Problem 87 - Chemistry The Central Science In Si Units

Question

The activation energy of an uncatalyzed reaction is \(95 \mathrm{~kJ} / \mathrm{mol}\). The addition of a catalyst lowers the activation energy to \(55 \mathrm{~kJ} / \mathrm{mol}\). Assuming that the collision factor remains the same, by what factor will the catalyst increase the rate of the reaction at (a) \(25^{\circ} \mathrm{C},(\mathbf{b}) 125^{\circ} \mathrm{C} ?\)

Step-by-Step Solution

Answer

(a) At 25°C, the catalyst increases the rate by about \(1.01 \times 10^7\) times. (b) At 125°C, it increases the rate by about \(1.74 \times 10^5\) times.

Step 1: Understanding the Arrhenius Equation

The Arrhenius equation is a formula that gives the rate constant \(k\) of a reaction as a function of temperature \(T\) and activation energy \(E_a\). It is given by \[ k = A e^{-E_a / RT} \]where \(A\) is the pre-exponential factor (or collision factor), \(E_a\) is the activation energy, \(R\) is the universal gas constant (\(8.314 \, \text{J/mol K}\)), and \(T\) is the temperature in Kelvin.

Step 2: Converting Temperature to Kelvin

To use the Arrhenius equation, temperatures must be in Kelvin. Convert the given temperatures:- For \(25^{\circ} \text{C}\), \[ T_1 = 25 + 273.15 = 298.15 \text{ K} \]- For \(125^{\circ} \text{C}\), \[ T_2 = 125 + 273.15 = 398.15 \text{ K} \]

Step 3: Calculate the Rate Constant Ratio at Each Temperature

Using the Arrhenius equation, the ratio of the rate constants for the catalyzed \((k_{cat})\) and uncatalyzed \((k_{uncat})\) reactions is:\[ \frac{k_{cat}}{k_{uncat}} = \frac{A e^{-E_{cat} / RT}}{A e^{-E_{uncat} / RT}} = e^{(E_{uncat} - E_{cat}) / RT} \]For:- \(E_{uncat} = 95,000 \text{ J/mol}\)- \(E_{cat} = 55,000 \text{ J/mol}\)

Step 4: Calculate the Factor Increase at 25°C

With \(T_1 = 298.15 \text{ K}\), the ratio is:\[ \frac{k_{cat}}{k_{uncat}} = e^{(95,000 - 55,000) / (8.314 \times 298.15)} = e^{40,000 / 2481.911} \approx e^{16.12} \]Evaluating this gives approximately \(1.01 \times 10^7\).

Step 5: Calculate the Factor Increase at 125°C

With \(T_2 = 398.15 \text{ K}\), the ratio is:\[ \frac{k_{cat}}{k_{uncat}} = e^{(95,000 - 55,000) / (8.314 \times 398.15)} = e^{40,000 / 3310.9991} \approx e^{12.08} \]Evaluating this gives approximately \(1.74 \times 10^5\).

Key Concepts

Arrhenius equation

The Arrhenius equation is a fundamental formula in chemistry that describes how the rate constant \( k \) of a reaction depends on temperature \( T \) and activation energy \( E_a \). It is written as: \[ k = A e^{-E_a / RT} \]where:

• \( A \) is the pre-exponential factor, also known as the collision factor.
• \( E_a \) is the activation energy, which is the energy barrier that reactants must overcome to form products.
• \( R \) is the universal gas constant (8.314 J/mol·K).
• \( T \) is the temperature in Kelvin.

This equation implies that as temperature increases or activation energy decreases, the rate constant \( k \) increases, leading to a faster reaction rate. Consequently, this helps in predicting how changes in conditions affect reaction rates.

Reaction rate

The reaction rate is a measure of how quickly a chemical reaction occurs. In essence, it illustrates the speed at which reactants turn into products. The rate can be influenced by various factors:

• Concentration of reactants: Higher concentration often leads to a faster reaction.
• Temperature: Increasing temperature generally increases the rate as molecules gain energy and collide more frequently.
• Catalysts: These lower the activation energy, thus increasing the reaction rate without being consumed in the process.
• Surface area: More exposed surface area can enhance the rate of reaction due to increased opportunities for collisions.

Understanding these factors can help predict and control reaction behaviors in practical applications.

Catalyst effect

Catalysts play a crucial role in increasing the reaction rate by lowering the activation energy of a reaction. For example, in the original exercise, a catalyst reduced the activation energy from 95 kJ/mol to 55 kJ/mol. This significant reduction means that less energy is required for the reactants to convert into products. By using the Arrhenius equation, the dramatic increase in the reaction rate can be calculated. For a given temperature, the rate of the catalyzed reaction can be several orders of magnitude faster than the uncatalyzed reaction. Catalysts are invaluable in both industrial and biological processes as they accelerate reactions while remaining unchanged, thus allowing them to be used repeatedly.

Temperature conversion

Converting temperature to Kelvin is essential when using the Arrhenius equation since it requires absolute temperature rather than Celsius. The conversion is straightforward: simply add 273.15 to the Celsius temperature.For instance, if the temperature is 25°C, converting it to Kelvin means:\[ T = 25 + 273.15 = 298.15 \, \text{K} \]Likewise, for 125°C:\[ T = 125 + 273.15 = 398.15 \, \text{K} \]This conversion ensures accurate calculations in the Arrhenius equation, which helps predict how the reaction rate varies with temperature, emphasizing the exponential effect temperature has on rate constants and subsequently, on reaction rates.