For a hypothetical isomerization that is first order in both directions with kf=17.7 min^−1 and kr=32.2 min^−1, find the equilibrium composition if the initial concentration of A is equal to 0.175 mol L^−1 and the initial concentration of B is equal to zero. Find the half-life and the relaxation

Question

For a hypothetical isomerization that is first order in both directions with [latex]k_{f}=17.7 min^{−1}[/latex] and [latex]k_{r}=32.2 min^{−1}[/latex], find the equilibrium composition if the initial concentration of A is equal to [latex]0.175 mol L^{−1}[/latex] and the initial concentration of B is equal to zero. Find the half-life and the relaxation time.

Accepted Answer

[latex]K=\frac{k_{f}}{k_{r}} =\frac{17.7 min^{−1}}{32.2 min^{−1}} =0.528[/latex]

We let

[latex]x=\frac{[B]_{eq}}{1 mol L^{-1}} =\frac{[A]_{0}-[A]_{eq}}{1 mol L^{-1}} [/latex]

[latex]0.528=\frac{x}{0.150−x}[/latex]

x=(0.528)(0.150-x)=0.0792-0.528x

[latex]x=\frac{0.0792}{1.528}=0.0518[/latex]

[latex][B]_{eq}=0.0518 mol L^{−1}[/latex]

[latex][A]_{eq}=0.0982 mol L^{−1}[/latex]

[latex]t_{1/2}=\frac{\ln(2)}{k_{f}+k_{r}} =\frac{\ln(2)}{17.7 min^{−1}+32.2 min^{−1}}=0.0139 min[/latex]

[latex]τ=\frac{1}{k_{f}+k_{r}} =\frac{1}{17.7 min^{−1}+32.2 min^{−1}}=0.0200 min[/latex]

Question 11.11: For a hypothetical isomerization that is first order in both...

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