Question

For the reaction \(\mathrm{A} \longrightarrow\) Products, it is found that the rate of reaction increases by a factor of \(6.25\), when the concentration of \(\mathrm{A}\) is increased by a factor of \(2.5 .\) The order of reaction with respect to \(\mathrm{A}\) is (a) \(0.5\) (b) 1 (c) 2 (d) 3

Short answer

The order of reaction with respect to \( \mathrm{A} \) is 2.

Step-by-step solution

Write the Rate Law Expression

The rate law for the reaction \( \mathrm{A} \rightarrow \) Products is expressed as: \[ \text{Rate} = k [A]^n \] where \( k \) is the rate constant, \( [A] \) is the concentration of \( \mathrm{A} \), and \( n \) is the order of reaction with respect to \( \mathrm{A} \).

Setup the Rate Increase Equation

We know that the rate increases by a factor of 6.25 when the concentration of \( \mathrm{A} \) is increased by a factor of 2.5. Substitute these values into the rate equation to get: \[ 6.25 = (2.5)^n \]

Solve for the Order \( n \)

Take the logarithm on both sides to solve for \( n \):\[ \log(6.25) = n \cdot \log(2.5) \]Calculate:\[ n = \frac{\log(6.25)}{\log(2.5)} \]Evaluating this, we find:\[ n = 2 \]

Confirm the Order of Reaction

The calculated \( n = 2 \) indicates that the order of the reaction with respect to \( \mathrm{A} \) is 2, which matches one of the given options.

Key concepts

Rate Law

The rate law of a chemical reaction describes how the rate of a reaction depends on the concentration of reactants. It is an expression that provides important insights into the dynamics of the reaction.
For the reaction \( \mathrm{A} \rightarrow \) Products, the rate law can be written as: \[ \text{Rate} = k [A]^n \]Here:

• \(k\) is the rate constant, a unique value for each reaction at a given temperature.
• \([A]\) represents the molar concentration of reactant \(A\).
• \(n\) is the order of the reaction with respect to \(A\), indicating how changes in \([A]\) affect the reaction rate.

The rate law is determined experimentally, as it is specific to each reaction and cannot be deduced from the stoichiometry alone.

Order of Reaction

The order of a reaction is a key part of the rate law equation, represented by the exponent of the concentration term. It provides information on how variations in the concentration of reactants influence the rate.
In the given exercise, the order \(n\) was found to be \(2\). This means that the rate of reaction is proportional to the square of the concentration of \(A\). So, doubling \([A]\) will increase the rate by a factor of four, as \((2)^2 = 4\). Similarly, increasing the concentration by a factor of \(2.5\) raises the rate by \(6.25\), since \[ (2.5)^2 = 6.25 \] Orders of reaction can be whole numbers, fractions, or even zero. They reflect:

• The pathway or mechanism of the reaction.
• The concentration effects on reaction speed.

It is crucial to understand that the order is independent of the stoichiometric coefficients of the reactants in the balanced equation.

Rate Constant

The rate constant, \(k\), is a crucial component of the rate law. It provides the relation between the reaction rate and the reactant concentrations.
While the order of the reaction \(n\) tells us how the rate depends on concentrations, \(k\) contains essential information about the reaction conditions such as:

• Temperature
• Presence of a catalyst
• The specific nature of the reactants

The units of \(k\) vary depending on the reaction order. For a second-order reaction, like in this exercise, the units will be \(M^{-1} \cdot s^{-1}\), reflecting the dependency on concentration and time.
The value of \(k\) is determined experimentally and varies with temperature according to the Arrhenius equation. This makes rate constants valuable for comparing reaction dynamics under different conditions. They are often used to calculate predicted reaction rates for given reactant concentrations.