 0.247531.

Josh Silverman by Josh Silverman

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2 solutions

Chew-Seong Cheong
Jun 23, 2014

I have got perfect 0.25 m M 0.25 mM as answer.

The law of mass action gives that for a reversible reaction 2 Y ⇌ Y 2 , 2Y\rightleftharpoons Y_2,

  • the forward reaction rate = k f [ Y ] 2 = k_f[Y]^{2}

  • the backward reaction rate = k b [ Y 2 ] = k_b[Y_2]

where [ Y ] [Y] and [ Y 2 ] [Y_2] are the concentrations of Y Y and Y 2 Y_2 respectively.

At equilibrium, the forward and back reaction rates are equal, therefore,

k f [ Y ] 2 = k b [ Y 2 ] k_f[Y]^{2}=k_b[Y_2] ⟹ [ Y 2 ] = k f [ Y ] 2 k b = 5 × 10 − 5 × 100 2 2 = 0.25 \Longrightarrow [Y_2]=\frac { { k }_{ f }{ [Y] }^{ 2 } }{ { k }_{ b } }=\frac { 5\times { 10 }^{ -5 }\times { 100 }^{ 2 } }{ 2 }=\boxed{0.25}

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But I think my solution was wrong. [ Y ] [Y] is not equal to 100 m M 100 mM but ( 100 − x ) m M (100 - x) mM .

Chew-Seong Cheong - 6 years, 11 months ago
Himanshu Arora
Jun 10, 2014

Let the concentration of dimer at equilibrium be x m M x mM . This implies that concentration of monomer would be ( 100 − 2 x ) m M (100-2x)mM , because 2 molecules of monomers give one dimer. Now equate the rates of reactions to get the equation 5 × 1 0 − 5 ( 100 − 2 x ) 2 = 2 x 5 \times 10^{-5}(100-2x)^{2}=2x . Solve the quadratic to get the answer 0.24753 \boxed{0.24753} .

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Seriously, we have to solve the quadratic? Why can't I approx it and take 100 − 2 x 100-2x as 100 100 only?

Avineil Jain - 7 years ago

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Simply due to the fact that the extent of dimerization(of the acid) is non-negligible and needs to be. Evaluated For better accuracy

Suhas Sheikh - 3 years ago

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