Question

For a reaction the initial rate is given as: \(R_{0}=k[A]_{0}^{2}[B]_{0}\) by what factor, the initial rate of reaction will increase if initial concentration of \(A\) is taken \(1.5\) times and of \(B\) is tripled? (a) \(4.5\) (b) \(2.25\) (c) \(6.75\) (d) None of these

Short answer

The initial rate of reaction will increase by a factor of 6.75.

Step-by-step solution

Identify the Rate Law

The given rate law is expressed as an equation relating the rate of reaction, \( R_0 \), to the concentration of the reactants \( [A]_0 \) and \( [B]_0 \). According to the rate law, \( R_0 = k[A]_0^2[B]_0 \), where \( k \) is the rate constant, \( [A]_0 \) is the initial concentration of reactant A, and \( [B]_0 \) is the initial concentration of reactant B.

Calculate the Effect of Changing Concentrations

To find out by what factor the initial rate of reaction will increase, substitute \( 1.5[A]_0 \) for the initial concentration of A and \( 3[B]_0 \) for the initial concentration of B in the rate law. The new rate, \( R_{new} \), will be \( R_{new} = k(1.5[A]_0)^2(3[B]_0) \).

Simplify the New Rate Expression

Simplify the expression for \( R_{new} \) to find the factor of increase: \( R_{new} = k \cdot 1.5^2 \cdot [A]_0^2 \cdot 3 \cdot [B]_0 = 2.25 \cdot k[A]_0^2 \cdot 3[B]_0 = 6.75 \cdot k[A]_0^2[B]_0 = 6.75R_0 \).

Determine the Factor of Increase

The initial rate of reaction increases by a factor equal to the constant multiplied by the initial rate: \( R_{new} = 6.75R_0 \). Therefore, the rate increases by a factor of 6.75.

Key concepts

Understanding Rate Law Chemistry

Rate law chemistry is a fundamental concept in chemical kinetics, which helps us understand how the rate of a chemical reaction is related to the concentration of its reactants. In its simplest form, a rate law can be written as an equation that relates the reaction rate to the concentration of each reactant raised to a power, which is referred to as the order of the reaction with respect to that reactant.

For example, if the rate law for a reaction is given by the equation
\( R = k[A]^{m}[B]^{n} \), then the rate (R) is directly proportional to the concentration of reactant A raised to the power of m and to the concentration of reactant B raised to the power of n. Here, k represents the rate constant, a value that is specific to the reaction at a given temperature and depends on various factors such as the presence of a catalyst.

Exploring Chemical Reaction Rates

The rate of a chemical reaction indicates how quickly reactants are transformed into products over time. This rate can be affected by various factors including temperature, pressure, concentration of reactants, surface area, and the presence of catalysts.

By conducting experiments and observing how the concentration of reactants decreases over time, chemists determine the rate of reactions. In general, for most reactions, as the concentration of reactants increases, the frequency of collision between the reactant particles also increases, which typically leads to an increase in the reaction rate. However, the exact relationship between concentration and rate needs to be determined experimentally for each specific reaction and is expressed through the rate law.

The Concentration Effect on Reaction Rate

The concentration of reactants is a crucial factor in determining the rate of a chemical reaction. According to the collision theory of chemical kinetics, a higher concentration of reactant molecules increases the likelihood of collisions, which can lead to a higher rate of reaction.

In the context of our exercise, when the concentration of reactant A was increased by 1.5 times, and that of reactant B was tripled, the rate increased significantly. This change in concentration raises the rate of reaction by a specific factor, which can be calculated using the rate law equation. It is important for students to understand that changes in concentration have a multiplicative effect on the rate of reaction, depending on the reaction order with respect to each reactant. This concept is a cornerstone of chemical kinetics and is critical when optimizing conditions for industrial chemical processes or even in everyday situations like cooking or taking medicines.