Question

The rate for the oxidation of iron(II) by cerium(IV)

${\mathbf{Ce}}^{\mathbf{+}\mathbf{4}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{Fe}}^{\mathbf{2}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{}\mathbf{\to }{\mathbf{Ce}}^{\mathbf{3}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{+}{\mathbf{Fe}}^{\mathbf{3}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}$

is measured at several different initial concentrations of the two reactants:

(a) Write the rate expression for this reaction.

(b) Calculate the rate constant k and give its units.

(c) Predict the initial reaction rate for a solution in which[Ce⁴⁺] is 2.6 × 1025 M and [Fe²⁺] is 1.3 × 1025 M.

Short answer

a) Rate expression is given $\mathrm{Rate}=\mathrm{k}\left[{\mathrm{Fe}}^{2+}\right]\left[{\mathrm{Ce}}^{4+}\right]$

b) Rate constant is given by K = 1.2 x 10¹⁸ Mol^-1 L s^-1

C) Rate of the reaction is found to be 4.0 x 10¹⁸

Step-by-step solution

Rate expression

Rate expression for the reaction is given as

a) The equation for the oxidation of iron with cerium is given as follows

${\mathbf{Ce}}^{\mathbf{+}\mathbf{4}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{Fe}}^{\mathbf{2}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{}\mathbf{\to }{\mathbf{Ce}}^{\mathbf{3}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}\mathbf{+}{\mathbf{Fe}}^{\mathbf{3}\mathbf{+}}\mathbf{\left(}\mathbf{aq}\mathbf{\right)}$

The rate expression for the oxidation of with can be given as follows.

$\mathrm{Rate}=\mathrm{k}{\left[{\mathrm{Fe}}^{2+}\right]}^{\mathrm{m}}{\left[{\mathrm{Ce}}^{4+}\right]}^n....................\left(1\right)$

In the above equation, m is the order with respect to iron and n is the order with respect to cerium.

Order of Reaction

When the concentration of cerium is tripled from 1.1 x 10^-5to 3.4 x 10^-5the reaction is tripled so it is concluded that the rate of reaction with respect to cerium is 1^st order ie n=1, so the rate expression can be written as

$\mathrm{Rate}=\mathrm{k}{\left[{\mathrm{Fe}}^{2+}\right]}^{\mathrm{m}}{\left[{\mathrm{Ce}}^{4+}\right]}^{}....................\left(2\right)$
The order of reaction with respect to iron is calculated in the following steps

Substituting concentrations  values

Substituting the Fe ²⁺and Ce⁺⁴concentrations in equation 2

$$\begin{gathered} \mathrm{Rate}1=\mathrm{k}{(1.8\times {10}^{-5})}^{\mathrm{m}}(1.1\times {10}^{-5}) \\ 2.0\times {10}^{-7}=\mathrm{k}{(1.8\times {10}^{-5})}^{\mathrm{m}}(1.1\times {10}^{-5})..............\left(3\right) \\ \mathrm{Rate}2=\mathrm{k}{(2.8\times {10}^{-5})}^{\mathrm{m}}(1.1\times {10}^{-5}) \\ 3.1\times {10}^{-7}=\mathrm{k}{(2.8\times {10}^{-5})}^{\mathrm{m}}(1.1\times {10}^{-5})...............\left(4\right) \end{gathered}$$

Overall order of the reaction

Dividing the equation 4 by 3

$$\begin{gathered} \frac{3.1\times {10}^{-7}}{2.1\times {10}^{-7}}=\frac{\mathrm{K}{(2.8\times {10}^{-5})}^{\mathrm{m}}}{\mathrm{k}{(1.8\times {10}^{-5})}^{\mathrm{m}}}\frac{(1.1\times {10}^{-5})}{(1.1\times {10}^{-5})}.......\left(5\right) \\ 1.55={(1.5)}^{\mathrm{m}}\mathrm{m}=1 \end{gathered}$$

The order of the reaction is 1 with respect to m and hence the rate expression for the reaction is

$\mathrm{Rate}=\mathrm{k}\left[{\mathrm{Fe}}^{2+}\right]\left[{\mathrm{Ce}}^{4+}\right].................\left(2\right)$

Rate Constant K

b) Substituting the values in rate expression to find the value of rate constant K:

$$\begin{gathered} \mathrm{Rate}=\mathrm{k}\left[{\mathrm{Fe}}^{2+}\right]\left[{\mathrm{Ce}}^{4+}\right].....................\left(2\right) \\ \mathrm{Rate}=2.0\times {10}^{-7} \\ \left[{\mathrm{Fe}}^{2+}\right]=1.8\times {10}^{-5} \\ \left[{\mathrm{Ce}}^{4+}\right]=1.1\times {10}^{-5} \\ \mathrm{Substituting}\mathrm{above}\mathrm{values}\mathrm{in}\mathrm{equation}2 \\ 2.0\times {10}^{-7}=\mathrm{K}1.8\times {10}^{-5}\times 1.1\times {10}^{-5} \\ \mathrm{K}=\frac{2.0\times {10}^{-7}}{1.8\times {10}^{-5}\times 1.1\times {10}^{-5}} \\ \mathrm{K}=1.01\times {10}^3{\mathrm{Mol}}^{-1}\mathrm{L}{\mathrm{s}}^{-1} \end{gathered}$$

The units of rate constant K is $1.01\times {10}^3{\mathrm{Mol}}^{-1}\mathrm{L}{\mathrm{s}}^{-1}$

Rate of Reaction

C) To find out the rate of the reaction one has to substitute the concentration values of Fe and Ce and rate constant values in equation 2,

$$\begin{gathered} \mathrm{Rate}=\mathrm{k}\left[{\mathrm{Fe}}^{2+}\right]\left[{\mathrm{Ce}}^{4+}\right].....................\left(2\right) \\ \mathrm{Rate}=1.01\times {10}^3\times 2.6\times {10}^{-5}\times 1.3\times {10}^{-5} \\ \mathrm{Rate}=3.41\times {10}^{-7} \end{gathered}$$

The rate of the reaction is found to be$3.41\times {10}^{-7}$