Question

By what factor does the rate change in each of the following cases (assuming constant temperature)? (a) A reaction is first order in reactant \(\mathrm{A}\), and \([\mathrm{A}]\) is doubled. (b) A reaction is second order in reactant \(\mathrm{B},\) and \([\mathrm{B}]\) is halved. (c) A reaction is second order in reactant \(\mathrm{C},\) and \([\mathrm{C}]\) is tripled.

Short answer

a) 2, b) 1/4, c) 9.

Step-by-step solution

Understand Rate Laws

The rate law for a reaction tells us how the rate of reaction depends on the concentration of reactants. For a reaction that is first order in reactant \(\text{A}\), the rate law is \( \text{rate} = k[\text{A}] \) where k is the rate constant. For a reaction that is second order in reactant \(\text{B}\), the rate law is \( \text{rate} = k[\text{B}]^2 \). For a reaction that is second order in reactant \(\text{C}\), the rate law is \( \text{rate} = k[\text{C}]^2 \).

Analyze Case (a)

For a reaction that is first order in reactant \(\text{A}\), the rate law is \( \text{rate} = k[\text{A}] \). If the concentration of \(\text{A}\) is doubled, the new rate will be \( k(2[\text{A}]) = 2k[\text{A}] \). Thus, the rate doubles. The factor by which the rate changes is 2.

Analyze Case (b)

For a reaction that is second order in reactant \(\text{B}\), the rate law is \( \text{rate} = k[\text{B}]^2 \). If the concentration of \(\text{B}\) is halved, the new rate will be \( k\bigg( \frac{[\text{B}]}{2} \bigg)^2 = k\frac{[\text{B}]^2}{4} \). Thus, the rate is reduced by a factor of 4. The factor by which the rate changes is \(\frac{1}{4} \).

Analyze Case (c)

For a reaction that is second order in reactant \(\text{C}\), the rate law is \( \text{rate} = k[\text{C}]^2 \). If the concentration of \(\text{C}\) is tripled, the new rate will be \( k(3[\text{C}])^2 = k9[\text{C}]^2 \). Thus, the rate increases by a factor of 9. The factor by which the rate changes is 9.

Key concepts

First Order Reaction

In first order reactions, the reaction rate is directly proportional to the concentration of one reactant. This means that if the concentration of reactant \(A\) changes, the reaction rate will change in the same way. The rate law for a first order reaction is given by \( \text{rate} = k[\text{A}] \). Here, \(k\) is the rate constant, and \[ \text{A} \] is the concentration of the reactant. For example, if you double the concentration of \[ \text{A} \], the rate of the reaction will also double. Similarly, halving the concentration of \[ \text{A} \] will halve the reaction rate.

Key Takeaways:

• The rate of a first order reaction is proportional to the reactant concentration.
• The rate constant \(k\) remains unaffected by changes in concentration.
• First order reactions have a characteristic exponential decay of reactant concentration over time.

Second Order Reaction

Second order reactions are characterized by a reaction rate that is proportional to the square of the concentration of one reactant or to the product of the concentrations of two different reactants. The general rate law for a second order reaction is \( \text{rate} = k[\text{B}]^2 \) for a single reactant or \( \text{rate} = k[\text{A}][\text{B}] \) when two different reactants are involved. In our example, consider reactant \[ \text{B} \]:

• If \[ \text{B} \] is halved, the rate becomes \[\frac{ \text{rate} = k\bigg(\frac{[\text{B}]}{2}\bigg)^2}{ \text{rate} = k\frac{[\text{B}]^2}{4} \].
• If \[ \text{B} \] is doubled, the rate becomes \[k(2[\text{B}])^2 = k4[\text{B}]^2 \].
  

Key Points:

• The rate of second order reactions is more sensitive to changes in reactant concentration compared to first order reactions.
• If the concentration of the reactant is changed, the effect on the rate is squared.
• Second order reactions often have different units for the rate constant \(\text{k}\) compared to first order reactions.

Rate Constant

The rate constant, denoted as \(k\), is a crucial component in determining the speed of a reaction according to a rate law. For a given reaction, the rate constant is a proportionality factor that links the concentration of reactants to the reaction rate. It remains constant as long as the temperature is constant and does not change with reactant concentration changes.

Characteristics of the Rate Constant:

• It is specific to a particular reaction at a given temperature.
• The units of \(k\) vary depending on the reaction order. For first order reactions, the units are \( \text{time}^{-1} \), while for second order reactions, the units are \( \text{concentration}^{-1} \text{time}^{-1} \).
• The value of the rate constant gives insight into the speed of the reaction – higher \(k\) values indicate faster reactions.

The rate constant can also change with temperature, following the Arrhenius equation \[ k = Ae^{-\frac{E_a}{RT}} \], where \(A\) is the pre-exponential factor, \(E_a\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin.