Question

The reaction between ethyl bromide \(\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}\right)\) and hydroxide ion in ethyl alcohol at 330 \(\mathrm{K}\) , \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}(a l c)+\mathrm{OH}^{-}(a l c) \longrightarrow \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(l)+\mathrm{Br}^{-}(a l c)\) is first order each in ethyl bromide and hydroxide ion. When \(\left[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}\right]\) is 0.0477 \(\mathrm{M}\) and \(\left[\mathrm{OH}^{-}\right]\) is \(0.100 \mathrm{M},\) the rate of disappearance of ethyl bromide is \(1.7 \times 10^{-7} \mathrm{M} / \mathrm{s}\) (a) What is the value of the rate constant? (b) What are the units of the rate constant? (c) How would the rate of disappearance of ethyl bromide change if the solution were diluted by adding an equal volume of pure ethyl alcohol to the solution?

Short answer

(a) The value of the rate constant, k, is approximately \(3.57 \times 10^{-6} \mathrm{M}^{-1} \mathrm{s}^{-1}\). (b) The units of the rate constant, k, are \(\mathrm{M}^{-1} \cdot \mathrm{s}^{-1}\). (c) Upon dilution, the rate of disappearance of ethyl bromide would change to approximately \(4.25 \times 10^{-9} M \cdot s^{-1}\).

Step-by-step solution

Write the rate equation for the reaction

The given reaction is first order each in ethyl bromide and hydroxide ion. Thus, the rate equation for this reaction can be written as: \(rate = k[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}][\mathrm{OH}^-]\) Now we have to calculate the rate constant, k.

Calculate the rate constant, k

We have the rate of disappearance of ethyl bromide, which is equal to the rate of the reaction. We also have the initial concentrations of the reactants. Using these values, we can calculate the rate constant as: \(k = \frac{rate}{[\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Br}][\mathrm{OH}^-]}\) \(k = \frac{1.7 \times 10^{-7} \mathrm{M} / \mathrm{s}}{(0.0477 \mathrm{M})(0.100 \mathrm{M})}\) \(k ≈ 3.57 \times 10^{-6} \mathrm{M}^{-1} \mathrm{s}^{-1}\) The value of the rate constant, k, is approximately \(3.57 \times 10^{-6} \mathrm{M}^{-1} \mathrm{s}^{-1}\). Now that we have the rate constant, let's determine its units.

Determine the units of the rate constant

Looking at the rate equation and the units of the concentrations, we can determine the units of the rate constant by analyzing the dimensions: \(\frac{\mathrm{M} \cdot \mathrm{s}^{-1}}{\mathrm{M}^2} = \mathrm{M}^{-1} \cdot \mathrm{s}^{-1}\) Thus, the units of the rate constant, k, are \(\mathrm{M}^{-1} \cdot \mathrm{s}^{-1}\). Finally, we have to find how the rate of disappearance of ethyl bromide would change upon dilution.

Determine the change in the rate of disappearance of ethyl bromide upon dilution

When the solution is diluted by adding an equal volume of pure ethyl alcohol, the concentrations of both reactants are halved. So, the new concentrations are: \([C2H5Br]_{new} = 0.5 [C2H5Br]_{initial} = 0.5 (0.0477 M) = 0.02385 M\) \([OH^-]_{new} = 0.5 [OH^-]_{initial} = 0.5 (0.100 M) = 0.0500 M\) Now, we can calculate the new rate of disappearance of ethyl bromide using the rate equation and the rate constant: \(rate_{new} = k [C2H5Br]_{new} [OH^-]_{new}\) \(rate_{new} = (3.57 \times 10^{-6} M^{-1}s^{-1})(0.02385 M)(0.0500 M)\) \(rate_{new} ≈ 4.25 \times 10^{-9} M \cdot s^{-1}\) Upon dilution, the rate of disappearance of ethyl bromide would change to approximately \(4.25 \times 10^{-9} M \cdot s^{-1}\).

Key concepts

Reaction Kinetics

Understanding reaction kinetics is essential for grasping how chemical reactions occur and how different factors affect the speed of these reactions. Reaction kinetics focuses on the rate at which reactants transform into products over time in a chemical reaction.

• **Rate of Reaction** is the measure of how quickly the concentration of a reactant decreases or the concentration of a product increases over time. In the provided exercise, the rate of disappearance of ethyl bromide is given as a measure of this speed.
• **Rate Equation** provides a mathematical expression that relates the rate of reaction to the concentration of reactants. In the exercise, the rate equation is given as: \(rate = k[\mathrm{C}_2 \mathrm{H}_5 \mathrm{Br}][\mathrm{OH}^-]\), indicating this is a second-order reaction (first-order with respect to each reactant).
• **Factors Affecting Reaction Rate** include concentration, temperature, presence of a catalyst, and physical state of reactants. In this exercise, concentration plays a role as dilution affects the reaction rate.

Understanding these principles can help predict and control the speed of chemical reactions, making them valuable in both industrial and laboratory settings.

Rate Constant

The rate constant, often denoted as \(k\), plays a crucial role in the rate equation of a chemical reaction. It provides insight into the speed of a reaction under specific conditions and is unique for each reaction.
- **Definition**: The rate constant is a proportionality factor that links the reaction rate with the concentrations of the reactants. In simple terms, it describes how fast a reaction occurs when all reactant concentrations are at a standard level.- **Calculation**: Using the rate equation, the rate constant can be calculated as seen in the exercise: \(k = \frac{rate}{[\mathrm{C}_2 \mathrm{H}_5 \mathrm{Br}][\mathrm{OH}^-]}\). This calculation gives \(k \approx 3.57 \times 10^{-6} \mathrm{M}^{-1} \mathrm{s}^{-1}\).- **Units**: For reactions of different orders, the units of the rate constant vary. In the given reaction, which is second order overall, the units are \(\mathrm{M}^{-1} \mathrm{s}^{-1}\).
The rate constant offers valuable information about how favorable or rapid a reaction is under given conditions, thus aiding in the design of chemical processes.

Order of Reaction

The order of reaction gives insight into how the concentration of one or more reactants affects the rate of reaction. It can be zero, first, second, or even higher, depending on the relationship between concentration and rate.
- **First Order Reaction**: A reaction is first order with respect to a reactant if the rate is directly proportional to the concentration of that reactant. In the exercise, the reaction is first order with respect to both ethyl bromide and hydroxide ion. This implies that doubling the concentration of either reactant will double the reaction rate.- **Second Order Overall Reaction**: When a reaction is first order in two different reactants, it is a second-order reaction overall. The rate depends on the product of the concentrations of the two involved reactants, as seen in the rate equation: \(rate = k[\mathrm{C}_2 \mathrm{H}_5 \mathrm{Br}][\mathrm{OH}^-]\).- **Impact of Dilution**: When the solution is diluted, as shown in the exercise, both reactants' concentrations are halved, thus reducing the overall reaction rate substantially. This directly follows from the reaction's overall order.
Understanding the order of reaction is essential for controlling and optimizing chemical reactions in various applications, from industrial to biological systems.