Question

65.Manfred Eigen, a German physical chemist working during the 1970s and 1980s, earned a Nobel Prize for developing the “temperature-jump” method for studying kinetics of very rapid reactions in solution, such as proton transfer. Eigen and his co-workers found that the specific rate of proton transfer from a water molecule to an ammonia molecule in a dilute aqueous solution is . The equilibrium constant K_b for the reaction of ammonia with water is 1.8 × 10^-5 . What, if anything, can be deduced from this information about the rate of transfer of a proton from NH₄⁺to a hydroxide ion? Write equations for any reactions you mention, making it clear to which reaction(s) any quoted constant(s) apply.

Short answer

The rate constant of reverse reaction is $1x{10}^{10}{\mathrm{s}}^{-1}{\mathrm{M}}^{-1}$

Step-by-step solution

Formation of ammonium ion:

The reaction between ammonia and water is given by the following reaction.

${\mathrm{NH}}_3+{\mathrm{H}}_2\mathrm{O}\underset{}{\overset{}{\leftrightharpoons }}\mathrm{N}{{\mathrm{H}}_4}^{+}+{\mathrm{OH}}^{-}........\left(1\right)$

Where k_f and k_b are the rate constant of forward and backward reaction.In the above reaction proton transfer occurs from H₂O to ammonia to form ammonium ion.

Rate constant of reaction:

From the above reaction equilbrium, constant is calculated as following

${\mathrm{K}}_{\mathrm{b}}=\frac{{\mathrm{k}}_{\mathrm{f}}}{{\mathrm{k}}_{\mathrm{r}}}........\left(2\right)$

Where K_b is the equilbrium rate constant,k_f is the rate constant of forward reaction,k_r is the rate constant of backward reaction

Substituting the values of K_b and k_f in the equation.

$$\begin{gathered} {\mathrm{k}}_{\mathrm{f}}=2\mathrm{x}{10}^5\backslash \mathrm{hfill} \\ {\mathrm{K}}_{\mathrm{b}}=1.8\mathrm{x}{10}^{-5}\backslash \mathrm{hfill} \\ 1.8\mathrm{x}{10}^{-5}=\frac{2\mathrm{x}{10}^5}{{\mathrm{k}}_{\mathrm{r}}}........\left(2\right) \\ {\mathrm{k}}_{\mathrm{r}}=\frac{2\mathrm{x}{10}^5}{1.8\mathrm{x}{10}^{-5}}{\mathrm{s}}^{-1}{\mathrm{M}}^{-1} \\ {\mathrm{k}}_{\mathrm{r}}=1\mathrm{x}{10}^{10}{\mathrm{s}}^{-1}{\mathrm{M}}^{-1} \end{gathered}$$

The rate constant of reverse reaction is $1\mathrm{x}{10}^{10}{\mathrm{s}}^{-1}{\mathrm{M}}^{-1}$