Question

90\. Consider the reaction between methyl acetate and water: $$ \mathrm{CH}_{3} \mathrm{COOCH}_{3}(a q)+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{CH}_{3} \mathrm{OH}(a q) $$ The rate law for the reaction at \(25^{\circ} \mathrm{C}\) is: $$ \text { rate }=k\left[\mathrm{CH}_{3} \mathrm{COOCH}_{3}\right]\left[\mathrm{H}^{+}\right] $$ The reaction is carried out in one liter of a solution that is \(0.48 \mathrm{M} \mathrm{CH}_{3} \mathrm{COOCH}_{3}\) and \(0.122 \mathrm{M} \mathrm{H}^{+} .\) Fill in the blanks with increases, decreases, or remains the same. (a) The rate of the reaction ________ when \(125 \mathrm{~mL}\) of water is added. (b) The value of \(k\) _______ when the temperature is increased to \(38^{\circ} \mathrm{C}\). (c) The value of \(E_{\mathrm{a}}\) _______ when \(\mathrm{HCl}\) is added. (d) The rate of the reaction ________ when \(\mathrm{NaOH}\) is added. (e) The value of \(k\) _______ when a catalyst is added.

Short answer

Question: Analyze the effect of the following factors on the rate of the reaction between methyl acetate and water based on the given rate law: (a) Adding 125 mL of water (b) Increasing the temperature to 38°C (c) Adding HCl (d) Adding NaOH (e) Adding a catalyst Explain how each factor affects the reaction rate based on the principles of chemical kinetics, rate law, and the concentration of reacting species.

Step-by-step solution

(a) Effect of adding water

When we add 125 mL of water to the reaction mixture, the volume of the solution increases, but the number of moles of methyl acetate (\(\mathrm{CH_3COOCH_3}\)) and H+ ions does not change. As a result, their molar concentrations (and thus the rate of the reaction) will decrease. So, the rate of the reaction decreases when 125 mL of water is added.

(b) Effect of increasing temperature

According to the Arrhenius equation, the rate constant \(k\) is directly related to the temperature of the reaction: $$k=Ae^{-\frac{E_a}{RT}}$$ 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. When the temperature increases, the exponent term in the equation becomes less negative, leading to an increase in the rate constant \(k\). So, the value of \(k\) increases when the temperature is increased to \(38^{\circ} \mathrm{C}\).

(c) Effect of adding HCl

Since the rate law involves the concentration of H+ ions (\([H^+]\)), the addition of an acid like HCl would increase the concentration of H+ ions in the solution. However, the activation energy (\(E_a\)) is an intrinsic property of the reaction and does not depend on the concentration of the reactants. Thus, the value of \(E_a\) remains the same when HCl is added.

(d) Effect of adding NaOH

Adding NaOH, a strong base, to the solution will decrease the concentration of H+ ions as the hydroxide ions (OH-) present in NaOH will react with the H+ ions to form water. As the rate law depends on the concentration of H+ ions, the decrease in H+ ions concentration will lead to a decrease in the reaction rate. So, the rate of the reaction decreases when NaOH is added.

(e) Effect of adding a catalyst

A catalyst works by providing an alternate reaction pathway with a lower activation energy, allowing the reaction to proceed at a faster rate. The rate constant \(k\) is directly related to the activation energy, as seen in the Arrhenius equation. So, when a catalyst is added, it lowers the activation energy, causing the rate constant \(k\) to increase. Thus, the value of \(k\) increases when a catalyst is added.

Key concepts

Rate Law

Understanding the rate law is key to grasping the dynamics of a chemical reaction. Indeed, the rate law for a reaction is an equation that links the rate of the reaction to the concentration of its reactants. In the context of the given exercise, the rate law is expressed as:
\[\text{rate} = k[\mathrm{CH}_3\mathrm{COOCH}_3][\mathrm{H}^+]\]
This indicates that the reaction rate is directly proportional to the concentrations of methyl acetate and the hydrogen ion. Simplifying things, if either of these concentrations increases, the rate of the reaction ramps up, and if they decrease, the reaction slows down. The rate constant (\(k\)) ties together temperature and reaction rate but stands independent of reactant concentrations.

Activation Energy

Activation energy, or \(E_a\), is the minimum energy necessary for a chemical reaction to proceed. It's like the 'entrance fee' for molecules to transform into products. In the exercise, adding an acid like HCl doesn't alter the activation energy, since \(E_a\) is an inherent characteristic of the reaction pathway itself—a consequence of the molecular interactions and the energy barrier they must overcome. To visualize this, imagine a ball (reactants) needing a certain amount of energy to roll over a hill (the energy barrier) to reach the other side where it becomes a different ball (products). No matter how many balls you have, the height of the hill doesn't change.

Factors Affecting Activation Energy

While reactant concentrations don't impact activation energy, introducing a catalyst can provide a shortcut—creating a new, lower-hill path.

Effects of Temperature on Reaction Rate

Think of temperature as the 'behind the scenes' force turbocharging chemical reactions. When the temperature climbs, molecules move faster, collide more frequently, and do so with greater energy. Subsequently, more molecules muster the minimum energy—activation energy—needed to react. The Arrhenius equation shines a light on this relationship:
\[k=Ae^{-\frac{E_a}{RT}}\]
As the temperature (\(T\)) rises, the negative exponent in the Arrhenius equation shrinks, leading to a spike in the rate constant (\(k\)), and thus, a faster reaction rate. It's as if we're making the hill lower or the balls more energetic, allowing more of them to get over the hill sooner. For our exercise, when we crank up the temperature from 25°C to 38°C, the reaction gets a boost, because \(k\) and therefore, the rate goes up.

Catalysis

A catalyst can be likened to a chemical 'matchmaker' that hastens a reaction without being consumed by it. By offering a shortcut, a reaction can proceed with a lower activation energy threshold. In the provided exercise, adding a catalyst presents an alternative pathway for transforming methyl acetate and water into products.

How Catalysts Accelerate Reactions

A catalyst may offer a surface for reactants to adhere to, bringing them closer together, or it might destabilize some bonds within the reactants, making it easier for the reaction to take place. Ultimately, the presence of a catalyst increases the rate constant (\(k\))—as seen through the Arrhenius equation lens, due to the reduced \(E_a\). This is why, in the textbook's problem, the rate of reaction gears up in the presence of a catalyst.