Question

Chemists commonly use a rule of thumb that an increase of \(10 \space\mathrm{K}\) in temperature doubles the rate of a reaction. What must the activation energy be for this statement to be true for a temperature increase from 25 to \(35^{\circ} \mathrm{C} ?\)

Short answer

For the rule of thumb to be true for a temperature increase from 25 to \(35^{\circ} \mathrm{C}\), the activation energy must be approximately \(5284.4 \space\mathrm{J\;mol^{-1}}\).

Step-by-step solution

Convert temperatures to Kelvin scale

First, we need to convert the given temperatures from Celsius to Kelvin. To do this, we will add 273.15 to each temperature: \( T_1 = 25^{\circ} \mathrm{C} + 273.15 = 298.15 \space\mathrm{K} \) \( T_2 = 35^{\circ} \mathrm{C} + 273.15 = 308.15 \space\mathrm{K} \)

Write the rule of thumb as a mathematical expression

The rule of thumb states that an increase of 10 K in temperature doubles the rate of the reaction. We can write this as: \( \frac{rate_2}{rate_1} = 2 \)

Apply the Arrhenius equation

The Arrhenius equation states that the rate constant k of a reaction depends upon the activation energy E and temperature T as follows: \( k = A \times e^{\frac{-E}{RT}} \) Here, A is the pre-exponential or frequency factor, E is the activation energy, R is the gas constant (\(8.314 \space\mathrm{J\;mol^{-1}K^{-1}}\)), and T is the temperature.

Write the equation for the ratio of rate constants

Now, we can write the equation for the ratio of rate constants at the two different temperatures: \( \frac{k_2}{k_1} = \frac{A \times e^{\frac{-E}{R \times 308.15}}}{A \times e^{\frac{-E}{R \times 298.15}}} \) We know that \( \frac{rate_2}{rate_1} = 2 \), and since the rate constant is directly proportional to the rate of reaction, we can substitute it in: \( 2 = \frac{A \times e^{\frac{-E}{R \times 308.15}}}{A \times e^{\frac{-E}{R \times 298.15}}} \)

Solve for the activation energy (E)

Now, we need to solve the equation for E. First, we can simplify the equation by canceling the pre-exponential factor A: \( 2 = \frac{e^{\frac{-E}{R \times 308.15}}}{e^{\frac{-E}{R \times 298.15}}} \) Next, we take the natural logarithm of both sides: \( \ln{2} = \ln{\left(\frac{e^{\frac{-E}{R \times 308.15}}}{e^{\frac{-E}{R \times 298.15}}}\right)} \) \( \ln{2} = \frac{-E}{R \times 308.15} + \frac{E}{R \times 298.15} \) Multiplying the whole equation by \(R \times 308.15 \times 298.15\) to get rid of the fractions: \( \ln{2} \times R \times 308.15 \times 298.15 = -E \times 298.15 + E \times 308.15 \) Now, combining the terms with E: \( E(308.15 - 298.15) = \ln{2} \times R \times 308.15 \times 298.15 \) Finally, we can solve for the activation energy E: \( E = \frac{\ln{2} \times R \times 308.15 \times 298.15}{10} \) Inserting the value of R (\(8.314 \space\mathrm{J\;mol^{-1}K^{-1}}\)) and calculating the result: \( E \approx 5284.4 \space\mathrm{J\;mol^{-1}} \) So, the activation energy must be approximately \(5284.4 \space\mathrm{J\;mol^{-1}}\) for the rule of thumb to be true for a temperature increase from 25 to \(35^{\circ} \mathrm{C}\).

Key concepts

Arrhenius Equation

The Arrhenius equation is a fundamental concept in chemistry used to determine how the rate constant \( k \) of a reaction changes with temperature. This equation describes how the likelihood of molecules reacting increases at higher temperatures due to greater molecular motion and collisions. It is denoted as:

• \( k = A \times e^{\frac{-E}{RT}} \)

In this equation:

• \( k \) is the rate constant, reflecting the reaction rate.
• \( A \) is the pre-exponential factor, also known as the frequency factor, indicating the frequency of collisions.
• \( E \) is the activation energy, the minimum energy required for a reaction to occur.
• \( R \) is the universal gas constant, \(8.314 \mathrm{J\,mol^{-1}K^{-1}}\).
• \( T \) is the absolute temperature in Kelvin.

By manipulating this equation, scientists can explore how changes in temperature influence the speed of chemical reactions.

Temperature Conversion

In chemistry, it is essential to use the Kelvin scale for temperature when applying the Arrhenius equation. The Kelvin scale begins at absolute zero and is used to ensure uniformity in calculations. To convert Celsius to Kelvin, you add 273.15 to the Celsius temperature:

• \( T_\mathrm{K} = T_\mathrm{C} + 273.15 \)

For example, to convert \(25^{\circ} \mathrm{C}\) and \(35^{\circ} \mathrm{C}\) to Kelvin:

• \( T_1 = 25^{\circ} \mathrm{C} + 273.15 = 298.15 \,\mathrm{K} \)
• \( T_2 = 35^{\circ} \mathrm{C} + 273.15 = 308.15 \,\mathrm{K} \)

Using Kelvin allows for accurate comparisons of reaction rates at different temperatures.

Rate Constants

Rate constants play a crucial role in understanding reaction kinetics. They define the speed of a chemical reaction at a certain temperature, encapsulating all complicated interactions within a reaction. Rate constants are highly dependent on temperature, as shown by the Arrhenius equation.

• Higher rate constants indicate faster reactions.
• These constants change with temperature changes: they increase with temperature due to increased molecular energy and collisions.
• In the context of this problem, a rate constant's ratio can illustrate how doubling the reaction rate is achieved by a \(10 \, \text{K}\) increase in temperature.

Understanding rate constants helps predict how reactions will progress under different conditions.

Reaction Rate

The reaction rate refers to how fast a reaction occurs. A common rule of thumb in chemistry is that a \(10 \degree \mathrm{K}\) increase in temperature doubles the reaction rate. This is because higher temperatures increase molecules' kinetic energy, leading to more frequent and effective collisions.

• Reaction rate can be linked to the rate constant; if the rate constant doubles, so does the rate.
• The relationship between temperature and reaction speed is exponential; small temperature changes can cause significant reaction rate changes.
• This principle is used to estimate the activation energy needed for doubling the rate when the rule of thumb holds, as demonstrated in the Arrhenius equation manipulation.

By investigating the reaction rate and its temperature dependency, chemists can better control and optimize chemical processes.