Question

Some bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of \({\bf{3 \times 1}}{{\bf{0}}^{\bf{4}}}\)g/mole that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalysed reactions can be complex, at low concentrations this reaction can be described by a rate equation that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 µg (\({\bf{0}}{\bf{.15 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\)g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.

(Penicillin) (M)	Rate (mole/L/min)
\({\bf{2}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\) \(\)	\({\bf{1}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}\)
\({\bf{3}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\)	\({\bf{1}}{\bf{.5 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}\)
\({\bf{4}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 6}}}}\)	\({\bf{2}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 10}}}}\)

Short answer

The order of the reaction with respect to penicillin is first order reaction which can be determined by the use of hit and trial method and the rate constant of the reaction is \({\bf{5}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 5}}}}{\bf{L/mole/min}}\).

Step-by-step solution

Reaction Rate

Reaction involved the effective collision of two reactants to produce the desired products. Reactions can be natural which are occurring in the surrounding environment whereas it can be artificially done in the laboratory to form a desired required product.

The reaction rate can be defined as the speed of reaction to produce the products. The reaction rate can be slow, fast or moderate. The reaction can take less than a millisecond to produce products or it can take years to produce a desired product.

\({\bf{Rate = K }}{\left( {\bf{A}} \right)^{\bf{n}}}\)

A = concentration of reactant.

K = Rate constant.

Explanation

Three rates are given there regarding the same reaction at given time.

From the table given in the question there are three rates \({\bf{Rat}}{{\bf{e}}_{\bf{1}}}{\bf{, Rat}}{{\bf{e}}_{\bf{2}}}{\bf{, Rat}}{{\bf{e}}_{\bf{3}}}\).

\(\begin{align}\frac{{Rate3}}{{Rate1}}{\bf{ }} &= {\bf{ }}\frac{{k \times {{\left( {A3} \right)}^n}}}{{k \times {{\left( {A1} \right)}^n}}}{\bf{ }}\\\frac{{4.0{\bf{ }} \times {\bf{ }}{{10}^{ - 6}}}}{{2.0{\bf{ }} \times {\bf{ }}{{10}^{ - 6}}}}{\bf{ }} &= {\bf{ }}\frac{{k \times {{\left( {2.0{\bf{ }} \times {\bf{ }}{{10}^{ - 10}}} \right)}^n}}}{{k \times {{\left( {1.0{\bf{ }} \times {\bf{ }}{{10}^{ - 10}}} \right)}^n}}}\\2{\bf{ }} &= {\bf{ }}{\left( 2 \right)^n}\\n &= 1\end{align}\)

The reaction of penicillin follows first-order reaction.

The rate constant of the reaction as follows:

\(\begin{align}Rate{\bf{ }} &= K{\bf{ }}{\left( A \right)^n}\\1.0{\bf{ }} \times {\bf{ }}{10^{ - 10}}{\bf{ }} &= K{\bf{ }}{\left( {2.0{\bf{ }} \times {\bf{ }}{{10}^{ - 6}}} \right)^1}\\K &= \frac{{1.0{\bf{ }} \times {\bf{ }}{{10}^{ - 10}}}}{{2.0{\bf{ }} \times {\bf{ }}{{10}^{ - 6}}}}\\K &= 5.0{\bf{ }} \times {\bf{ }}{10^{ - 5}}\end{align}\)

Hence, the rate constant of the reaction is \({\bf{5}}{\bf{.0 \times 1}}{{\bf{0}}^{{\bf{ - 5}}}}{\bf{L/mole/min}}\).